Step 1: Understanding the Concept:

The number of subsets of a finite set with \(k\) elements is given by the formula \(2^k\). We are given a relationship between the number of subsets of two sets, A and B, and we need to find the number of elements in each set.


Step 2: Key Formula or Approach:

Let \(n(A) = M\) and \(n(B) = N\).

Number of subsets of A = \(2^M\).

Number of subsets of B = \(2^N\).

The given condition is: Number of subsets of A = Number of subsets of B + 48.
\[ 2^M = 2^N + 48 \]


Step 3: Detailed Explanation:

We need to solve the equation \(2^M - 2^N = 48\) for integers M and N.

First, we can factor out the smaller power of 2, which is \(2^N\) (assuming \(M > N\), which must be true since \(2^M > 2^N\)).
\[ 2^N (2^{M-N} - 1) = 48 \]
Now, we find the prime factorization of 48.
\[ 48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3 \]
Comparing the factored equation with the prime factorization of 48:
\[ 2^N (2^{M-N} - 1) = 2^4 \times 3 \]
The term \(2^N\) must be a power of 2, and the term \((2^{M-N} - 1)\) will be an odd number (since any power of 2 is even, and subtracting 1 makes it odd).

By comparing the factors on both sides of the equation, we can equate the even and odd parts.

Equating the power of 2 part:
\[ 2^N = 2^4 \implies N = 4 \]
Equating the odd part:
\[ 2^{M-N} - 1 = 3 \]
Now, we solve for M using this second equation.
\[ 2^{M-N} = 3 + 1 = 4 \] \[ 2^{M-N} = 2^2 \]
This implies that the exponents are equal:
\[ M - N = 2 \]
We already found that \(N = 4\). Substituting this value:
\[ M - 4 = 2 \implies M = 6 \]

Step 4: Final Answer:

The values are \(M = 6\) and \(N = 4\). This corresponds to option (B).
Quick Tip: When solving equations of the form \(a^x - a^y = k\), factoring out the smaller power (\(a^y\)) is a very effective strategy. Then, use prime factorization on the constant \(k\) to compare factors.

Step 1: Understanding the Concept:

This question requires the use of De Morgan's Laws and the formula for the cardinality of the union of two sets. We are asked to find the number of elements that are neither in A nor in B.


Step 2: Key Formula or Approach:

De Morgan's Law states that \(A' \cap B' = (A \cup B)'\).

The number of elements in the complement of a set is given by \(n(S') = n(U) - n(S)\).

Therefore, \(n(A' \cap B') = n((A \cup B)') = n(U) - n(A \cup B)\).

The formula for the union of two sets is \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).


Step 3: Detailed Explanation:

The problem statement as transcribed from the OCR is incomplete. It provides \(n(U) = 800\) and \(n(A) = 300\), but the values for \(n(B)\) and \(n(A \cap B)\) are missing.

To calculate \(n(A' \cap B')\), we would need to first find \(n(A \cup B)\) using the formula: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Since \(n(B)\) and \(n(A \cap B)\) are not provided in the question, it is impossible to compute a numerical value for \(n(A \cup B)\) and subsequently for \(n(A' \cap B')\).

For this reason, the question is considered invalid or cancelled due to missing information.


Step 4: Final Answer:

The question cannot be solved as stated. The provided answer "Question Cancelled" is correct.
Quick Tip: In competitive exams, always check if the question provides all the necessary data. If crucial information is missing, the question may be flawed, and you shouldn't waste time trying to guess the missing values.

Step 1: Understanding the Concept:

This problem involves function notation and algebraic manipulation. We need to evaluate the function \(f(x)\) at two different points, \(a\) and \(a+1\), find their ratio, and then see which of the given options matches the resulting expression.


Step 2: Key Formula or Approach:

The given function is \(f(x) = \frac{x}{x-1}\).

We need to compute the expression \(\frac{f(a)}{f(a+1)}\).


Step 3: Detailed Explanation:

First, let's find the expressions for \(f(a)\) and \(f(a+1)\).

For \(f(a)\), we substitute \(x=a\) into the function definition:
\[ f(a) = \frac{a}{a-1} \]
For \(f(a+1)\), we substitute \(x=a+1\):
\[ f(a+1) = \frac{a+1}{(a+1)-1} = \frac{a+1}{a} \]
Now, we compute the ratio \(\frac{f(a)}{f(a+1)}\):
\[ \frac{f(a)}{f(a+1)} = \frac{\frac{a}{a-1}}{\frac{a+1}{a}} \]
To simplify this complex fraction, we multiply the numerator by the reciprocal of the denominator:
\[ \frac{f(a)}{f(a+1)} = \frac{a}{a-1} \times \frac{a}{a+1} = \frac{a \times a}{(a-1)(a+1)} \]
Using the difference of squares formula, \((p-q)(p+q) = p^2 - q^2\), in the denominator:
\[ \frac{f(a)}{f(a+1)} = \frac{a^2}{a^2 - 1} \]
Now we must check which of the options is equal to this expression. Let's evaluate option (A), \(f(a^2)\).

To find \(f(a^2)\), we substitute \(x = a^2\) into the original function \(f(x) = \frac{x}{x-1}\):
\[ f(a^2) = \frac{a^2}{a^2 - 1} \]

Step 4: Final Answer:

The expression we derived, \(\frac{a^2}{a^2 - 1}\), is exactly equal to \(f(a^2)\). Therefore, the correct answer is option (A).
Quick Tip: When dealing with function manipulations, always perform the substitutions carefully. For complex fractions, remember the "keep-change-flip" rule: keep the numerator, change division to multiplication, and flip the denominator (take its reciprocal).

Step 1: Understanding the Concept:

The given functional relation, \(f(x + y) = f(x) + f(y)\), is known as Cauchy's functional equation. For functions over real numbers, a continuous solution is of the form \(f(x) = kx\) for some constant \(k\). We can also solve this problem by deducing the values of the function for integer inputs step-by-step.


Step 2: Key Formula or Approach:

We are given:

1. \(f(x + y) = f(x) + f(y)\)

2. \(f(1) = 3\)

We need to find the values of \(f(0), f(2)\), and \(f(3)\) and then compute the sum \(f(0) + f(1) + f(2) + f(3)\).


Step 3: Detailed Explanation:

Finding f(0):

Let \(x = 0\) and \(y = 0\) in the functional equation.
\[ f(0 + 0) = f(0) + f(0) \implies f(0) = 2f(0) \]
Subtracting \(f(0)\) from both sides gives \(f(0) = 0\).


Finding f(2):

Let \(x = 1\) and \(y = 1\).
\[ f(1 + 1) = f(1) + f(1) \implies f(2) = 3 + 3 = 6 \]

Finding f(3):

Let \(x = 2\) and \(y = 1\).
\[ f(2 + 1) = f(2) + f(1) \implies f(3) = 6 + 3 = 9 \]

Calculating the sum:

Now we compute the required sum:
\[ f(0) + f(1) + f(2) + f(3) = 0 + 3 + 6 + 9 \] \[ Sum = 18 \]

Alternative Method (using \(f(x) = kx\)):

Since \(f(x+y) = f(x)+f(y)\), we can assume \(f(x) = kx\).

Using the given condition \(f(1) = 3\):
\[ f(1) = k \times 1 = 3 \implies k = 3 \]
So, the function is \(f(x) = 3x\).

Now we can find the required values:
\(f(0) = 3 \times 0 = 0\)
\(f(1) = 3 \times 1 = 3\)
\(f(2) = 3 \times 2 = 6\)
\(f(3) = 3 \times 3 = 9\)

The sum is \(0 + 3 + 6 + 9 = 18\).


Step 4: Final Answer:

The sum is 18, which corresponds to option (D).
Quick Tip: Recognizing Cauchy's functional equation \(f(x+y)=f(x)+f(y)\) can save time. If a function satisfies this and you're given \(f(1)=k\), you can generally deduce that \(f(n) = nk\) for any integer \(n\), and often \(f(x)=kx\) for any real \(x\) if continuity is assumed.

Step 1: Understanding the Concept:

This problem involves operations with complex numbers. We are given a complex number \(z\) and asked to evaluate a polynomial expression involving \(z\). A direct substitution is possible, but a more elegant method involves creating a quadratic equation from the definition of \(z\).


Step 2: Key Formula or Approach:

Method 1: Direct Substitution

Substitute \(z = 2+i\) directly into the expression \(z^2 - 4z + 15\).

Method 2: Creating a Quadratic Equation

Rearrange \(z = 2+i\) to isolate \(i\), then square both sides to eliminate \(i\) and form a quadratic equation in \(z\).


Step 3: Detailed Explanation:

Method 1: Direct Substitution

We are given \(z = 2+i\) and need to evaluate \(z^2 - 4z + 15\).

First, calculate \(z^2\):
\[ z^2 = (2+i)^2 = 2^2 + 2(2)(i) + i^2 = 4 + 4i - 1 = 3 + 4i \]
Next, calculate \(4z\):
\[ 4z = 4(2+i) = 8 + 4i \]
Now substitute these back into the expression:
\[ z^2 - 4z + 15 = (3 + 4i) - (8 + 4i) + 15 \] \[ = 3 + 4i - 8 - 4i + 15 \]
Group the real and imaginary parts:
\[ = (3 - 8 + 15) + (4i - 4i) = 10 + 0i = 10 \]

Method 2: Creating a Quadratic Equation (More Efficient)

Start with the given complex number:
\[ z = 2 + i \]
Isolate the imaginary part by moving the real part to the other side:
\[ z - 2 = i \]
Square both sides to eliminate \(i\):
\[ (z - 2)^2 = i^2 \]
Expand the left side and use the fact that \(i^2 = -1\):
\[ z^2 - 4z + 4 = -1 \]
Rearrange the equation to match the expression we need to evaluate:
\[ z^2 - 4z = -1 - 4 \implies z^2 - 4z = -5 \]
Now look at the target expression: \(z^2 - 4z + 15\).

We can substitute the value we just found for \(z^2 - 4z\):
\[ (z^2 - 4z) + 15 = (-5) + 15 = 10 \]

Step 4: Final Answer:

Both methods yield the value 10. This corresponds to option (E).
Quick Tip: When asked to evaluate a polynomial in \(z\) where \(z\) is a complex number, the method of creating a quadratic equation is often faster and less prone to calculation errors than direct substitution.

Step 1: Understanding the Concept:

The modulus of a complex number \(z = a + bi\) is its distance from the origin in the Argand plane, given by \(|z| = \sqrt{a^2 + b^2}\). The modulus of a quotient of two complex numbers is the quotient of their moduli: \(|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}\).


Step 2: Key Formula or Approach:

For the reconstructed question, we have \(z = \frac{4+3i}{2i}\). We will use the property \(|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}\).

Let \(z_1 = 4+3i\) and \(z_2 = 2i\).


Step 3: Detailed Explanation:

First, we find the modulus of the numerator, \(|z_1| = |4+3i|\).
\[ |4+3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Next, we find the modulus of the denominator, \(|z_2| = |2i|\). The complex number \(2i\) can be written as \(0 + 2i\).
\[ |2i| = |0+2i| = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]
Now, we find the modulus of the quotient:
\[ |z| = \left| \frac{4+3i}{2i} \right| = \frac{|4+3i|}{|2i|} = \frac{5}{2} \]

Alternative Method (Simplifying First):

We can first simplify the expression for \(z\) by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of \(2i\) is \(-2i\). \[ z = \frac{4+3i}{2i} \times \frac{-2i}{-2i} = \frac{(4+3i)(-2i)}{(2i)(-2i)} \] \[ z = \frac{-8i - 6i^2}{-4i^2} = \frac{-8i - 6(-1)}{-4(-1)} = \frac{6-8i}{4} \] \[ z = \frac{6}{4} - \frac{8i}{4} = \frac{3}{2} - 2i \]
Now, we find the modulus of this simplified complex number:
\[ |z| = \left| \frac{3}{2} - 2i \right| = \sqrt{\left(\frac{3}{2}\right)^2 + (-2)^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{9}{4} + \frac{16}{4}} = \sqrt{\frac{25}{4}} = \frac{5}{2} \]

Step 4: Final Answer:

Both methods give the modulus as \(\frac{5}{2}\), which corresponds to option (E).
Quick Tip: Using the property \(|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}\) is often quicker for finding the modulus of a complex fraction than first rationalizing the denominator.

Step 1: Understanding the Concept:

The locus of a complex number \(z\) is the set of points in the complex plane that satisfy a given condition. We need to translate the given condition about the real and imaginary parts into a Cartesian equation involving \(x\) and \(y\), where \(z = x + iy\).


Step 2: Key Formula or Approach:

Let the complex number be \(z = x + iy\), where \(x\) and \(y\) are real numbers representing the coordinates in the Argand plane.

We need to find the expression for \(z - 2 - 3i\), identify its real and imaginary parts, and then set them equal to each other.

Real part: \(Re(z)\)

Imaginary part: \(Im(z)\)

The condition is \(Re(z - 2 - 3i) = Im(z - 2 - 3i)\).


Step 3: Detailed Explanation:

Substitute \(z = x + iy\) into the expression \(z - 2 - 3i\):
\[ z - 2 - 3i = (x + iy) - 2 - 3i \]
Group the real and imaginary terms together:
\[ z - 2 - 3i = (x - 2) + i(y - 3) \]
From this expression, we can identify the real and imaginary parts:

The real part is \(Re(z - 2 - 3i) = x - 2\).

The imaginary part is \(Im(z - 2 - 3i) = y - 3\).

Now, we apply the given condition that these two parts are equal:
\[ x - 2 = y - 3 \]
To better identify the geometric shape, we can rearrange this equation into a standard form, like \(y = mx + c\).
\[ y = x - 2 + 3 \] \[ y = x + 1 \]

Step 4: Final Answer:

The equation \(y = x + 1\) is the equation of a straight line in the Cartesian plane (which corresponds to the Argand plane for complex numbers). Therefore, the locus of \(z\) is a straight line. This is option (B).
Quick Tip: Problems involving the locus of a complex number can almost always be solved by substituting \(z = x + iy\) and converting the given complex equation into a Cartesian equation in \(x\) and \(y\).

Step 1: Understanding the Concept:

The powers of the imaginary unit \(i\) are cyclic with a period of 4. We need to evaluate \(i^k\) where \(k\) is of the form \(4n+3\). This form directly relates to the remainder when \(k\) is divided by 4.


Step 2: Key Formula or Approach:

The powers of \(i\) follow a cycle:
\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = i^2 \cdot i = -i\)
\(i^4 = (i^2)^2 = (-1)^2 = 1\)

This cycle repeats every four powers. In general, for an integer \(m\), \(i^m = i^{m \pmod 4}\). The expression \(k = 4n+3\) means that \(k\) leaves a remainder of 3 when divided by 4.

We will use the law of exponents: \(a^{m+n} = a^m \cdot a^n\).


Step 3: Detailed Explanation:

We need to calculate \(i^k\) where \(k = 4n + 3\).

Substitute the expression for \(k\):
\[ i^k = i^{4n + 3} \]
Using the law of exponents, we can split this into two parts:
\[ i^{4n+3} = i^{4n} \cdot i^3 \]
Now, let's evaluate each part separately.

For \(i^{4n}\), we can write it as \((i^4)^n\). Since we know \(i^4 = 1\):
\[ i^{4n} = (i^4)^n = 1^n = 1 \quad (for any integer n) \]
For \(i^3\), we know its value from the cycle:
\[ i^3 = -i \]
Now, multiply the two parts together:
\[ i^k = 1 \cdot (-i) = -i \]

Step 4: Final Answer:

The value of \(i^k\) is \(-i\). This corresponds to option (E).
Quick Tip: To find the value of \(i^k\) for any large integer \(k\), simply find the remainder of \(k\) when divided by 4. Let the remainder be \(r\). Then \(i^k = i^r\). For \(k = 4n+3\), the remainder is 3, so \(i^k = i^3 = -i\).

Step 1: Understanding the Concept:

This problem deals with a Geometric Progression (G.P.). We are given information about the sum of certain terms and asked to find a specific term (the 100th term).


Step 2: Key Formula or Approach:

Let the G.P. have the first term \(a\) and the common ratio \(r\). The terms are \(a, ar, ar^2, \dots\).

The \(n\)-th term of a G.P. is given by \(T_n = ar^{n-1}\).

We are given:

1. Sum of first three terms: \(a + ar + ar^2 = 14\)

2. Sum of next three terms (4th, 5th, 6th): \(ar^3 + ar^4 + ar^5 = 112\)

We need to find \(T_{100}\).


Step 3: Detailed Explanation:

First, we write the given information as equations.

Equation 1: From the sum of the first three terms, we factor out \(a\).
\[ a(1 + r + r^2) = 14 \quad \cdots (1) \]
Equation 2: From the sum of the next three terms, we factor out \(ar^3\).
\[ ar^3(1 + r + r^2) = 112 \quad \cdots (2) \]
Now we have a system of two equations with two unknowns, \(a\) and \(r\). A good strategy is to divide Equation (2) by Equation (1) to eliminate \(a\) and the common bracketed term.
\[ \frac{ar^3(1 + r + r^2)}{a(1 + r + r^2)} = \frac{112}{14} \]
Assuming \(a \neq 0\) and \(1+r+r^2 \neq 0\), we can cancel the terms:
\[ r^3 = \frac{112}{14} \]
To simplify the fraction: \(112 = 14 \times 8\). So,
\[ r^3 = 8 \]
Taking the cube root of both sides, we get the common ratio:
\[ r = 2 \]
Now that we have \(r\), we can find \(a\) by substituting \(r=2\) back into Equation (1).
\[ a(1 + 2 + 2^2) = 14 \] \[ a(1 + 2 + 4) = 14 \] \[ a(7) = 14 \implies a = \frac{14}{7} = 2 \]
So, the first term is \(a=2\) and the common ratio is \(r=2\).

Finally, we need to find the 100th term, \(T_{100}\).

Using the formula \(T_n = ar^{n-1}\):
\[ T_{100} = a \cdot r^{100-1} = a \cdot r^{99} \]
Substitute the values of \(a\) and \(r\):
\[ T_{100} = 2 \cdot (2)^{99} = 2^1 \cdot 2^{99} = 2^{1+99} = 2^{100} \]

Step 4: Final Answer:

The 100th term of the G.P. is \(2^{100}\). This corresponds to option (C).
Quick Tip: When given sums of consecutive blocks of terms in a G.P., setting up equations and then dividing them is a standard and very effective method to quickly find the common ratio \(r\).

Step 1: Understanding the Concept:

The problem involves the properties of a Geometric Progression (G.P.). We are given the product of the first three terms and the product of the first four terms, and we need to determine the first term, \(a\).


Step 2: Key Formula or Approach:

Let the terms of the G.P. be \(a, ar, ar^2, ar^3, \dots\).

Product of the first three terms (\(P_3\)): \(a \cdot ar \cdot ar^2 = a^3r^3\).

Product of the first four terms (\(P_4\)): \(a \cdot ar \cdot ar^2 \cdot ar^3 = a^4r^6\).

Notice that \(P_4 = P_3 \times (the 4th term)\). The 4th term is \(ar^3\).


Step 3: Detailed Explanation:

We are given:
\(P_3 = 216\)
\(P_4 = 324\)

From the relationship \(P_4 = P_3 \times (4th term)\), we can find the fourth term of the G.P.
\[ 324 = 216 \times (T_4) \] \[ T_4 = \frac{324}{216} \]
We can simplify this fraction. Both are divisible by 108. \(324 = 3 \times 108\) and \(216 = 2 \times 108\).
\[ T_4 = ar^3 = \frac{3}{2} \quad \cdots (1) \]
Now, let's use the information about \(P_3\).
\[ P_3 = a \cdot ar \cdot ar^2 = a^3r^3 = (ar)^3 = 216 \]
Taking the cube root of both sides:
\[ \sqrt[3]{(ar)^3} = \sqrt[3]{216} \]
Since \(6^3 = 216\), we get:
\[ ar = 6 \quad \cdots (2) \]
This tells us that the second term of the G.P. is 6.

Now we have two equations: \(ar^3 = 3/2\) and \(ar = 6\). We can find \(r\) by dividing the first by the second.
\[ \frac{ar^3}{ar} = \frac{3/2}{6} \] \[ r^2 = \frac{3}{2 \times 6} = \frac{3}{12} = \frac{1}{4} \]
Taking the square root, we get \(r = \pm \frac{1}{2}\).

Now we can find the first term, \(a\), using equation (2): \(a = \frac{6}{r}\).

Case 1: If \(r = \frac{1}{2}\)
\[ a = \frac{6}{1/2} = 6 \times 2 = 12 \]
Case 2: If \(r = -\frac{1}{2}\)
\[ a = \frac{6}{-1/2} = 6 \times (-2) = -12 \]
Since all the options provided are positive, we choose \(a=12\).


Step 4: Final Answer:

The first term is 12. This corresponds to option (E).
Quick Tip: When dealing with products of terms in a G.P., remember that the product of the first \(n+1\) terms is simply the product of the first \(n\) terms multiplied by the \((n+1)\)-th term. This can often simplify the problem significantly.

Step 1: Understanding the Concept:

This problem relies on a key property of Geometric Progressions (G.P.). For any finite G.P., the product of terms equidistant from the beginning and the end is constant and equal to the product of the first and last terms.


Step 2: Key Formula or Approach:

Let the first four terms of the G.P. be \(T_1, T_2, T_3, T_4\), which are \(a, ar, ar^2, ar^3\).

We are given the product of these four terms: \(P_4 = T_1 \cdot T_2 \cdot T_3 \cdot T_4 = \frac{1}{1024}\).

We need to find the product of the second and third terms: \(P_{2,3} = T_2 \cdot T_3\).

Property: In a G.P., \(T_1 \cdot T_4 = T_2 \cdot T_3\).


Step 3: Detailed Explanation:

Let's verify the property first.

Product of the first and fourth terms: \(T_1 \cdot T_4 = a \cdot (ar^3) = a^2r^3\).

Product of the second and third terms: \(T_2 \cdot T_3 = (ar) \cdot (ar^2) = a^2r^3\).

The property holds true.

Now, let's use this property to solve the problem. The product of the four terms can be regrouped:
\[ P_4 = (T_1 \cdot T_4) \cdot (T_2 \cdot T_3) \]
Since \(T_1 \cdot T_4 = T_2 \cdot T_3\), let's call this product \(P\). So, \(P = T_2 \cdot T_3\).

Then the total product is:
\[ P_4 = P \cdot P = P^2 \]
We are given that \(P_4 = \frac{1}{1024}\).

So, we have the equation:
\[ P^2 = \frac{1}{1024} \]
To find \(P\), we take the square root of both sides:
\[ P = \sqrt{\frac{1}{1024}} \]
We know that \(32^2 = 1024\). Therefore, \(\sqrt{1024} = 32\).
\[ P = \frac{1}{32} \]
(Assuming the product is positive, which is consistent with the options).


Step 4: Final Answer:

The product of the second and third terms is \(\frac{1}{32}\). This corresponds to option (D).
Quick Tip: For a G.P. with an even number of terms, say \(2k\), the product of all terms is equal to (Product of the two middle terms)\(^k\). Here, we have 4 terms (so \(k=2\)), and the product is \((T_2 \cdot T_3)^2\). This is a useful shortcut.

Step 1: Understanding the Concept:

The problem involves the definitions of Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of two numbers. We need to use the given A.M. to find the unknown number, and then calculate the G.M. of the two numbers.


Step 2: Key Formula or Approach:

For two numbers \(x\) and \(y\):

Arithmetic Mean (A.M.) = \(\frac{x+y}{2}\)

Geometric Mean (G.M.) = \(\sqrt{xy}\)


Step 3: Detailed Explanation:

We are given that the A.M. of \(a\) and \(c\) is 16.

Using the formula for A.M.:
\[ \frac{a+c}{2} = 16 \]
We are also given that \(a = 8\). We can substitute this value into the equation to find \(c\).
\[ \frac{8+c}{2} = 16 \]
Multiply both sides by 2:
\[ 8 + c = 32 \]
Subtract 8 from both sides to solve for \(c\):
\[ c = 32 - 8 = 24 \]
Now we have both numbers, \(a=8\) and \(c=24\). We need to find their Geometric Mean (G.M.).

Using the formula for G.M.:
\[ G.M. = \sqrt{a \cdot c} = \sqrt{8 \times 24} \]
To simplify the square root, we can find the prime factors or look for perfect squares.
\[ 8 \times 24 = 8 \times (8 \times 3) = 64 \times 3 \]
Now, substitute this back into the G.M. formula:
\[ G.M. = \sqrt{64 \times 3} = \sqrt{64} \times \sqrt{3} \]
Since \(\sqrt{64} = 8\), we have:
\[ G.M. = 8\sqrt{3} \]

Step 4: Final Answer:

The Geometric Mean of \(a\) and \(c\) is \(8\sqrt{3}\). This corresponds to option (A).
Quick Tip: When calculating square roots of products, like \(\sqrt{xy}\), it's often easier to find the prime factorization of \(x\) and \(y\) or to break them down into factors that include perfect squares, rather than multiplying them into a large number first.

Step 1: Understanding the Concept:

This question involves permutations. We need to solve an equation involving the permutation notation \(^n P_r\), which represents the number of ways to arrange \(r\) items from a set of \(n\) distinct items.


Step 2: Key Formula or Approach:

The formula for permutations is:
\[ ^n P_r = \frac{n!}{(n-r)!} \]
We will apply this formula to both sides of the given equation and solve for \(n\).

A useful expansion is \(n! = n \cdot (n-1) \cdot (n-2) \cdots 1\). This allows for cancellation of factorial terms. For example, \((n-3)! = (n-3)(n-4)(n-5)!\).


Step 3: Detailed Explanation:

The given equation is \(^n P_5 = 42 \cdot ^n P_3\).

First, let's write out the expressions using the permutation formula.

For the left side:
\[ ^n P_5 = \frac{n!}{(n-5)!} \]
For the right side:
\[ ^n P_3 = \frac{n!}{(n-3)!} \]
Now, substitute these into the equation:
\[ \frac{n!}{(n-5)!} = 42 \cdot \frac{n!}{(n-3)!} \]
For the permutation to be defined, we must have \(n \geq 5\), so \(n!\) is non-zero. We can divide both sides by \(n!\):
\[ \frac{1}{(n-5)!} = \frac{42}{(n-3)!} \]
To solve for \(n\), we can cross-multiply or rearrange to get the factorial terms on one side:
\[ \frac{(n-3)!}{(n-5)!} = 42 \]
Now, expand the larger factorial, \((n-3)!\), until we get the smaller factorial, \((n-5)!\).
\[ (n-3)! = (n-3) \cdot (n-4) \cdot (n-5)! \]
Substitute this expansion back into the equation:
\[ \frac{(n-3)(n-4)(n-5)!}{(n-5)!} = 42 \]
Cancel the \((n-5)!\) terms:
\[ (n-3)(n-4) = 42 \]
We are looking for two consecutive integers, \((n-4)\) and \((n-3)\), whose product is 42. We know that \(6 \times 7 = 42\).

By comparison, the larger number is \((n-3)\) and the smaller is \((n-4)\).
\[ n-3 = 7 \implies n = 10 \] \[ n-4 = 6 \implies n = 10 \]
Both give the same result, \(n = 10\).

This also satisfies the initial condition that \(n \geq 5\).


Step 4: Final Answer:

The value of \(n\) is 10. This corresponds to option (E).
Quick Tip: When solving equations with factorials like \(\frac{n!}{(n-k)!}\), remember that this is equivalent to the product \(n(n-1)\cdots(n-k+1)\). In this problem, \((n-3)(n-4) = 42\). Instead of expanding into a quadratic equation, look for consecutive integers whose product matches the constant. It's often much faster.

Step 1: Understanding the Concept:

This is a problem of permutations with repetitions under a constraint. We need to find the number of ways to arrange the letters of a word where some letters are repeated, and the positions of two specific letters are fixed.


Step 2: Key Formula or Approach:

The number of permutations of \(n\) objects, where there are \(p_1\) objects of type 1, \(p_2\) objects of type 2, , \(p_k\) objects of type k, is given by the formula: \[ \frac{n!}{p_1! p_2! \cdots p_k!} \]

Step 3: Detailed Explanation:

The given word is INDEPENDENCE.

Total number of letters is 12.

The letters are I, N, D, E, P, E, N, D, E, N, C, E.

The frequency of each letter is:

E: 4 times

N: 3 times

D: 2 times

I: 1 time

P: 1 time

C: 1 time

The constraint is that the arrangement must start with 'I' and end with 'P'.

So, we fix the first position for 'I' and the last position for 'P'.
\[ I \quad \underbrace{\_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_ \,\, \_\_}_{10 positions to fill} \quad P \]
The remaining letters to be arranged in the 10 middle positions are: N, D, E, E, N, D, E, N, C, E.

Total number of letters to arrange, \(n = 10\).

The repetitions among these 10 letters are:

E: 4 times (\(p_1 = 4\))

N: 3 times (\(p_2 = 3\))

D: 2 times (\(p_3 = 2\))

C: 1 time (\(p_4 = 1\))

Using the formula for permutations with repetitions:
\[ Number of arrangements = \frac{10!}{4! \cdot 3! \cdot 2! \cdot 1!} \]
Now, we calculate the value:
\[ \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}{4! \times (3 \times 2 \times 1) \times (2 \times 1)} \]
Cancel out \(4!\):
\[ \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5}{(3 \times 2 \times 1) \times (2 \times 1)} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5}{6 \times 2} \]
Cancel out the 6:
\[ \frac{10 \times 9 \times 8 \times 7 \times 5}{2} \]
Divide 8 by 2:
\[ 10 \times 9 \times 4 \times 7 \times 5 = 90 \times 28 \times 5 = 90 \times 140 = 12600 \]

Step 4: Final Answer:

The total number of arrangements is 12600. This corresponds to option (D).
Quick Tip: In permutation problems with constraints, always handle the fixed positions first. This simplifies the problem to arranging the remaining objects in the remaining available positions.

Step 1: Understanding the Concept:

This is a problem of combinations. We need to find the number of ways to select positions for heads in a sequence of four coin tosses. The word "or" indicates that we need to find the number of ways for each case (2 heads and 3 heads) and then add them together.


Step 2: Key Formula or Approach:

The number of ways to choose \(r\) items from a set of \(n\) distinct items is given by the combination formula:
\[ ^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
The total number of ways for event A "or" event B is the sum of the number of ways for A and the number of ways for B, since they are mutually exclusive events.

Total ways = (Ways of getting 2 heads) + (Ways of getting 3 heads).


Step 3: Detailed Explanation:

We have 4 coins being tossed, so \(n = 4\).

Case 1: Getting exactly 2 heads.

We need to choose 2 positions for the heads out of the 4 available positions. The other positions will be tails.

Number of ways = \(^4C_2\).
\[ ^4C_2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} = \frac{24}{4} = 6 \]
Case 2: Getting exactly 3 heads.

We need to choose 3 positions for the heads out of the 4 available positions. The remaining position will be a tail.

Number of ways = \(^4C_3\).
\[ ^4C_3 = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(1)} = \frac{24}{6} = 4 \]
Total number of ways:

The total number of ways of getting 2 or 3 heads is the sum of the ways from Case 1 and Case 2.
\[ Total ways = ^4C_2 + ^4C_3 = 6 + 4 = 10 \]

Step 4: Final Answer:

The number of possible ways of getting 2 or 3 heads is 10. This corresponds to option (B).
Quick Tip: Remember the property of combinations \(^nC_r = ^nC_{n-r}\). This can simplify calculations. For example, \(^4C_3\) is the same as \(^4C_{4-3} = ^4C_1 = 4\).

Step 1: Understanding the Concept:

This problem requires knowledge of the properties of binomial coefficients (\(^nC_r\)). Specifically, we need to know for which value of \(r\) the term \(^nC_r\) is maximized. Then we evaluate the required ratio using these maximum values.


Step 2: Key Formula or Approach:

The value of \(^nC_r\) is greatest when:

\(r = \frac{n}{2}\) if \(n\) is even.
\(r = \frac{n-1}{2}\) or \(r = \frac{n+1}{2}\) if \(n\) is odd. (The values are equal in these two cases).


Step 3: Detailed Explanation:

Maximizing the Numerator (\(^5C_r\)):

Here, \(n=5\), which is an odd number.

The greatest value of \(^5C_r\) occurs when \(r = \frac{5-1}{2} = 2\) or \(r = \frac{5+1}{2} = 3\).

Let's calculate both:
\[ ^5C_2 = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10 \] \[ ^5C_3 = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10 \]
So, the greatest value of the numerator is 10. This occurs at \(r=2\) or \(r=3\).


Maximizing the Denominator (\(^6C_r\)):

Here, \(n=6\), which is an even number.

The greatest value of \(^6C_r\) occurs when \(r = \frac{6}{2} = 3\).

Let's calculate this value:
\[ ^6C_3 = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
So, the greatest value of the denominator is 20. This occurs at \(r=3\).


Calculating the Ratio:

The question asks for the value of the ratio when both numerator and denominator take their greatest value. This implies we need to find a value of \(r\) that maximizes both simultaneously if possible, or use the respective maximum values. The wording "when the numerator and denominator take their greatest value" implies we should use the maximum value of each.

Maximum value of numerator = 10.

Maximum value of denominator = 20.

The ratio is:
\[ \frac{Greatest value of ^5C_r}{Greatest value of ^6C_r} = \frac{10}{20} = \frac{1}{2} \]
Note that if we must use the same \(r\) for both, we would choose \(r=3\), which maximizes the denominator and is one of the values that maximizes the numerator. In this case, the ratio would be \(\frac{^5C_3}{^6C_3} = \frac{10}{20} = \frac{1}{2}\). Both interpretations lead to the same result.


Step 4: Final Answer:

The value of the ratio is \(\frac{1}{2}\). This corresponds to option (B).
Quick Tip: The values of \(^nC_r\) for a fixed \(n\) increase from \(r=0\) to the middle value(s) and then decrease. This pattern is symmetric and is visualized in Pascal's triangle.

Step 1: Understanding the Concept:

This problem involves the properties of coefficients in a polynomial expansion. We can find sums of coefficients by substituting specific values for \(x\) (like 1 and -1) into the polynomial identity.


Step 2: Key Formula or Approach:

Let \(P(x) = (1 + x - 2x^2)^6 = a_0 + a_1x + a_2x^2 + \dots + a_{12}x^{12}\).

1. To get the sum of all coefficients (\(a_0 + a_1 + a_2 + \dots\)), we set \(x=1\).

2. To get the alternating sum of coefficients (\(a_0 - a_1 + a_2 - \dots\)), we set \(x=-1\).

3. The sum of even-indexed coefficients is \(\frac{P(1) + P(-1)}{2}\).

4. The sum of odd-indexed coefficients is \(\frac{P(1) - P(-1)}{2}\).


Step 3: Detailed Explanation:

Let \(P(x) = (1 + x - 2x^2)^6\).

Step 3.1: Calculate P(1).

Set \(x=1\) in the identity:
\[ P(1) = (1 + 1 - 2(1)^2)^6 = (2 - 2)^6 = 0^6 = 0 \]
This means the sum of all coefficients is zero:
\[ a_0 + a_1 + a_2 + a_3 + \dots + a_{12} = 0 \quad \cdots (1) \]
Step 3.2: Calculate P(-1).

Set \(x=-1\) in the identity:
\[ P(-1) = (1 + (-1) - 2(-1)^2)^6 = (1 - 1 - 2(1))^6 = (-2)^6 = 64 \]
This means the alternating sum of coefficients is 64:
\[ a_0 - a_1 + a_2 - a_3 + \dots + a_{12} = 64 \quad \cdots (2) \]
Step 3.3: Find the sum of even-indexed coefficients.

We want to find \(a_2 + a_4 + \dots + a_{12}\). Let's first find \(a_0 + a_2 + a_4 + \dots + a_{12}\). We can get this by adding equations (1) and (2).
\[ (a_0 + a_1 + \dots) + (a_0 - a_1 + \dots) = 0 + 64 \] \[ 2a_0 + 2a_2 + 2a_4 + \dots + 2a_{12} = 64 \]
Divide by 2:
\[ a_0 + a_2 + a_4 + \dots + a_{12} = 32 \]
Step 3.4: Find \(a_0\) and the final sum.

The question asks for the sum starting from \(a_2\). So we need to find \(a_0\) and subtract it from the sum we just calculated.
\(a_0\) is the constant term of the expansion, which is obtained by setting \(x=0\).
\[ a_0 = P(0) = (1 + 0 - 2(0)^2)^6 = 1^6 = 1 \]
Now, we can find the required sum:
\[ a_2 + a_4 + \dots + a_{12} = (a_0 + a_2 + a_4 + \dots + a_{12}) - a_0 \] \[ = 32 - 1 = 31 \]

Step 4: Final Answer:

The value of the sum is 31. This corresponds to option (A).
Quick Tip: This technique of substituting \(x=1\) and \(x=-1\) is very powerful for finding the sum of all coefficients, the sum of even-powered coefficients, and the sum of odd-powered coefficients in any polynomial expansion.

Step 1: Understanding the Concept:

A square matrix \(A\) is called a symmetric matrix if it is equal to its transpose, i.e., \(A = A^T\). The transpose of a matrix is obtained by interchanging its rows and columns.


Step 2: Key Formula or Approach:

The condition for a matrix \(A\) to be symmetric is that its elements \(a_{ij}\) must satisfy \(a_{ij} = a_{ji}\) for all \(i\) and \(j\). This means the element in the \(i\)-th row and \(j\)-th column must be equal to the element in the \(j\)-th row and \(i\)-th column.


Step 3: Detailed Explanation:

The given matrix is: \[ A = \begin{pmatrix} 5 & 2 & x
y & 2 & -3
4 & t & -7 \end{pmatrix} \]
We apply the condition for symmetry, \(a_{ij} = a_{ji}\), to the off-diagonal elements.

1. For \(i=1, j=2\): \(a_{12} = a_{21}\)

The element in the 1st row, 2nd column is 2.

The element in the 2nd row, 1st column is \(y\).

Therefore, \(y = 2\).

2. For \(i=1, j=3\): \(a_{13} = a_{31}\)

The element in the 1st row, 3rd column is \(x\).

The element in the 3rd row, 1st column is 4.

Therefore, \(x = 4\).

3. For \(i=2, j=3\): \(a_{23} = a_{32}\)

The element in the 2nd row, 3rd column is -3.

The element in the 3rd row, 2nd column is \(t\).

Therefore, \(t = -3\).


The values are \(x=4\), \(y=2\), and \(t=-3\).


Step 4: Final Answer:

The values of \(x, y, t\) respectively are 4, 2, -3. This corresponds to option (B).
Quick Tip: For a 3x3 symmetric matrix, you can quickly find the unknown values by visually matching the elements that are mirrored across the main diagonal (from top-left to bottom-right).

Step 1: Understanding the Concept:

This problem involves matrix multiplication and solving a matrix equation. We need to first compute the square of matrix \(A\), which is \(A \times A\), and then equate the resulting matrix with matrix \(B\) to find the unknown value \(x\).


Step 2: Key Formula or Approach:

For two 2x2 matrices \(M = \begin{pmatrix} a & b
c & d \end{pmatrix}\) and \(N = \begin{pmatrix} p & q
r & s \end{pmatrix}\), their product is: \[ MN = \begin{pmatrix} ap+br & aq+bs
cp+dr & cq+ds \end{pmatrix} \]
Two matrices are equal if and only if their corresponding elements are equal.


Step 3: Detailed Explanation:

We are given the matrix \(A = \begin{pmatrix} x & 0
1 & 1 \end{pmatrix}\).

First, we calculate \(A^2\):
\[ A^2 = A \times A = \begin{pmatrix} x & 0
1 & 1 \end{pmatrix} \begin{pmatrix} x & 0
1 & 1 \end{pmatrix} \]
Using the rule for matrix multiplication:
\[ A^2 = \begin{pmatrix} (x)(x) + (0)(1) & (x)(0) + (0)(1)
(1)(x) + (1)(1) & (1)(0) + (1)(1) \end{pmatrix} \] \[ A^2 = \begin{pmatrix} x^2 & 0
x+1 & 1 \end{pmatrix} \]
We are given that \(A^2 = B\), where \(B = \begin{pmatrix} 16 & 0
5 & 1 \end{pmatrix}\).

So, we equate the two matrices:
\[ \begin{pmatrix} x^2 & 0
x+1 & 1 \end{pmatrix} = \begin{pmatrix} 16 & 0
5 & 1 \end{pmatrix} \]
For these matrices to be equal, their corresponding elements must be equal. This gives us a system of equations:

1. \(x^2 = 16\)

2. \(0 = 0\) (This is consistent)

3. \(x+1 = 5\)

4. \(1 = 1\) (This is consistent)

From equation (1), \(x^2 = 16\), we get two possible solutions: \(x = 4\) or \(x = -4\).

From equation (3), \(x+1 = 5\), we get \(x = 4\).

We need the value of \(x\) that satisfies both equations simultaneously. The common solution is \(x = 4\).


Step 4: Final Answer:

The value of \(x\) is 4. This corresponds to option (C).
Quick Tip: When solving matrix equations, equating the elements often gives multiple equations for the same variable. Always check that your solution is consistent across all equations derived from the matrix equality.

Step 1: Understanding the Concept:

This problem involves evaluating a determinant using its properties. A key property is the linearity of the determinant with respect to its columns (or rows).


Step 2: Key Formula or Approach:

The property of determinants states that if a column (or row) of a determinant is expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants. \[ \begin{vmatrix} a & b & c+d
e & f & g+h
i & j & k+l \end{vmatrix} = \begin{vmatrix} a & b & c
e & f & g
i & j & k \end{vmatrix} + \begin{vmatrix} a & b & d
e & f & h
i & j & l \end{vmatrix} \]
Another key concept is the Vandermonde determinant.


Step 3: Detailed Explanation:

Let the given determinant be \(\Delta\). The third column consists of a difference of terms. We can use the linearity property to split the determinant into two: \[ \Delta = \begin{vmatrix} e^\alpha & e^{2\alpha} & e^{3\alpha}
e^\beta & e^{2\beta} & e^{3\beta}
e^\gamma & e^{2\gamma} & e^{3\gamma} \end{vmatrix} - \begin{vmatrix} e^\alpha & e^{2\alpha} & 1
e^\beta & e^{2\beta} & 1
e^\gamma & e^{2\gamma} & 1 \end{vmatrix} \]
Let's call the first determinant \(\Delta_1\) and the second \(\Delta_2\).

Evaluating \(\Delta_1\):
\[ \Delta_1 = \begin{vmatrix} e^\alpha & (e^\alpha)^2 & (e^\alpha)^3
e^\beta & (e^\beta)^2 & (e^\beta)^3
e^\gamma & (e^\gamma)^2 & (e^\gamma)^3 \end{vmatrix} \]
We can factor out \(e^\alpha\) from the first row, \(e^\beta\) from the second row, and \(e^\gamma\) from the third row.
\[ \Delta_1 = e^\alpha e^\beta e^\gamma \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} \]
Using the property of exponents, \(e^\alpha e^\beta e^\gamma = e^{\alpha+\beta+\gamma}\). We are given that \(\alpha+\beta+\gamma=0\).
\[ e^{\alpha+\beta+\gamma} = e^0 = 1 \]
So, \(\Delta_1\) simplifies to: \[ \Delta_1 = \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} \]
This is a Vandermonde determinant.

Evaluating \(\Delta_2\):
\[ \Delta_2 = \begin{vmatrix} e^\alpha & (e^\alpha)^2 & 1
e^\beta & (e^\beta)^2 & 1
e^\gamma & (e^\gamma)^2 & 1 \end{vmatrix} \]
We can rearrange the columns to match the form of \(\Delta_1\). Swapping two columns negates the determinant's value.

Swap Column 2 and Column 3: \[ \Delta_2 = - \begin{vmatrix} e^\alpha & 1 & (e^\alpha)^2
e^\beta & 1 & (e^\beta)^2
e^\gamma & 1 & (e^\gamma)^2 \end{vmatrix} \]
Swap Column 1 and Column 2: \[ \Delta_2 = -(-1) \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} = \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} \]
Final Calculation:

The original determinant is \(\Delta = \Delta_1 - \Delta_2\).
\[ \Delta = \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} - \begin{vmatrix} 1 & e^\alpha & (e^\alpha)^2
1 & e^\beta & (e^\beta)^2
1 & e^\gamma & (e^\gamma)^2 \end{vmatrix} = 0 \]

Step 4: Final Answer:

The value of the determinant is 0. This corresponds to option (E).
Quick Tip: When a column or row in a determinant is a sum or difference, immediately think of splitting the determinant. This often reveals underlying structures, like identical determinants or determinants with proportional columns/rows which simplify to zero.

Step 1: Understanding the Concept:

Three points are collinear if they lie on the same straight line. This geometric condition can be translated into an algebraic one. Two common methods are using the area of a triangle (which must be zero for collinear points) or by equating the slopes between pairs of points.


Step 2: Key Formula or Approach:

Method 1: Slope Formula

If points \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) are collinear, then the slope of line segment AB must be equal to the slope of line segment AC (or BC). \[ Slope = \frac{y_2 - y_1}{x_2 - x_1} \]
Method 2: Area of Triangle Formula

The area of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by: \[ Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]
For collinear points, the area is 0.


Step 3: Detailed Explanation (Using Slopes):

Let the points be \(A(2, -3)\), \(B(x, 1)\), and \(C(0, 5)\).

We will calculate the slope of AB and the slope of AC and set them equal.

Slope of AB: \[ m_{AB} = \frac{1 - (-3)}{x - 2} = \frac{1 + 3}{x - 2} = \frac{4}{x - 2} \]
Slope of AC: \[ m_{AC} = \frac{5 - (-3)}{0 - 2} = \frac{5 + 3}{-2} = \frac{8}{-2} = -4 \]
Since the points are collinear, \(m_{AB} = m_{AC}\).
\[ \frac{4}{x - 2} = -4 \]
Multiply both sides by \((x-2)\):
\[ 4 = -4(x - 2) \]
Divide both sides by 4:
\[ 1 = -(x - 2) \] \[ 1 = -x + 2 \]
Rearrange to solve for \(x\):
\[ x = 2 - 1 = 1 \]

Step 4: Final Answer:

The value of \(x\) is 1. This corresponds to option (D).
Quick Tip: Using the slope method is often quicker and involves less computation than the area of a triangle method, especially when one of the coordinates is zero.

Step 1: Understanding the Concept:

This problem requires solving a rational inequality. It is crucial not to multiply both sides by the denominator directly, as the sign of the denominator is unknown. The correct procedure is to move all terms to one side and find a common denominator.


Step 2: Key Formula or Approach:

1. Move all terms to one side to compare the expression with 0.
\[ \frac{x-2}{x-5} - 4 > 0 \]
2. Combine the terms into a single fraction.
3. Find the critical points where the numerator and denominator are zero.
4. Use a sign chart (or test points in intervals) to determine where the inequality holds true.


Step 3: Detailed Explanation:

We start with the inequality: \[ \frac{x-2}{x-5} > 4 \]
Subtract 4 from both sides: \[ \frac{x-2}{x-5} - 4 > 0 \]
Find a common denominator, which is \((x-5)\): \[ \frac{x-2 - 4(x-5)}{x-5} > 0 \]
Distribute the -4 in the numerator: \[ \frac{x-2 - 4x + 20}{x-5} > 0 \]
Combine like terms in the numerator: \[ \frac{-3x + 18}{x-5} > 0 \]
To make the leading coefficient of \(x\) positive, we can multiply the numerator by -1 and flip the inequality sign, or factor out -3. Let's factor out -3: \[ \frac{-3(x - 6)}{x-5} > 0 \]
Divide both sides by -3 and reverse the inequality sign: \[ \frac{x - 6}{x-5} < 0 \]
The critical points are the values of \(x\) that make the numerator or denominator zero.
Numerator: \(x - 6 = 0 \implies x = 6\).

Denominator: \(x - 5 = 0 \implies x = 5\).

These points divide the number line into three intervals: \((-\infty, 5)\), \((5, 6)\), and \((6, \infty)\).

We need to find the interval where the expression \(\frac{x-6}{x-5}\) is negative.


Interval 1: \(x < 5\) (e.g., let \(x=0\)). \(\frac{0-6}{0-5} = \frac{-6}{-5} = \frac{6}{5} > 0\). (Not a solution)
Interval 2: \(5 < x < 6\) (e.g., let \(x=5.5\)). \(\frac{5.5-6}{5.5-5} = \frac{-0.5}{0.5} = -1 < 0\). (This is a solution)
Interval 3: \(x > 6\) (e.g., let \(x=7\)). \(\frac{7-6}{7-5} = \frac{1}{2} > 0\). (Not a solution)

The solution set is the interval \((5, 6)\).


Step 4: Final Answer:

The interval in which \(x\) lies is \((5, 6)\). This corresponds to option (C).
Quick Tip: Never multiply across an inequality by an expression involving the variable unless you know its sign. The safest method for rational inequalities is always to bring everything to one side and simplify into a single fraction.

Step 1: Understanding the Concept:

This problem involves solving an inequality with an absolute value. The inequality \(|u| < a\) is equivalent to the compound inequality \(-a < u < a\).


Step 2: Key Formula or Approach:

We use the property \(|u| < a \iff -a < u < a\).

Let \(u = \frac{1}{x} - 2\) and \(a = 4\). The inequality becomes: \[ -4 < \frac{1}{x} - 2 < 4 \]
We need to solve this compound inequality for \(x\).


Step 3: Detailed Explanation:

Start with the compound inequality: \[ -4 < \frac{1}{x} - 2 < 4 \]
Add 2 to all parts of the inequality to isolate the term with \(x\): \[ -4 + 2 < \frac{1}{x} < 4 + 2 \] \[ -2 < \frac{1}{x} < 6 \]
This represents two simultaneous inequalities:
1. \(\frac{1}{x} > -2\)
2. \(\frac{1}{x} < 6\)

We solve each one separately, considering the cases \(x>0\) and \(x<0\).

Solving \(\frac{1}{x} > -2\):


Case (a): If \(x > 0\), we can multiply by \(x\) without changing the inequality sign: \(1 > -2x \implies -\frac{1}{2} < x\). The intersection of \(x>0\) and \(x > -1/2\) is \(x > 0\). So, the solution is \((0, \infty)\).
Case (b): If \(x < 0\), we multiply by \(x\) and reverse the inequality sign: \(1 < -2x \implies -\frac{1}{2} > x\), or \(x < -\frac{1}{2}\). The intersection of \(x<0\) and \(x < -1/2\) is \(x < -1/2\). So, the solution is \((-\infty, -1/2)\).

The solution for the first inequality is \((-\infty, -1/2) \cup (0, \infty)\).


Solving \(\frac{1}{x} < 6\):


Case (a): If \(x > 0\), we multiply by \(x\): \(1 < 6x \implies \frac{1}{6} < x\). The intersection of \(x>0\) and \(x > 1/6\) is \(x > 1/6\). So, the solution is \((1/6, \infty)\).
Case (b): If \(x < 0\), we multiply by \(x\) and reverse the inequality: \(1 > 6x \implies \frac{1}{6} > x\). The intersection of \(x<0\) and \(x < 1/6\) is \(x < 0\). So, the solution is \((-\infty, 0)\).

The solution for the second inequality is \((-\infty, 0) \cup (1/6, \infty)\).


Finding the final solution set:

We need the intersection of the solution sets from both inequalities: \[ ((-\infty, -1/2) \cup (0, \infty)) \cap ((-\infty, 0) \cup (1/6, \infty)) \]
Let's find the intersection term by term:

\((-\infty, -1/2)\) intersected with \((-\infty, 0)\) is \((-\infty, -1/2)\).
\((0, \infty)\) intersected with \((1/6, \infty)\) is \((1/6, \infty)\).

The union of these results gives the final solution set.


Step 4: Final Answer:

The solution set is \((-\infty, -1/2) \cup (1/6, \infty)\). This corresponds to option (A).
Quick Tip: When solving an inequality involving \(1/x\), it's essential to consider two cases: \(x>0\) and \(x<0\). This is because the direction of the inequality sign flips when you multiply by a negative number.

Step 1: Understanding the Concept:

This problem requires the use of the trigonometric half-angle identity for cosine. We also need to determine the correct sign of the result based on the quadrant in which the half-angle \(\frac{x}{2}\) lies.


Step 2: Key Formula or Approach:

The half-angle formula for cosine is: \[ \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \]
The sign (\(+\) or \(-\)) depends on the quadrant of the angle \(\frac{x}{2}\).


Step 3: Detailed Explanation:

We are given \(\cos x = \frac{4}{5}\).

We are also given that \(x\) is in the interval \([0, \frac{\pi}{2}]\). This means \(x\) is in the first quadrant.

To determine the sign for the half-angle formula, we need to find the interval for \(\frac{x}{2}\).

If \(0 \leq x \leq \frac{\pi}{2}\), then dividing by 2 gives \(0 \leq \frac{x}{2} \leq \frac{\pi}{4}\).

The angle \(\frac{x}{2}\) lies in the first quadrant (between 0 and 45 degrees). In the first quadrant, the cosine function is positive. Therefore, we will use the positive root in the formula.
\[ \cos\left(\frac{x}{2}\right) = + \sqrt{\frac{1 + \cos x}{2}} \]
Now, substitute the given value of \(\cos x\): \[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \frac{4}{5}}{2}} \]
Simplify the expression inside the square root: \[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{\frac{5}{5} + \frac{4}{5}}{2}} = \sqrt{\frac{\frac{9}{5}}{2}} \] \[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{9}{10}} \]
Take the square root of the numerator and denominator: \[ \cos\left(\frac{x}{2}\right) = \frac{\sqrt{9}}{\sqrt{10}} = \frac{3}{\sqrt{10}} \]

Step 4: Final Answer:

The value of \(\cos(\frac{x}{2})\) is \(\frac{3}{\sqrt{10}}\). This corresponds to option (C).
Quick Tip: Always pay attention to the given interval for the angle when using half-angle or double-angle formulas. The interval determines the quadrant, which in turn determines the sign (\(+\) or \(-\)) of the trigonometric function.

Step 1: Understanding the Concept:

This problem involves evaluating a product of trigonometric functions. We can use the product-to-sum formulas to simplify the expression, or we can find the individual values of the sine functions and multiply them.


Step 2: Key Formula or Approach:

Method 1: Product-to-Sum Formula

The relevant formula is: \(2\sin A \sin B = \cos(A-B) - \cos(A+B)\).

Therefore, \(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\).

Method 2: Special Angle Values

Convert angles from radians to degrees and use sum/difference formulas for sine.
\(\frac{\pi}{12}\) radians = \(\frac{180^\circ}{12} = 15^\circ\).
\(\frac{5\pi}{12}\) radians = \(5 \times 15^\circ = 75^\circ\).


Step 3: Detailed Explanation (Using Method 1):

Let \(A = \frac{5\pi}{12}\) and \(B = \frac{\pi}{12}\).

We use the formula \(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\).

First, calculate \(A-B\) and \(A+B\):
\[ A - B = \frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \] \[ A + B = \frac{5\pi}{12} + \frac{\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]
Now substitute these back into the formula:
\[ \sin\frac{5\pi}{12}\sin\frac{\pi}{12} = \frac{1}{2}\left[\cos\left(\frac{\pi}{3}\right) - \cos\left(\frac{\pi}{2}\right)\right] \]
We know the values of these standard angles:
\(\cos(\frac{\pi}{3}) = \cos(60^\circ) = \frac{1}{2}\)
\(\cos(\frac{\pi}{2}) = \cos(90^\circ) = 0\)

Substitute these values:
\[ \frac{1}{2}\left[\frac{1}{2} - 0\right] = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

Detailed Explanation (Using Method 2):

We need to calculate \(\sin(75^\circ) \sin(15^\circ)\).
\(\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\)
\[ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6} + \sqrt{2}}{4} \] \(\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ\)
\[ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6} - \sqrt{2}}{4} \]
Now, multiply the two results:
\[ \sin(75^\circ) \sin(15^\circ) = \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) \]
This is of the form \((a+b)(a-b) = a^2 - b^2\).
\[ = \frac{(\sqrt{6})^2 - (\sqrt{2})^2}{4 \times 4} = \frac{6 - 2}{16} = \frac{4}{16} = \frac{1}{4} \]

Step 4: Final Answer:

The value of the expression is \(\frac{1}{4}\). This corresponds to option (B).
Quick Tip: The product-to-sum and sum-to-product formulas are very efficient for simplifying expressions involving products or sums of sines and cosines, often leading to simpler calculations than finding individual values.

Step 1: Understanding the Concept:

This problem involves simplifying a trigonometric expression using fundamental identities, particularly the Pythagorean identity and double-angle formulas.


Step 2: Key Formula or Approach:

We will use the following identities:

Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)
Algebraic identity: \(a^2+b^2 = (a+b)^2 - 2ab\)
Sine double-angle formula: \(\sin(2\theta) = 2\sin\theta\cos\theta\)
Tangent identity: \(\tan x = \frac{\sin x}{\cos x}\)


Step 3: Detailed Explanation:

Let's first simplify the numerator: \(1 - \sin^4\theta - \cos^4\theta\).

We can rewrite it as \(1 - (\sin^4\theta + \cos^4\theta)\).

Now, let's simplify the term in the parenthesis. Let \(a = \sin^2\theta\) and \(b = \cos^2\theta\). Then \(\sin^4\theta + \cos^4\theta = a^2 + b^2\).
Using the identity \(a^2+b^2 = (a+b)^2 - 2ab\): \[ \sin^4\theta + \cos^4\theta = (\sin^2\theta + \cos^2\theta)^2 - 2\sin^2\theta\cos^2\theta \]
Since \(\sin^2\theta + \cos^2\theta = 1\), this simplifies to: \[ 1 - 2\sin^2\theta\cos^2\theta \]
Now substitute this back into the numerator expression: \[ Numerator = 1 - (1 - 2\sin^2\theta\cos^2\theta) = 1 - 1 + 2\sin^2\theta\cos^2\theta = 2\sin^2\theta\cos^2\theta \]
We can relate this to the sine double-angle formula, \(\sin(2\theta) = 2\sin\theta\cos\theta\).
\[ 2\sin^2\theta\cos^2\theta = 2(\sin\theta\cos\theta)^2 = 2\left(\frac{\sin(2\theta)}{2}\right)^2 = 2\left(\frac{\sin^2(2\theta)}{4}\right) = \frac{1}{2}\sin^2(2\theta) \]
Now, we can write the full expression with the simplified numerator and the given denominator: \[ \frac{Numerator}{Denominator} = \frac{\frac{1}{2}\sin^2(2\theta)}{\cos^2(2\theta)} \] \[ = \frac{1}{2} \left(\frac{\sin(2\theta)}{\cos(2\theta)}\right)^2 = \frac{1}{2} \tan^2(2\theta) \]

Step 4: Final Answer:

The value of the expression is \(\frac{1}{2}\tan^2(2\theta)\). This corresponds to option (B).
Quick Tip: The expression \(\sin^4\theta + \cos^4\theta\) is a common one in trigonometry problems. It's useful to remember its simplified form: \(1 - \frac{1}{2}\sin^2(2\theta)\). This can save time in calculations.

Step 1: Understanding the Concept:

This problem involves algebraic manipulation combined with trigonometric sum-to-product identities. The expression \(\frac{\alpha+1}{\alpha-1}\) suggests the use of the Componendo and Dividendo rule, or direct substitution.


Step 2: Key Formula or Approach:

We will use direct substitution and then apply the sum-to-product formulas:

\(\cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)\)
\(\cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)\)

The Componendo and Dividendo rule states that if \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+b}{a-b} = \frac{c+d}{c-d}\).


Step 3: Detailed Explanation:

We are given \(\alpha = \frac{\cos A}{\cos B}\). We need to find \(\frac{\alpha+1}{\alpha-1}\).

Let's substitute the expression for \(\alpha\): \[ \frac{\alpha+1}{\alpha-1} = \frac{\frac{\cos A}{\cos B} + 1}{\frac{\cos A}{\cos B} - 1} \]
To simplify this complex fraction, multiply the numerator and denominator by \(\cos B\): \[ = \frac{\cos A + \cos B}{\cos A - \cos B} \]
Now we apply the sum-to-product formulas to the numerator and the denominator.

For the numerator (\(\cos A + \cos B\)): \[ \cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \]
For the denominator (\(\cos A - \cos B\)): \[ \cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \]
Now form the ratio: \[ \frac{2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}{-2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)} \]
Cancel the 2s and group the terms: \[ = - \left(\frac{\cos\left(\frac{A+B}{2}\right)}{\sin\left(\frac{A+B}{2}\right)}\right) \left(\frac{\cos\left(\frac{A-B}{2}\right)}{\sin\left(\frac{A-B}{2}\right)}\right) \]
Using the definition of the cotangent function, \(\cot x = \frac{\cos x}{\sin x}\): \[ = -\cot\left(\frac{A+B}{2}\right)\cot\left(\frac{A-B}{2}\right) \]

Step 4: Final Answer:

The expression is equal to \(-\cot\left(\frac{A+B}{2}\right)\cot\left(\frac{A-B}{2}\right)\). This corresponds to option (D).
Quick Tip: Recognizing the structure \(\frac{a+b}{a-b}\) is a strong hint to use Componendo and Dividendo. If we write the given relation as \(\frac{\alpha}{1} = \frac{\cos A}{\cos B}\), applying the rule gives \(\frac{\alpha+1}{\alpha-1} = \frac{\cos A + \cos B}{\cos A - \cos B}\) directly, which saves a step of algebraic manipulation.

Step 1: Understanding the Concept:

This problem requires solving an equation involving inverse trigonometric functions. The key is to use the addition formula for the inverse tangent function.


Step 2: Key Formula or Approach:

The sum of two inverse tangent functions is given by the formula: \[ \tan^{-1} A + \tan^{-1} B = \tan^{-1}\left(\frac{A+B}{1-AB}\right), \quad if AB < 1 \]
We will apply this formula with \(A=2x\) and \(B=3x\), and then solve the resulting equation.


Step 3: Detailed Explanation:

The given equation is \(\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\).

Applying the sum formula to the left side: \[ \tan^{-1}\left(\frac{2x+3x}{1-(2x)(3x)}\right) = \frac{\pi}{4} \] \[ \tan^{-1}\left(\frac{5x}{1-6x^2}\right) = \frac{\pi}{4} \]
For this formula to be valid, the product \((2x)(3x) = 6x^2\) must be less than 1. We will check this condition later.

Now, take the tangent of both sides of the equation: \[ \frac{5x}{1-6x^2} = \tan\left(\frac{\pi}{4}\right) \]
We know that \(\tan(\frac{\pi}{4}) = 1\).
\[ \frac{5x}{1-6x^2} = 1 \]
Now, solve this algebraic equation for \(x\). \[ 5x = 1 - 6x^2 \]
Rearrange the terms to form a quadratic equation: \[ 6x^2 + 5x - 1 = 0 \]
We can solve this by factoring: \[ 6x^2 + 6x - x - 1 = 0 \] \[ 6x(x+1) - 1(x+1) = 0 \] \[ (6x-1)(x+1) = 0 \]
This gives two possible solutions: \(x = \frac{1}{6}\) or \(x = -1\).


Step 3.1: Check the validity of the solutions.

We must check the condition \(AB < 1\), which is \(6x^2 < 1\).

For \(x = \frac{1}{6}\): \(6\left(\frac{1}{6}\right)^2 = 6\left(\frac{1}{36}\right) = \frac{1}{6}\). Since \(\frac{1}{6} < 1\), this solution is valid.
For \(x = -1\): \(6(-1)^2 = 6(1) = 6\). Since \(6 > 1\), this solution is extraneous and arose because we used a formula outside its domain of validity. If \(AB > 1\), the formula has an additional \(\pi\) term.

Let's verify \(x=-1\) in the original equation: \(\tan^{-1}(-2) + \tan^{-1}(-3) = -\tan^{-1}(2) - \tan^{-1}(3)\). Since \(\tan^{-1}(2)\) and \(\tan^{-1}(3)\) are both positive, their sum is positive, and the left side is negative. It cannot be equal to the positive value \(\frac{\pi}{4}\).


Step 4: Final Answer:

The only valid solution is \(x = \frac{1}{6}\). This corresponds to option (A).
Quick Tip: Always remember to check your solutions for inverse trigonometric equations against the conditions required for the formulas you used. Extraneous solutions are common.

Step 1: Understanding the Concept:

The domain of a function is the set of all possible input values (\(x\)-values) for which the function is defined. For the inverse cosine function, \(\cos^{-1}(u)\), the argument \(u\) must be in the range \([-1, 1]\). In this problem, the argument is the greatest integer function, \([x]\).


Step 2: Key Formula or Approach:

The domain of \(f(u) = \cos^{-1}(u)\) is \(-1 \leq u \leq 1\).

For the given function \(f(x) = \cos^{-1}([x])\), we must have: \[ -1 \leq [x] \leq 1 \]
We need to find all values of \(x\) that satisfy this condition.


Step 3: Detailed Explanation:

The greatest integer function, \([x]\), outputs an integer. The condition \(-1 \leq [x] \leq 1\) means that the possible integer values for \([x]\) are -1, 0, and 1.

We need to find the values of \(x\) that result in these integer outputs.

Case 1: \([x] = -1\)

By the definition of the greatest integer function, if \([x] = n\), then \(n \leq x < n+1\).

For \([x] = -1\), this means \(-1 \leq x < -1+1\), which simplifies to \(-1 \leq x < 0\).

The solution for this case is the interval \([-1, 0)\).


Case 2: \([x] = 0\)

For \([x] = 0\), this means \(0 \leq x < 0+1\), which simplifies to \(0 \leq x < 1\).

The solution for this case is the interval \([0, 1)\).


Case 3: \([x] = 1\)

For \([x] = 1\), this means \(1 \leq x < 1+1\), which simplifies to \(1 \leq x < 2\).

The solution for this case is the interval \([1, 2)\).


Finding the total domain:

The domain of the function is the union of all possible values of \(x\) from the three cases. \[ Domain = [-1, 0) \cup [0, 1) \cup [1, 2) \]
Combining these adjacent intervals gives: \[ Domain = [-1, 2) \]

Step 4: Final Answer:

The domain of the function is the interval \([-1, 2)\). This corresponds to option (B).
Quick Tip: When finding the domain of a composite function like \(f(g(x))\), first identify the domain requirement of the outer function \(f\), and then solve for \(x\) using the inner function \(g(x)\). Here, the outer function is \(\cos^{-1}\) and the inner is \([x]\).

Step 1: Understanding the Concept:

This problem involves the properties of the inverse sine function. The key is to understand the condition under which \(\sin^{-1}(u)\) equals \(\frac{\pi}{2}\).


Step 2: Key Formula or Approach:

The inverse sine function, \(\sin^{-1}(u)\), has a range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

The equation \(\sin^{-1}(u) = \frac{\pi}{2}\) holds if and only if the argument \(u\) is equal to 1.

So, we need to set the argument of the \(\sin^{-1}\) function equal to 1 and solve.


Step 3: Detailed Explanation:

We are given the equation: \[ \sin^{-1}\left(\frac{3\sin(2a)}{5+4\cos(2a)}\right) = \frac{\pi}{2} \]
Based on the property of the inverse sine function, this implies that the expression inside the parentheses must be equal to 1. \[ \frac{3\sin(2a)}{5+4\cos(2a)} = 1 \]
Now, we can solve this equation by multiplying both sides by the denominator: \[ 3\sin(2a) = 5+4\cos(2a) \]
The question asks for the value of the expression \(3\sin(2a) - 4\cos(2a)\).

We can rearrange the equation we just derived to match the required expression. Subtract \(4\cos(2a)\) from both sides: \[ 3\sin(2a) - 4\cos(2a) = 5 \]

Step 4: Final Answer:

The value of the expression \(3\sin(2a) - 4\cos(2a)\) is 5. This corresponds to option (E).
Quick Tip: When an inverse trigonometric function is equated to one of its boundary range values (like \(\pm\frac{\pi}{2}\) for \(\sin^{-1}\) or \(0, \pi\) for \(\cos^{-1}\)), the problem usually simplifies significantly by setting the argument of the function to the corresponding value (\(\pm 1\) or \(\pm 1\)).

Step 1: Understanding the Concept:

The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using a standard formula involving the tangent function. We are given the angle and one slope, and we need to find the possible values for the other slope.


Step 2: Key Formula or Approach:

The formula for the angle \(\theta\) between two lines is: \[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
We are given \(\theta = \frac{\pi}{4}\) and let's say \(m_1 = \frac{1}{2}\). We need to solve for \(m_2\).


Step 3: Detailed Explanation:

First, find the value of \(\tan(\theta)\).
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Now substitute the given values into the formula: \[ 1 = \left| \frac{m_2 - \frac{1}{2}}{1 + \frac{1}{2} m_2} \right| \]
The absolute value equation implies two possible cases:

Case 1: The expression inside the absolute value is 1. \[ \frac{m_2 - \frac{1}{2}}{1 + \frac{1}{2} m_2} = 1 \] \[ m_2 - \frac{1}{2} = 1 + \frac{1}{2} m_2 \] \[ m_2 - \frac{1}{2} m_2 = 1 + \frac{1}{2} \] \[ \frac{1}{2} m_2 = \frac{3}{2} \implies m_2 = 3 \]
Case 2: The expression inside the absolute value is -1. \[ \frac{m_2 - \frac{1}{2}}{1 + \frac{1}{2} m_2} = -1 \] \[ m_2 - \frac{1}{2} = -1\left(1 + \frac{1}{2} m_2\right) \] \[ m_2 - \frac{1}{2} = -1 - \frac{1}{2} m_2 \] \[ m_2 + \frac{1}{2} m_2 = -1 + \frac{1}{2} \] \[ \frac{3}{2} m_2 = -\frac{1}{2} \implies m_2 = -\frac{1}{3} \]
So, the two possible slopes for the other line are 3 and \(-\frac{1}{3}\).


Step 4: Final Answer:

The slope of the other line is 3 or \(-\frac{1}{3}\). This corresponds to option (A).
Quick Tip: When solving an equation involving an absolute value, such as \(|x|=a\), always remember to consider both the positive and negative cases: \(x=a\) and \(x=-a\). This is why there are typically two possible slopes for the second line.

Step 1: Understanding the Concept:

To find the y-intercept of a line, we first need to determine the equation of the line. We can find the equation using the two given points. The y-intercept is the y-coordinate of the point where the line crosses the y-axis (i.e., where \(x=0\)).


Step 2: Key Formula or Approach:

1. Find the slope (\(m\)) of the line using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

2. Use the point-slope form of a linear equation: \(y - y_1 = m(x - x_1)\).

3. To find the y-intercept, set \(x=0\) in the equation and solve for \(y\).


Step 3: Detailed Explanation:

Let the two points be \((x_1, y_1) = (-\frac{1}{2}, 1)\) and \((x_2, y_2) = (1, 2)\).

1. Calculate the slope (m):
\[ m = \frac{2 - 1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]
2. Find the equation of the line:

Using the point-slope form with the point (1,2): \[ y - 2 = \frac{2}{3}(x - 1) \]
3. Find the y-intercept:

Set \(x=0\) in the equation: \[ y - 2 = \frac{2}{3}(0 - 1) \] \[ y - 2 = -\frac{2}{3} \] \[ y = 2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3} \]
The y-intercept is \(\frac{4}{3}\).


Step 4: Final Answer:

The y-intercept of the straight line is \(\frac{4}{3}\). This corresponds to option (E).
Quick Tip: Once you have the equation in point-slope form, \(y - y_1 = m(x - x_1)\), you can rearrange it to slope-intercept form, \(y = mx + c\). The constant term \(c\) is the y-intercept. In this case, \(y = \frac{2}{3}x - \frac{2}{3} + 2 \implies y = \frac{2}{3}x + \frac{4}{3}\). The y-intercept is clearly \(\frac{4}{3}\).

Step 1: Understanding the Concept:

The altitude (or height) of an equilateral triangle is the perpendicular distance from a vertex to the opposite side (the base). We can calculate this distance using the formula for the perpendicular distance from a point to a line. Once we have the altitude, we can find the length of the side of the equilateral triangle using the relationship between them.


Step 2: Key Formula or Approach:

1. The perpendicular distance (\(d\)) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by: \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\). This distance is the altitude (\(h\)) of the triangle.

2. In an equilateral triangle with side length \(s\), the altitude \(h\) is related to the side by the formula: \(h = \frac{\sqrt{3}}{2}s\).


Step 3: Detailed Explanation:

1. Calculate the altitude (h):

The given vertex is \((x_1, y_1) = (-1, 2)\).

The equation of the base is \(2x - y = 1\), which can be written as \(2x - y - 1 = 0\).

Here, \(A=2\), \(B=-1\), and \(C=-1\).

Using the distance formula: \[ h = \frac{|2(-1) + (-1)(2) - 1|}{\sqrt{2^2 + (-1)^2}} = \frac{|-2 - 2 - 1|}{\sqrt{4 + 1}} = \frac{|-5|}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5} \]
So, the altitude of the triangle is \(\sqrt{5}\) units.


2. Calculate the side length (s):

We use the relationship \(h = \frac{\sqrt{3}}{2}s\). We need to solve for \(s\).
\[ s = \frac{2h}{\sqrt{3}} \]
Substitute the value of \(h = \sqrt{5}\): \[ s = \frac{2\sqrt{5}}{\sqrt{3}} \]
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\): \[ s = \frac{2\sqrt{5}\sqrt{3}}{3} = \frac{2\sqrt{15}}{3} \]
Let's check the options. Option (B) is \(2\sqrt{\frac{5}{3}}\). Let's see if this is equivalent. \[ 2\sqrt{\frac{5}{3}} = 2 \frac{\sqrt{5}}{\sqrt{3}} \]
This is exactly what we found. So, the side length is \(2\sqrt{\frac{5}{3}}\).


Step 4: Final Answer:

The length of the side of the triangle is \(2\sqrt{\frac{5}{3}}\) units. This corresponds to option (B).
Quick Tip: For an equilateral triangle, remembering the relationship between side length \(s\), altitude \(h = \frac{\sqrt{3}}{2}s\), and area \(A = \frac{\sqrt{3}}{4}s^2\) is essential for solving problems quickly.

Step 1: Understanding the Concept:

The radius of a circle is the distance from its center to any point on its circumference. Since the center lies on the y-axis, its x-coordinate is 0. We can use the distance formula and the fact that the distances from the center to the two given points are equal (both are radii) to find the coordinates of the center.


Step 2: Key Formula or Approach:

1. Let the center of the circle be \(C(0, k)\) since it lies on the y-axis.

2. Let the given points be \(A(4, 0)\) and \(B(0, 2)\).

3. The distance from the center to any point on the circle is the radius \(r\). So, \(CA = CB = r\).

4. Use the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). The equation \(CA^2 = CB^2\) can be used to solve for \(k\).


Step 3: Detailed Explanation:

1. Set up the distance equality:

The square of the distance from the center \(C(0, k)\) to point \(A(4, 0)\) is: \[ CA^2 = (4-0)^2 + (0-k)^2 = 4^2 + (-k)^2 = 16 + k^2 \]
The square of the distance from the center \(C(0, k)\) to point \(B(0, 2)\) is: \[ CB^2 = (0-0)^2 + (2-k)^2 = 0^2 + (2-k)^2 = (2-k)^2 \]
Since both distances are equal to the radius squared (\(r^2\)), we have \(CA^2 = CB^2\): \[ 16 + k^2 = (2-k)^2 \]
2. Solve for k:

Expand the right side: \[ 16 + k^2 = 4 - 4k + k^2 \]
Subtract \(k^2\) from both sides: \[ 16 = 4 - 4k \]
Solve for \(k\): \[ 4k = 4 - 16 = -12 \] \[ k = -3 \]
So, the center of the circle is \((0, -3)\).

3. Calculate the radius (r):

We can use the distance \(CA\) (or \(CB\)) to find the radius. Using the expression for \(CA^2\): \[ r^2 = CA^2 = 16 + k^2 = 16 + (-3)^2 = 16 + 9 = 25 \] \[ r = \sqrt{25} = 5 \]

Step 4: Final Answer:

The radius of the circle is 5. This corresponds to option (A).
Quick Tip: A point on the y-axis is the perpendicular bisector of the chord connecting \((x_0, y_0)\) and \((-x_0, y_0)\). Here, the center is on the y-axis, which is the perpendicular bisector of the line segment connecting \((4,0)\) and \((-4,0)\). While this property is not directly used here, it's a useful concept. A more direct property is that the perpendicular bisector of any chord passes through the center.

Step 1: Understanding the Concept:

This problem relates the geometric properties of an ellipse (lengths of major and minor axes) to its eccentricity (\(e\)). The eccentricity measures how much the ellipse deviates from being a circle.


Step 2: Key Formula or Approach:

1. The length of the major axis of an ellipse is \(2a\).

2. The length of the minor axis is \(2b\).

3. The relationship between \(a\), \(b\), and the eccentricity \(e\) is given by the formula: \(b^2 = a^2(1 - e^2)\), which can be rearranged to \(e = \sqrt{1 - \frac{b^2}{a^2}}\).


Step 3: Detailed Explanation:

We are given that the length of the major axis is twice the length of the minor axis. \[ 2a = 2 \times (2b) \] \[ 2a = 4b \] \[ a = 2b \]
This gives us a relationship between \(a\) and \(b\). We can substitute this into the eccentricity formula. \[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substitute \(a = 2b\): \[ e = \sqrt{1 - \frac{b^2}{(2b)^2}} = \sqrt{1 - \frac{b^2}{4b^2}} \]
Cancel out the \(b^2\) term: \[ e = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} \] \[ e = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2} \]

Step 4: Final Answer:

The eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\). This corresponds to option (B).
Quick Tip: The ratio \(b/a\) determines the "roundness" of an ellipse. The eccentricity \(e\) is directly related to this ratio. When you're given a relationship between the axes, your goal is to find the value of \(b/a\) (or \(b^2/a^2\)) to plug into the eccentricity formula.

Step 1: Understanding the Concept:

This question asks for the lengths of the transverse and conjugate axes of a hyperbola given in its standard form. We need to identify the parameters \(a\) and \(b\) from the equation.


Step 2: Key Formula or Approach:

The standard form of a hyperbola with a horizontal transverse axis is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
For this hyperbola:

The length of the transverse axis is \(2a\).
The length of the conjugate axis is \(2b\).


Step 3: Detailed Explanation:

The given equation of the hyperbola is: \[ \frac{x^2}{9} - \frac{y^2}{25} = 1 \]
By comparing this to the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify \(a^2\) and \(b^2\).
\[ a^2 = 9 \implies a = \sqrt{9} = 3 \] \[ b^2 = 25 \implies b = \sqrt{25} = 5 \]
Now, we calculate the lengths of the axes.

Length of transverse axis = \(2a = 2 \times 3 = 6\).

Length of conjugate axis = \(2b = 2 \times 5 = 10\).


Step 4: Final Answer:

The lengths of the transverse and conjugate axes are 6 and 10, respectively. This corresponds to option (C).
Quick Tip: Be careful not to confuse \(a\) and \(a^2\). The values in the denominator of the standard equation are \(a^2\) and \(b^2\). Also, remember that for a hyperbola, \(a\) is always associated with the positive term, regardless of whether it's larger or smaller than \(b\).

Step 1: Understanding the Concept:

This question asks for the directrix of a parabola given in its vertex form. We need to identify the vertex and the focal length parameter \(p\) from the equation to determine the directrix.


Step 2: Key Formula or Approach:

The standard equation of a parabola that opens vertically is \((x - h)^2 = 4p(y - k)\), where:

\((h, k)\) is the vertex.
\(p\) is the distance from the vertex to the focus (and from the vertex to the directrix).
If \(p > 0\), the parabola opens upwards.
If \(p < 0\), the parabola opens downwards.

For a parabola opening upwards, the directrix is a horizontal line given by the equation \(y = k - p\).


Step 3: Detailed Explanation:

The given equation is \((x - 1)^2 = 2(y - 2)\).

Comparing this with the standard form \((x - h)^2 = 4p(y - k)\), we can identify the parameters:


\(h = 1\)
\(k = 2\)
\(4p = 2\)

From \(4p = 2\), we find \(p = \frac{2}{4} = \frac{1}{2}\).

The vertex is at \((h, k) = (1, 2)\).

Since \(p = \frac{1}{2}\) is positive, the parabola opens upwards.

The equation of the directrix is \(y = k - p\).

Substitute the values of \(k\) and \(p\): \[ y = 2 - \frac{1}{2} = \frac{3}{2} \]
To match the options, we can rearrange this equation: \[ 2y = 3 \] \[ 2y - 3 = 0 \]

Step 4: Final Answer:

The equation of the directrix is \(2y - 3 = 0\). This corresponds to option (A).
Quick Tip: Remember the orientation of the parabola. If the \(x\) term is squared, the parabola is vertical (opens up or down). If the \(y\) term is squared, it's horizontal (opens left or right). The sign of \(p\) determines the direction.

Step 1: Understanding the Concept:

The magnitude of the cross product of two vectors representing the adjacent sides of a parallelogram gives the area of that parallelogram.


Step 2: Key Formula or Approach:

Let the adjacent sides be vectors \(\vec{a}\) and \(\vec{b}\).

The area of the parallelogram is given by \(|\vec{a} \times \vec{b}|\).

The cross product can be calculated using the determinant formula: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
a_x & a_y & a_z
b_x & b_y & b_z \end{vmatrix} \]

Step 3: Detailed Explanation:

Let \(\vec{a} = -\hat{i} + \frac{1}{2}\hat{j} + 2\hat{k}\) and \(\vec{b} = \hat{i} + \frac{1}{2}\hat{j} + 2\hat{k}\).

First, calculate the cross product \(\vec{a} \times \vec{b}\): \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
-1 & \frac{1}{2} & 2
1 & \frac{1}{2} & 2 \end{vmatrix} \]
Expand the determinant: \[ = \hat{i}\left(\left(\frac{1}{2}\right)(2) - (2)\left(\frac{1}{2}\right)\right) - \hat{j}((-1)(2) - (2)(1)) + \hat{k}\left((-1)\left(\frac{1}{2}\right) - \left(\frac{1}{2}\right)(1)\right) \] \[ = \hat{i}(1 - 1) - \hat{j}(-2 - 2) + \hat{k}\left(-\frac{1}{2} - \frac{1}{2}\right) \] \[ = \hat{i}(0) - \hat{j}(-4) + \hat{k}(-1) \] \[ = 0\hat{i} + 4\hat{j} - 1\hat{k} \]
Now, find the magnitude of this resulting vector: \[ |\vec{a} \times \vec{b}| = \sqrt{0^2 + 4^2 + (-1)^2} = \sqrt{0 + 16 + 1} = \sqrt{17} \]
There seems to be a calculation error or a typo in the question or options as the calculated answer \(\sqrt{17}\) does not match the provided options involving \(\sqrt{65}\). Let's re-check the OCR and calculation.
Let me check the question transcription again. `vectors-i + 1/2j + 2k and i + 1/2j + 2k` seems correct. My calculation also seems correct. Let me re-calculate the cross product components again carefully.
i-component: `(1/2)2 - 2(1/2) = 1-1 = 0`. Correct.
j-component: `- ((-1)2 - 21) = -(-2-2) = -(-4) = 4`. Correct.
k-component: `(-1)(1/2) - (1/2)1 = -1/2 - 1/2 = -1`. Correct.
The result is \(4\hat{j} - \hat{k}\), magnitude is \(\sqrt{17}\).
Given the correct answer is (C) or (D) \(\frac{\sqrt{65}}{2}\), there must be a typo in the question's vectors. Let's assume one of the vectors was, for instance, \(\vec{b} = \hat{i} + 2\hat{j} + \frac{1}{2}\hat{k}\). This is just speculation to match the answer.
Given the provided solution, it is not possible to arrive at it with the provided vectors. Let's assume the question had a typo. For example, if the vectors were \(\vec{a} = -\hat{i} + 2\hat{j} + \frac{1}{2}\hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + \frac{1}{2}\hat{k}\).
Then \(\vec{a} \times \vec{b} = \hat{i}(1-1) - \hat{j}(-1/2 - 1/2) + \hat{k}(-2-2) = \hat{j} - 4\hat{k}\), magnitude \(\sqrt{17}\). Still no.
This question is likely flawed as written. However, to provide a solution path that could have been intended, let's assume the correct answer is indeed \(\frac{\sqrt{65}}{2}\) and work backwards. We need \(|\vec{a} \times \vec{b}|^2 = \frac{65}{4}\). This means the components of the cross product vector \((c_x, c_y, c_z)\) must satisfy \(c_x^2+c_y^2+c_z^2 = 16.25\).
It is impossible to proceed logically to the given answer. We will state this.

Step 4: Final Answer:

Following the standard procedure with the given vectors, the cross product is \(4\hat{j} - \hat{k}\) and its magnitude (the area) is \(\sqrt{17}\). This does not match any of the options. The question as stated in the text is likely incorrect. Options (C) and (D) are identical. Based on the provided "Correct Answer", there is a high probability of a typo in the vector components given in the problem statement.
Quick Tip: When your calculated answer doesn't match any of the multiple-choice options, double-check your arithmetic first. If the calculation is correct, the problem statement itself may be flawed, which can happen in exam papers.

Step 1: Understanding the Concept:

The magnitude of the cross product of two vectors \(\vec{a}\) and \(\vec{b}\) is related to the magnitudes of the vectors and the sine of the angle between them. We are given the magnitudes and the fact that the cross product is a unit vector (meaning its magnitude is 1).


Step 2: Key Formula or Approach:

The magnitude of the cross product is given by: \[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) \]
where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\).


Step 3: Detailed Explanation:

We are given:

\(|\vec{a}| = 3\)
\(|\vec{b}| = \frac{\sqrt{2}}{3}\)
\(\vec{a} \times \vec{b}\) is a unit vector, which means \(|\vec{a} \times \vec{b}| = 1\).

Substitute these values into the formula: \[ 1 = (3) \left(\frac{\sqrt{2}}{3}\right) \sin(\theta) \]
Simplify the equation: \[ 1 = \sqrt{2} \sin(\theta) \]
Solve for \(\sin(\theta)\): \[ \sin(\theta) = \frac{1}{\sqrt{2}} \]
The principal value for the angle \(\theta\) (where \(0 \le \theta \le \pi\)) for which \(\sin(\theta) = \frac{1}{\sqrt{2}}\) is \(\theta = \frac{\pi}{4}\). Another possible value would be \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\). Option (B) is \(\frac{\pi}{4}\). The calculation leads to a valid answer among the options.

The question is marked as "Cancelled". This could be due to a typo in the given values that was later corrected, or because there are two possible answers (\(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\)) and only one was intended to be correct. However, \(\frac{\pi}{4}\) is a standard answer. Without further clarification, the reason for cancellation is uncertain, but the mathematical procedure leads to a clear result.


Step 4: Final Answer:

The calculation yields \(\theta = \frac{\pi}{4}\), which is option (B). However, as the question is marked "Cancelled", we will adhere to that.
Quick Tip: Remember the geometric definition of the cross product magnitude: \(|\vec{a} \times \vec{b}|\) is the area of the parallelogram formed by vectors \(\vec{a}\) and \(\vec{b}\). This is different from the dot product, which is \(|\vec{a}||\vec{b}|\cos(\theta)\).

Step 1: Understanding the Concept:

The projection of a vector \(\vec{a}\) onto another vector \(\vec{b}\) is the scalar length of the "shadow" that \(\vec{a}\) casts on \(\vec{b}\). It is found using the dot product.


Step 2: Key Formula or Approach:

The scalar projection of vector \(\vec{a}\) on vector \(\vec{b}\) is given by the formula: \[ proj_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
We need to calculate the dot product \(\vec{a} \cdot \vec{b}\) and the magnitude of \(\vec{b}\).


Step 3: Detailed Explanation:

Given vectors: \(\vec{a} = 3\hat{i} - \hat{j} - 2\hat{k}\)
\(\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}\)

1. Calculate the dot product \(\vec{a} \cdot \vec{b}\):
\[ \vec{a} \cdot \vec{b} = (3)(1) + (-1)(2) + (-2)(-3) = 3 - 2 + 6 = 7 \]
2. Calculate the magnitude of \(\vec{b}\):
\[ |\vec{b}| = \sqrt{(1)^2 + (2)^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]
3. Calculate the projection:
\[ proj_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{7}{\sqrt{14}} \]
To simplify, we can rationalize the denominator or recognize that \(14 = 2 \times 7\): \[ \frac{7}{\sqrt{14}} = \frac{7}{\sqrt{2 \times 7}} = \frac{\sqrt{7} \times \sqrt{7}}{\sqrt{2} \times \sqrt{7}} = \frac{\sqrt{7}}{\sqrt{2}} \]
Multiplying the numerator and denominator by \(\sqrt{2}\) to rationalize: \[ \frac{\sqrt{7}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{14}}{2} \]

Step 4: Final Answer:

The projection of \(\vec{a}\) on \(\vec{b}\) is \(\frac{\sqrt{14}}{2}\). This corresponds to option (A).
Quick Tip: Distinguish between scalar projection (a length, which is a scalar) and vector projection (a vector). The vector projection of \(\vec{a}\) on \(\vec{b}\) is \(\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}\). The scalar projection is the magnitude of the vector projection.

Step 1: Understanding the Concept:

This problem deals with the magnitude of a scalar multiple of a vector. The magnitude \(|\lambda\vec{a}|\) is related to the scalar \(|\lambda|\) and the magnitude \(|\vec{a}|\).


Step 2: Key Formula or Approach:

The key property is: \[ |\lambda\vec{a}| = |\lambda| |\vec{a}| \]
We are given the range of \(\lambda\) and the value of \(|\vec{a}|\). We need to find the resulting range of \(|\lambda\vec{a}|\).


Step 3: Detailed Explanation:

We are given:

\(|\vec{a}| = 4\)
\(-1 \le \lambda \le 3\)

First, we need to find the range of \(|\lambda|\).
Since \(\lambda\) is in the interval \([-1, 3]\), the absolute value \(|\lambda|\) will take values from \(0\) (when \(\lambda=0\)) up to a maximum of \(|3|=3\). For instance, if \(\lambda=-1\), \(|\lambda|=1\). If \(\lambda=2\), \(|\lambda|=2\). The minimum value of \(|\lambda|\) is 0 and the maximum is 3.
So, the range of \(|\lambda|\) is \(0 \le |\lambda| \le 3\).

Now, we find the range of \(|\lambda\vec{a}| = |\lambda| |\vec{a}| = 4|\lambda|\).

Since \(0 \le |\lambda| \le 3\), we multiply the inequality by 4: \[ 4 \times 0 \le 4|\lambda| \le 4 \times 3 \] \[ 0 \le 4|\lambda| \le 12 \]
Therefore, \(|\lambda\vec{a}|\) lies in the interval \([0, 12]\).

None of the given options match this result. The closest option is (E) [4,12], but it misses the values between 0 and 4. For this reason, the question is likely flawed and was cancelled. For example, if the condition was \(1 \le \lambda \le 3\), then \(1 \le |\lambda| \le 3\) and the range would be \([4, 12]\).


Step 4: Final Answer:

The correct interval is \([0, 12]\). Since this is not among the options, the question is invalid, justifying its "Cancelled" status.
Quick Tip: When finding the range of a function of a variable, like \(f(\lambda) = |\lambda|\), be careful with the interval. For an interval that includes zero and both positive and negative numbers, the minimum of the absolute value will be 0.

Step 1: Understanding the Concept:

The angle between two lines in vector form is the angle between their direction vectors. The direction vectors are the vectors that are multiplied by the scalar parameters (\(\lambda\) and \(\mu\)).


Step 2: Key Formula or Approach:

If two lines have direction vectors \(\vec{b_1}\) and \(\vec{b_2}\), the angle \(\theta\) between them is given by the dot product formula: \[ \cos(\theta) = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}| |\vec{b_2}|} \]

Step 3: Detailed Explanation:

From the equations of the lines, we identify the direction vectors.

For the first line, the direction vector is \(\vec{b_1} = \hat{i} + 2\hat{j} + 2\hat{k}\).

For the second line, the direction vector is \(\vec{b_2} = 3\hat{i} + 2\hat{j} + 6\hat{k}\).

1. Calculate the dot product \(\vec{b_1} \cdot \vec{b_2}\):
\[ \vec{b_1} \cdot \vec{b_2} = (1)(3) + (2)(2) + (2)(6) = 3 + 4 + 12 = 19 \]
2. Calculate the magnitudes \(|\vec{b_1}|\) and \(|\vec{b_2}|\):
\[ |\vec{b_1}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \] \[ |\vec{b_2}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
3. Calculate \(\cos(\theta)\):
\[ \cos(\theta) = \frac{19}{3 \times 7} = \frac{19}{21} \]
4. Find the angle \(\theta\):
\[ \theta = \cos^{-1}\left(\frac{19}{21}\right) \]

Step 4: Final Answer:

The angle between the lines is \(\cos^{-1}\left(\frac{19}{21}\right)\). This corresponds to option (C).
Quick Tip: The position vectors (\(3\hat{i} + 2\hat{j} - 4\hat{k}\) and \(5\hat{i} - 2\hat{k}\)) only tell you a point on each line. They are irrelevant for finding the angle between the lines. Focus only on the direction vectors.

Step 1: Understanding the Concept:

To find the equation of a line in 3D space, we need a point on the line and a direction vector that is parallel to the line. We can find the direction vector by taking the difference between the coordinates of the two given points.


Step 2: Key Formula or Approach:

The Cartesian equation of a line passing through the point \((x_1, y_1, z_1)\) with direction ratios \((a, b, c)\) is: \[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
The direction ratios \((a, b, c)\) can be found from two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) as: \(a = x_2 - x_1\), \(b = y_2 - y_1\), \(c = z_2 - z_1\).


Step 3: Detailed Explanation:

Let the two points be \(P_1(-3, 4, 11)\) and \(P_2(1, -2, 7)\).

1. Find the direction ratios (a, b, c):
\[ a = x_2 - x_1 = 1 - (-3) = 4 \] \[ b = y_2 - y_1 = -2 - 4 = -6 \] \[ c = z_2 - z_1 = 7 - 11 = -4 \]
So, the direction ratios are \((4, -6, -4)\).

Note that we can use any scalar multiple of the direction vector. Dividing by -2 gives a simpler set of direction ratios: \[ (\frac{4}{-2}, \frac{-6}{-2}, \frac{-4}{-2}) = (-2, 3, 2) \]
2. Write the equation of the line:

Using the point \(P_1(-3, 4, 11)\) and the direction ratios \((-2, 3, 2)\): \[ \frac{x - (-3)}{-2} = \frac{y - 4}{3} = \frac{z - 11}{2} \] \[ \frac{x+3}{-2} = \frac{y-4}{3} = \frac{z-11}{2} \]
This matches option (B).


Step 4: Final Answer:

The equation of the line is \(\frac{x+3}{-2} = \frac{y-4}{3} = \frac{z-11}{2}\). This corresponds to option (B).
Quick Tip: The direction ratios of a line are not unique. Any non-zero scalar multiple represents the same direction. When checking options, be prepared to see simplified or negated versions of the direction ratios you calculated.

Step 1: Understanding the Concept:

Two lines in 3D space are coplanar if they lie on the same plane. This occurs if the lines are either parallel or they intersect. The condition for coplanarity can be expressed using a determinant involving the direction vectors of the lines and a vector connecting a point on each line.


Step 2: Key Formula or Approach:

For two lines \(\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}\) and \(\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}\) to be coplanar, the scalar triple product of the vector connecting the points and the two direction vectors must be zero. This is equivalent to the determinant condition: \[ \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1
a_1 & b_1 & c_1
a_2 & b_2 & c_2 \end{vmatrix} = 0 \]

Step 3: Detailed Explanation:

From the given equations, we extract the necessary information.

Line 1: Passes through \(P_1(1, -1, -10)\) with direction vector \(\vec{d_1} = (2, -3, 8)\).

Line 2: Passes through \(P_2(4, -3, -1)\) with direction vector \(\vec{d_2} = (1, k, 7)\).

First, find the vector connecting the two points, \(\vec{P_1P_2}\): \[ \vec{P_1P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (4-1, -3-(-1), -1-(-10)) = (3, -2, 9) \]
Now, set up the determinant for the coplanarity condition: \[ \begin{vmatrix} 3 & -2 & 9
2 & -3 & 8
1 & k & 7 \end{vmatrix} = 0 \]
Expand the determinant along the first row: \[ 3((-3)(7) - (8)(k)) - (-2)((2)(7) - (8)(1)) + 9((2)(k) - (-3)(1)) = 0 \] \[ 3(-21 - 8k) + 2(14 - 8) + 9(2k + 3) = 0 \] \[ -63 - 24k + 2(6) + 18k + 27 = 0 \] \[ -63 - 24k + 12 + 18k + 27 = 0 \]
Combine the terms with \(k\) and the constant terms: \[ (-24k + 18k) + (-63 + 12 + 27) = 0 \] \[ -6k + (-24) = 0 \] \[ -6k = 24 \] \[ k = \frac{24}{-6} = -4 \]

Step 4: Final Answer:

The value of k is -4. This corresponds to option (E).
Quick Tip: The condition for coplanarity essentially checks if the three vectors (the two direction vectors and the vector connecting the lines) are linearly dependent, meaning they can all lie on the same plane. The scalar triple product (or the determinant) is the standard tool to test for this.

Step 1: Understanding the Concept:

The vector equation of a line describes all the points on that line. A general point on the line can be represented by its position vector, which is obtained by choosing a specific value for the parameter \(\lambda\). To check if a given point lies on the line, we see if we can find a single value of \(\lambda\) that produces the coordinates of that point.


Step 2: Key Formula or Approach:

The position vector of any point on the line is given by \(\vec{r} = (1, 2, -3) + \lambda(4, 5, -7)\).
In coordinate form, a general point \((x, y, z)\) on the line is: \[ x = 1 + 4\lambda \] \[ y = 2 + 5\lambda \] \[ z = -3 - 7\lambda \]
We will test each option by trying to solve for a consistent value of \(\lambda\).


Step 3: Detailed Explanation:

Let's test option (D), the point \((9, 12, -17)\).
We set the coordinates of this point equal to the general coordinate expressions from the line's equation.
For x-coordinate: \[ 9 = 1 + 4\lambda \implies 8 = 4\lambda \implies \lambda = 2 \]
For y-coordinate: \[ 12 = 2 + 5\lambda \implies 10 = 5\lambda \implies \lambda = 2 \]
For z-coordinate: \[ -17 = -3 - 7\lambda \implies -14 = -7\lambda \implies \lambda = 2 \]
Since we found the same value of \(\lambda = 2\) for all three coordinates, the point \((9, 12, -17)\) lies on the line.


Let's quickly check another option, say (A) (9,12,-15), to see why it fails.
From the x and y coordinates, we get \(\lambda = 2\).
Using \(\lambda=2\) for the z-coordinate, we should get \(z = -3 - 7(2) = -17\).
However, the z-coordinate in option (A) is -15. Since \(-15 \neq -17\), this point is not on the line.


Step 4: Final Answer:

The point (9,12,-17) lies on the given line. This corresponds to option (D).
Quick Tip: To quickly check points, solve for \(\lambda\) using the simplest coordinate equation (e.g., the x-equation) and then substitute that value of \(\lambda\) into the expressions for the other two coordinates to see if they match the point's coordinates.

Step 1: Understanding the Concept:

The mean (or average) of a set of numbers is the sum of the numbers divided by the count of the numbers. We can use the information about the mean of the first set to find the value of \(x\), and then use that value to find the mean of the second set. Alternatively, we can use properties of the mean.


Step 2: Key Formula or Approach:

Mean = \(\frac{Sum of observations}{Number of observations}\).

Method 1: Direct Calculation

1. Set up an equation for the mean of the first set and solve for \(x\).

2. Substitute \(x\) into the second set of numbers.

3. Calculate the mean of the new set.

Method 2: Using Properties of Mean

If we add a constant \(k\) to each observation, the new mean is also increased by \(k\).


Step 3: Detailed Explanation (Method 1):

1. Find x:

The first set of numbers is \(12+x, 17+x, 25+x, 34+x\). There are 4 numbers.
The mean is given as 22. \[ \frac{(12+x) + (17+x) + (25+x) + (34+x)}{4} = 22 \] \[ \frac{12+17+25+34 + 4x}{4} = 22 \] \[ \frac{88 + 4x}{4} = 22 \] \[ 22 + x = 22 \implies x = 0 \]
2. Find the mean of the second set:

Now that we know \(x=0\), the second set of numbers is \(38, 42, 52, 60\).
Calculate the mean of this set: \[ Mean = \frac{38 + 42 + 52 + 60}{4} = \frac{192}{4} = 48 \]

Detailed Explanation (Method 2):

Let the first set of observations be \(y_i = z_i + x\), where \(z_i\) are \(12, 17, 25, 34\).
Mean(\(y_i\)) = Mean(\(z_i + x\)) = Mean(\(z_i\)) + \(x\).
Mean(\(z_i\)) = \(\frac{12+17+25+34}{4} = \frac{88}{4} = 22\).
So, Mean(\(y_i\)) = \(22+x\). We are given this is 22, so \(22+x=22 \implies x=0\).
The second set is \(w_i = v_i + x\), where \(v_i\) are \(38, 42, 52, 60\).
Mean(\(w_i\)) = Mean(\(v_i\)) + \(x\).
Mean(\(v_i\)) = \(\frac{38+42+52+60}{4} = \frac{192}{4} = 48\).
So, Mean(\(w_i\)) = \(48+x = 48+0 = 48\).


Step 4: Final Answer:

The mean of the second set is 48. This corresponds to option (C).
Quick Tip: The property that adding a constant to every number in a dataset adds the same constant to the mean is very useful. In this problem, one can see \(\frac{88+4x}{4} = 22+x\). Setting this to 22 immediately gives \(x=0\), simplifying the rest of the problem.

Step 1: Understanding the Concept:

This question tests the properties of standard deviation with respect to scaling of data. We need to understand how the standard deviation changes when every data point in a set is multiplied by a constant.


Step 2: Key Formula or Approach:

Let the original data set be \(x_1, x_2, \dots, x_n\) with standard deviation \(\sigma_x\).
If we create a new data set by multiplying each observation by a constant \(k\), so the new set is \(kx_1, kx_2, \dots, kx_n\), then the new standard deviation \(\sigma_{kx}\) is related to the old one by: \[ \sigma_{kx} = |k| \sigma_x \]
If we add a constant to each observation, the standard deviation does not change.


Step 3: Detailed Explanation:

Let the original data set be \(X = \{3, 8, 6, 10, 12, 9, 11, 10, 12, 7\}\).

The standard deviation of this set is given as \(\sigma_X = 2.71\).

The new data set is \(Y = \{30, 80, 60, 100, 120, 90, 110, 100, 120, 70\}\).

By observing the two sets, we can see that each element in set \(Y\) is 10 times the corresponding element in set \(X\). \[ y_i = 10 x_i \]
This means the data has been scaled by a constant factor \(k=10\).

Using the property of standard deviation, the new standard deviation \(\sigma_Y\) will be: \[ \sigma_Y = |10| \times \sigma_X = 10 \times 2.71 = 27.1 \]

Step 4: Final Answer:

The standard deviation of the new set is 27.1. This corresponds to option (C).
Quick Tip: Remember the effects of transformation on measures of central tendency and dispersion. - \textbf{Adding a constant \(c\):} Mean changes by \(c\), but Variance and Standard Deviation remain unchanged. - \textbf{Multiplying by a constant \(k\):} Mean is multiplied by \(k\), Variance is multiplied by \(k^2\), and Standard Deviation is multiplied by \(|k|\).

Note: The value for P(B) in the OCR is ambiguous. Based on the options and the correct answer, P(B) is assumed to be 1/5.


Step 1: Understanding the Concept:

Mutually exclusive events are events that cannot occur at the same time. This means their intersection is empty, and the probability of their intersection is zero. This simplifies the addition rule for probabilities.


Step 2: Key Formula or Approach:

For any two events A and B, the addition rule is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
If A and B are mutually exclusive, then \(P(A \cap B) = 0\). The formula simplifies to: \[ P(A \cup B) = P(A) + P(B) \]
We are given \(P(A \cup B)\) and \(P(B)\) and need to find \(P(A)\).


Step 3: Detailed Explanation:

We are given:

A and B are mutually exclusive events.
\(P(B) = \frac{1{5}\)
\(P(A \cup B) = \frac{13}{35}\)

Using the simplified addition rule for mutually exclusive events: \[ P(A) = P(A \cup B) - P(B) \]
Substitute the given values: \[ P(A) = \frac{13}{35} - \frac{1}{5} \]
To subtract the fractions, we need a common denominator, which is 35. \[ \frac{1}{5} = \frac{1 \times 7}{5 \times 7} = \frac{7}{35} \]
Now, perform the subtraction: \[ P(A) = \frac{13}{35} - \frac{7}{35} = \frac{13 - 7}{35} = \frac{6}{35} \]

Step 4: Final Answer:

The value of \(P(A)\) is \(\frac{6}{35}\). This corresponds to option (D).
Quick Tip: Always distinguish between "mutually exclusive" (\(P(A \cap B) = 0\)) and "independent" (\(P(A \cap B) = P(A)P(B)\)). The formula for \(P(A \cup B)\) changes depending on which condition is given.

Step 1: Understanding the Concept:

Independent events are events where the occurrence of one does not affect the probability of the other. For independent events, the probability of their intersection is the product of their individual probabilities. We need to use this property along with the general addition rule for probabilities.


Step 2: Key Formula or Approach:

1. The probability of the complement of an event A is \(P(A') = 1 - P(A)\).

2. The addition rule for probability is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

3. For independent events A and B, \(P(A \cap B) = P(A) \times P(B)\).

Combining (2) and (3) for independent events gives: \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\).


Step 3: Detailed Explanation:

We are given:

A and B are independent events.
\(P(A') = 0.8\)
\(P(B) = 0.6\)

First, we need to find \(P(A)\). \[ P(A) = 1 - P(A') = 1 - 0.8 = 0.2 \]
Now, we can use the formula for the union of two independent events: \[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]
Substitute the values of \(P(A)\) and \(P(B)\): \[ P(A \cup B) = 0.2 + 0.6 - (0.2)(0.6) \] \[ P(A \cup B) = 0.8 - 0.12 \] \[ P(A \cup B) = 0.68 \]

Step 4: Final Answer:

The value of \(P(A \cup B)\) is 0.68. This corresponds to option (C).
Quick Tip: An alternative formula for the union of independent events is \(P(A \cup B) = 1 - P(A' \cap B') = 1 - P(A')P(B')\). Given \(P(A') = 0.8\) and \(P(B') = 1 - 0.6 = 0.4\), we get \(P(A \cup B) = 1 - (0.8)(0.4) = 1 - 0.32 = 0.68\). This can sometimes be faster.

Step 1: Understanding the Concept:

This limit is in the indeterminate form \(\frac{0}{0}\) (as \(x \to 1\), the numerator becomes \(1+1+1+1+1 - 5 = 0\) and the denominator becomes \(1-1=0\)). We can solve this using either L'Hôpital's Rule or by recognizing the limit as the definition of a derivative.


Step 2: Key Formula or Approach:

Method 1: L'Hôpital's Rule

If \(\lim_{x \to c} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\).

Method 2: Definition of the Derivative

The derivative of a function \(f\) at a point \(c\) is defined as \(f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}\).


Step 3: Detailed Explanation (Using Method 2):

Let's define a function \(f(x) = x + x^2 + x^3 + x^4 + x^5\).

Now, let's find the value of this function at \(x=1\): \[ f(1) = 1 + 1^2 + 1^3 + 1^4 + 1^5 = 1+1+1+1+1 = 5 \]
The given limit can be rewritten by substituting \(f(x)\) and \(f(1)\): \[ \lim_{x \to 1} \frac{f(x) - f(1)}{x - 1} \]
This is exactly the definition of the derivative of \(f(x)\) at the point \(x=1\), i.e., \(f'(1)\).

First, we find the derivative of \(f(x)\): \[ f'(x) = \frac{d}{dx}(x + x^2 + x^3 + x^4 + x^5) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 \]
Now, we evaluate this derivative at \(x=1\): \[ f'(1) = 1 + 2(1) + 3(1)^2 + 4(1)^3 + 5(1)^4 = 1 + 2 + 3 + 4 + 5 = 15 \]

Detailed Explanation (Using Method 1):

Let \(f(x) = x + x^2 + x^3 + x^4 + x^5 - 5\) and \(g(x) = x - 1\).
\(f'(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4\)
\(g'(x) = 1\)

According to L'Hôpital's Rule: \[ \lim_{x \to 1} \frac{f'(x)}{g'(x)} = \lim_{x \to 1} \frac{1 + 2x + 3x^2 + 4x^3 + 5x^4}{1} \]
Now, substitute \(x=1\): \[ \frac{1 + 2(1) + 3(1)^2 + 4(1)^3 + 5(1)^4}{1} = 1 + 2 + 3 + 4 + 5 = 15 \]

Step 4: Final Answer:

The value of the limit is 15. This corresponds to option (E).
Quick Tip: Recognizing a limit problem as the definition of a derivative is often the fastest way to solve it. If you see the structure \(\frac{f(x)-f(c)}{x-c}\), immediately think of finding \(f'(c)\).

Step 1: Understanding the Concept:

This is a limit problem involving trigonometric functions, and it results in the indeterminate form \(\frac{0}{0}\) as \(x \to 0\). We can solve it using standard trigonometric limits or L'Hôpital's Rule.


Step 2: Key Formula or Approach:

Method 1: Using Standard Limits

The key standard limit is \(\lim_{u \to 0} \frac{\sin(u)}{u} = 1\). To use this, we can divide both the numerator and the denominator of the main expression by \(x\).

Method 2: L'Hôpital's Rule

Since the limit is of the form \(\frac{0}{0}\), we can differentiate the numerator and the denominator with respect to \(x\) and then take the limit.


Step 3: Detailed Explanation (Using Method 1):

Divide the numerator and the denominator by \(x\): \[ \lim_{x \to 0} \frac{\frac{\sin(2x)}{x} + \frac{3x}{x}}{\frac{4x}{x} + \frac{\sin(6x)}{x}} \] \[ = \lim_{x \to 0} \frac{\frac{\sin(2x)}{x} + 3}{4 + \frac{\sin(6x)}{x}} \]
Now we need to evaluate the limits of the sine terms. We can adjust the expressions to fit the standard form \(\lim_{u \to 0} \frac{\sin(u)}{u} = 1\).
For the numerator: \[ \lim_{x \to 0} \frac{\sin(2x)}{x} = \lim_{x \to 0} 2 \cdot \frac{\sin(2x)}{2x} = 2 \cdot \lim_{2x \to 0} \frac{\sin(2x)}{2x} = 2 \cdot 1 = 2 \]
For the denominator: \[ \lim_{x \to 0} \frac{\sin(6x)}{x} = \lim_{x \to 0} 6 \cdot \frac{\sin(6x)}{6x} = 6 \cdot \lim_{6x \to 0} \frac{\sin(6x)}{6x} = 6 \cdot 1 = 6 \]
Now, substitute these results back into the main limit expression: \[ \frac{2 + 3}{4 + 6} = \frac{5}{10} = \frac{1}{2} \]

Detailed Explanation (Using Method 2):

Let \(f(x) = \sin(2x) + 3x\) and \(g(x) = 4x + \sin(6x)\).

Find the derivatives: \[ f'(x) = \cos(2x) \cdot 2 + 3 = 2\cos(2x) + 3 \] \[ g'(x) = 4 + \cos(6x) \cdot 6 = 4 + 6\cos(6x) \]
Apply L'Hôpital's Rule: \[ \lim_{x \to 0} \frac{f'(x)}{g'(x)} = \lim_{x \to 0} \frac{2\cos(2x) + 3}{4 + 6\cos(6x)} \]
Now, substitute \(x=0\), remembering that \(\cos(0) = 1\): \[ \frac{2\cos(0) + 3}{4 + 6\cos(0)} = \frac{2(1) + 3}{4 + 6(1)} = \frac{2+3}{4+6} = \frac{5}{10} = \frac{1}{2} \]

Step 4: Final Answer:

The value of the limit is \(\frac{1}{2}\). This corresponds to option (C).
Quick Tip: For limits of the form \(\lim_{x \to 0} \frac{ax + \sin(bx)}{cx + \sin(dx)}\), a quick shortcut is to use the approximation \(\sin(\theta) \approx \theta\) for small \(\theta\). The limit becomes \(\lim_{x \to 0} \frac{2x + 3x}{4x + 6x} = \lim_{x \to 0} \frac{5x}{10x} = \frac{5}{10} = \frac{1}{2}\). This is a very fast method for multiple-choice questions.

Step 1: Understanding the Concept:

The domain of a function is the set of all possible input values (\(x\)) for which the function is defined. For a function involving square roots, the expression inside each square root must be non-negative (greater than or equal to zero).


Step 2: Key Formula or Approach:

For the function \(f(x) = \sqrt{|x|-1} + \sqrt{4-|x|}\) to be defined, two conditions must be met simultaneously:
1. \(|x|-1 \ge 0\)
2. \(4-|x| \ge 0\)
The domain of the function is the intersection of the solution sets of these two inequalities.


Step 3: Detailed Explanation:

Condition 1: \(|x|-1 \ge 0\)

This inequality simplifies to \(|x| \ge 1\).

This is true when \(x \ge 1\) or \(x \le -1\).

In interval notation, the solution set is \((-\infty, -1] \cup [1, \infty)\).


Condition 2: \(4-|x| \ge 0\)

This inequality simplifies to \(4 \ge |x|\), which is the same as \(|x| \le 4\).

This is true when \(-4 \le x \le 4\).

In interval notation, the solution set is \([-4, 4]\).


Finding the Intersection:

We need to find the values of \(x\) that satisfy both conditions. We find the intersection of the two solution sets: \((-\infty, -1] \cup [1, \infty)\) and \([-4, 4]\).

The intersection of \((-\infty, -1]\) and \([-4, 4]\) is \([-4, -1]\).
The intersection of \([1, \infty)\) and \([-4, 4]\) is \([1, 4]\).

The final domain is the union of these two resulting intervals.


Step 4: Final Answer:

The domain is \([-4, -1] \cup [1, 4]\). This corresponds to option (E).
Quick Tip: When solving intersections of intervals, it can be helpful to visualize the solution sets on a number line. Draw the intervals for each condition and identify the regions where they overlap.

Step 1: Understanding the Concept:

The range of a function is the set of all possible output values. For functions of the form \(a\sin x + b\cos x\), we can find the minimum and maximum values to determine the range.


Step 2: Key Formula or Approach:

For any expression of the form \(a\sin x + b\cos x\), its range is given by \([-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]\). We first find the range of \(\sin x + \cos x\) and then adjust it for the constant term "+3".


Step 3: Detailed Explanation:

Let's consider the part \(g(x) = \sin x + \cos x\).

This is of the form \(a\sin x + b\cos x\) with \(a=1\) and \(b=1\).

The minimum value of \(g(x)\) is \(-\sqrt{a^2+b^2} = -\sqrt{1^2+1^2} = -\sqrt{2}\).

The maximum value of \(g(x)\) is \(\sqrt{a^2+b^2} = \sqrt{1^2+1^2} = \sqrt{2}\).

So, the range of \(\sin x + \cos x\) is \([-\sqrt{2}, \sqrt{2}]\).

This means: \[ -\sqrt{2} \le \sin x + \cos x \le \sqrt{2} \]
The given function is \(f(x) = (\sin x + \cos x) + 3\). To find its range, we add 3 to all parts of the inequality: \[ -\sqrt{2} + 3 \le \sin x + \cos x + 3 \le \sqrt{2} + 3 \]
So, the range of \(f(x)\) is \([3-\sqrt{2}, 3+\sqrt{2}]\).


Step 4: Final Answer:

The range of the function is \([-\sqrt{2}+3, \sqrt{2}+3]\). This corresponds to option (B).
Quick Tip: You can also convert \(a\sin x + b\cos x\) to the form \(R\sin(x+\alpha)\) or \(R\cos(x-\alpha)\) where \(R = \sqrt{a^2+b^2}\). For \(\sin x + \cos x\), this is \(\sqrt{2}\sin(x+\pi/4)\). Since the range of \(\sin(\cdot)\) is \([-1, 1]\), the range of \(\sqrt{2}\sin(x+\pi/4)\) is \([-\sqrt{2}, \sqrt{2}]\).

Step 1: Understanding the Concept:

The given limit expression is the definition of the derivative of the function \(F(x)\) at the point \(x=1\).


Step 2: Key Formula or Approach:

The derivative of a function \(F\) at a point \(c\) is defined as: \[ F'(c) = \lim_{x \to c} \frac{F(x) - F(c)}{x - c} \]
So, the problem is asking us to find the value of \(F'(1)\). We need to differentiate \(F(x)\) and then evaluate the result at \(x=1\).


Step 3: Detailed Explanation:

The function is \(F(x) = -\sqrt{9-x^2} = -(9-x^2)^{1/2}\).

We will use the chain rule to find the derivative, \(F'(x)\).

Let \(u = 9-x^2\), then \(F(u) = -u^{1/2}\). \[ F'(x) = \frac{dF}{du} \cdot \frac{du}{dx} \] \[ \frac{dF}{du} = -\frac{1}{2}u^{-1/2} = -\frac{1}{2\sqrt{u}} \] \[ \frac{du}{dx} = -2x \]
So, \[ F'(x) = \left(-\frac{1}{2\sqrt{9-x^2}}\right) \cdot (-2x) = \frac{2x}{2\sqrt{9-x^2}} = \frac{x}{\sqrt{9-x^2}} \]
Now, we evaluate this derivative at \(x=1\): \[ F'(1) = \frac{1}{\sqrt{9-(1)^2}} = \frac{1}{\sqrt{9-1}} = \frac{1}{\sqrt{8}} \]
To simplify \(\sqrt{8}\): \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\). \[ F'(1) = \frac{1}{2\sqrt{2}} \]

Step 4: Final Answer:

The value of the limit is \(\frac{1}{2\sqrt{2}}\). This corresponds to option (D).
Quick Tip: Recognizing that a limit is in the form of the definition of a derivative can save a lot of time compared to using other methods like L'Hôpital's rule, especially when the function is straightforward to differentiate.

Step 1: Understanding the Concept:

This question involves differentiation. We are given an implicit relationship between \(x\) and \(y\). To find \(\frac{dy}{dx}\), it's easiest to first express \(y\) explicitly as a function of \(x\).


Step 2: Key Formula or Approach:

1. Convert the logarithmic equation \(\log_b a = c\) to its equivalent exponential form \(b^c = a\).
2. Differentiate the resulting exponential function. The derivative of \(a^x\) with respect to \(x\) is \(a^x \ln(a)\), where \(\ln(a)\) is the natural logarithm of \(a\), also written as \(\log_e a\).


Step 3: Detailed Explanation:

The given equation is \(\log_2 y = x\).

Converting this to exponential form (with base 2), we get: \[ y = 2^x \]
Now, we need to differentiate this function with respect to \(x\).

Using the standard formula for the derivative of an exponential function \(\frac{d}{dx}(a^x) = a^x \ln(a)\): \[ \frac{dy}{dx} = \frac{d}{dx}(2^x) = 2^x \ln(2) \]
The natural logarithm \(\ln(2)\) is the same as \(\log_e 2\).

So, the derivative is \(2^x \log_e 2\).


Step 4: Final Answer:

The value of \(\frac{dy}{dx}\) is \(2^x \log_e 2\). This corresponds to option (A).
Quick Tip: Be careful to distinguish between different logarithm bases. \(\log x\) often means \(\log_{10} x\), while \(\ln x\) means \(\log_e x\). The derivative formula \(\frac{d}{dx}(a^x) = a^x \ln a\) always uses the natural logarithm.

Step 1: Understanding the Concept:

We need to find the derivative of a function that is a product of four factors, and then evaluate it at a specific point. A key observation is that one of the factors becomes zero at the point of evaluation, which greatly simplifies the calculation.


Step 2: Key Formula or Approach:

We use the product rule for differentiation. For a product of four functions \(y = f_1 f_2 f_3 f_4\), the derivative is: \[ y' = f_1' f_2 f_3 f_4 + f_1 f_2' f_3 f_4 + f_1 f_2 f_3' f_4 + f_1 f_2 f_3 f_4' \]
We need to evaluate this at \(x = \frac{1}{2}\).


Step 3: Detailed Explanation:

Let the four factors be: \(f_1(x) = x-1\)
\(f_2(x) = 2x-1\)
\(f_3(x) = 3-x\)
\(f_4(x) = 4-x\)

Let's evaluate the second factor at \(x = \frac{1}{2}\): \[ f_2\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right) - 1 = 1 - 1 = 0 \]
Now consider the expanded form of the derivative using the product rule. The factor \(f_2(x) = (2x-1)\) appears in the first, third, and fourth terms of the sum. When we substitute \(x = \frac{1}{2}\), these three terms will become zero. \[ y'\left(\frac{1}{2}\right) = 0 + f_1\left(\frac{1}{2}\right) f_2'\left(\frac{1}{2}\right) f_3\left(\frac{1}{2}\right) f_4\left(\frac{1}{2}\right) + 0 + 0 \]
So we only need to calculate the second term.
First, find the derivative of the second factor: \(f_2'(x) = \frac{d}{dx}(2x-1) = 2\). So, \(f_2'(\frac{1}{2}) = 2\).

Now, evaluate the other factors at \(x=\frac{1}{2}\): \[ f_1\left(\frac{1}{2}\right) = \frac{1}{2} - 1 = -\frac{1}{2} \] \[ f_3\left(\frac{1}{2}\right) = 3 - \frac{1}{2} = \frac{5}{2} \] \[ f_4\left(\frac{1}{2}\right) = 4 - \frac{1}{2} = \frac{7}{2} \]
Now, multiply these values together: \[ y'\left(\frac{1}{2}\right) = \left(-\frac{1}{2}\right) \cdot (2) \cdot \left(\frac{5}{2}\right) \cdot \left(\frac{7}{2}\right) = -1 \cdot \frac{35}{4} = -\frac{35}{4} \]

Step 4: Final Answer:

The derivative at \(x=\frac{1}{2}\) is \(-\frac{35}{4}\). This corresponds to option (B).
Quick Tip: Before applying a lengthy differentiation rule like the product rule for multiple terms, always check if any of the terms become zero at the point of evaluation. This can save a significant amount of calculation.

Step 1: Understanding the Concept:

To find the derivative of a function involving an absolute value, we must first resolve the absolute value. This is done by determining the sign of the expression inside the absolute value bars in the neighborhood of the point of interest.


Step 2: Key Formula or Approach:

1. Determine the sign of \(\cos x - \sin x\) at \(x = \frac{\pi}{6}\).
2. Based on the sign, rewrite \(f(x)\) without the absolute value sign for \(x\) near \(\frac{\pi}{6}\).
3. Differentiate the simplified function.
4. Evaluate the derivative at \(x = \frac{\pi}{6}\).


Step 3: Detailed Explanation:

1. Check the sign of the expression:

At \(x = \frac{\pi}{6}\) (or \(30^\circ\)): \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) \(\sin(\frac{\pi}{6}) = \frac{1}{2}\)
The expression inside the absolute value is \(\cos(\frac{\pi}{6}) - \sin(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3}-1}{2}\).
Since \(\sqrt{3} \approx 1.732\), this value is positive. Therefore, for values of \(x\) close to \(\frac{\pi}{6}\), the expression \(\cos x - \sin x\) is positive.


2. Simplify the function:

Since the expression is positive, we can remove the absolute value bars: \[ f(x) = \cos x - \sin x \quad (for x near \(\frac{\pi{6}\))} \]
3. Differentiate:
\[ f'(x) = \frac{d}{dx}(\cos x - \sin x) = -\sin x - \cos x \]
4. Evaluate the derivative:

Substitute \(x = \frac{\pi}{6}\) into \(f'(x)\): \[ f'\left(\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) - \cos\left(\frac{\pi}{6}\right) = -\frac{1}{2} - \frac{\sqrt{3}}{2} = -\frac{1+\sqrt{3}}{2} \]

Step 4: Final Answer:

The value of \(f'(\frac{\pi}{6})\) is \(-\frac{\sqrt{3}+1}{2}\). This corresponds to option (A).
Quick Tip: The sign of \(\cos x - \sin x\) changes at \(x=\frac{\pi}{4}\). For \(x < \frac{\pi}{4}\) in the first quadrant, \(\cos x > \sin x\). For \(x > \frac{\pi}{4}\), \(\sin x > \cos x\). Since \(\frac{\pi}{6} < \frac{\pi}{4}\), we know \(\cos x - \sin x\) is positive.

Step 1: Understanding the Concept:

This question is a direct application of the First Fundamental Theorem of Calculus. This theorem provides a link between differentiation and integration, stating that the derivative of an integral function with a variable upper limit is the integrand evaluated at that limit.


Step 2: Key Formula or Approach:

The First Fundamental Theorem of Calculus states that if a function \(F\) is defined by \(F(x) = \int_a^x f(t) dt\) for some constant \(a\), then the derivative of \(F(x)\) is \(f(x)\). \[ F'(x) = \frac{d}{dx} \int_a^x f(t) dt = f(x) \]
So, to find \(f(x)\), we need to differentiate \(F(x)\).


Step 3: Detailed Explanation:

We are given the integral function: \[ F(x) = \int_0^x f(t) dt \]
We are also given the explicit form of \(F(x)\): \[ F(x) = x^2(1+x) = x^2 + x^3 \]
According to the Fundamental Theorem of Calculus, \(f(x) = F'(x)\).
We differentiate \(F(x)\) with respect to \(x\): \[ f(x) = \frac{d}{dx}(x^2 + x^3) \]
Using the power rule for differentiation: \[ f(x) = 2x + 3x^2 \]
The question asks for the value of \(f(2)\). We substitute \(x=2\) into the expression for \(f(x)\): \[ f(2) = 2(2) + 3(2)^2 = 4 + 3(4) = 4 + 12 = 16 \]

Step 4: Final Answer:

The value of \(f(2)\) is 16. This corresponds to option (D).
Quick Tip: This theorem is a cornerstone of calculus. Whenever you see a function defined as an integral with \(x\) in the upper limit, immediately think of differentiation to find the original integrand function.

Step 1: Understanding the Concept:

Similar to question 57, to find the derivative of a function involving an absolute value, we first need to determine the sign of the expression inside the absolute value bars at the point of interest. This allows us to rewrite the function without the absolute value for differentiation.


Step 2: Key Formula or Approach:

1. Evaluate the expression \(x^2-1\) at \(x = \frac{3}{2}\) to determine its sign.
2. Rewrite \(f(x)\) without the absolute value for \(x\) near \(\frac{3}{2}\).
3. Differentiate the simplified function.
4. Evaluate the derivative at \(x = \frac{3}{2}\).


Step 3: Detailed Explanation:

1. Check the sign of the expression:

At \(x = \frac{3}{2}\), the expression inside the absolute value is: \[ x^2 - 1 = \left(\frac{3}{2}\right)^2 - 1 = \frac{9}{4} - 1 = \frac{9}{4} - \frac{4}{4} = \frac{5}{4} \]
Since \(\frac{5}{4}\) is positive, the expression \(x^2-1\) is positive in the neighborhood of \(x=\frac{3}{2}\).


2. Simplify the function:

Because the expression is positive, we can remove the absolute value bars: \[ f(x) = x^2 - 1 \quad (for x near \(\frac{3{2}\))} \]
3. Differentiate:
\[ f'(x) = \frac{d}{dx}(x^2 - 1) = 2x \]
4. Evaluate the derivative:

Substitute \(x = \frac{3}{2}\) into \(f'(x)\): \[ f'\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right) = 3 \]

Step 4: Final Answer:

The value of \(f'(\frac{3}{2})\) is 3. This corresponds to option (A).
Quick Tip: The function \(f(x) = |x^2-1|\) is not differentiable at \(x=1\) and \(x=-1\) because the graph has sharp corners there. For any other point, you can differentiate by first removing the absolute value.

Step 1: Understanding the Concept:

Critical points of a function are points in the domain where the first derivative is either zero or undefined. For a polynomial function, the derivative is always defined, so we only need to find where the derivative is zero. A critical point is given as a coordinate pair \((x, f(x))\).


Step 2: Key Formula or Approach:

1. Find the first derivative of the function, \(f'(x)\).
2. Set \(f'(x) = 0\) and solve for the critical x-values (also called critical numbers).
3. For each critical x-value, calculate the corresponding y-value by substituting it back into the original function \(f(x)\).


Step 3: Detailed Explanation:

The given function is \(f(x) = \frac{x^3}{3} + 3x^2 - 7x\).

1. Find the derivative:
\[ f'(x) = \frac{d}{dx}\left(\frac{x^3}{3} + 3x^2 - 7x\right) = \frac{3x^2}{3} + 6x - 7 = x^2 + 6x - 7 \]
2. Find the critical x-values:

Set \(f'(x) = 0\): \[ x^2 + 6x - 7 = 0 \]
Factor the quadratic equation: \[ (x+7)(x-1) = 0 \]
This gives two critical x-values: \(x = -7\) and \(x = 1\).

3. Find the corresponding y-values:

We need to find the critical point that matches one of the options. Let's test \(x=1\). \[ f(1) = \frac{(1)^3}{3} + 3(1)^2 - 7(1) = \frac{1}{3} + 3 - 7 = \frac{1}{3} - 4 = \frac{1-12}{3} = -\frac{11}{3} \]
So, one critical point is \((1, -\frac{11}{3})\). This matches option (A).

For completeness, let's find the other critical point for \(x=-7\): \[ f(-7) = \frac{(-7)^3}{3} + 3(-7)^2 - 7(-7) = \frac{-343}{3} + 3(49) + 49 = -\frac{343}{3} + 147 + 49 = -\frac{343}{3} + 196 = \frac{-343+588}{3} = \frac{245}{3} \]
The other critical point is \((-7, \frac{245}{3})\).


Step 4: Final Answer:

The point \((1, -\frac{11}{3})\) is a critical point of the function. This corresponds to option (A).
Quick Tip: Critical points are candidates for local maxima or minima. You can use the second derivative test to classify them. \(f''(x) = 2x+6\). \(f''(1)=8>0\), so \((1, -11/3)\) is a local minimum. \(f''(-7)=-8<0\), so \((-7, 245/3)\) is a local maximum.

Step 1: Understanding the Concept:

A function is decreasing on an interval if its first derivative is negative on that interval. To find such intervals, we first find the critical points (where the derivative is zero or undefined) and then test the sign of the derivative in the intervals defined by these points.


Step 2: Key Formula or Approach:

1. Find the derivative \(f'(x)\).
2. Find the critical points by solving \(f'(x) = 0\).
3. Create a sign chart for \(f'(x)\) using the critical points.
4. Identify the interval(s) where \(f'(x) < 0\).


Step 3: Detailed Explanation:

The function is \(f(x) = 2x^3 + 9x^2 + 12x - 1\).

1. Find the derivative:
\[ f'(x) = \frac{d}{dx}(2x^3 + 9x^2 + 12x - 1) = 6x^2 + 18x + 12 \]
2. Find the critical points:

Set \(f'(x) = 0\): \[ 6x^2 + 18x + 12 = 0 \]
Divide by 6 to simplify: \[ x^2 + 3x + 2 = 0 \]
Factor the quadratic: \[ (x+2)(x+1) = 0 \]
The critical points are \(x = -2\) and \(x = -1\).

3. Analyze the sign of \(f'(x)\):

The critical points divide the number line into three intervals: \((-\infty, -2)\), \((-2, -1)\), and \((-1, \infty)\). We test a point in each interval to determine the sign of \(f'(x) = 6(x+2)(x+1)\).

Interval \((-\infty, -2)\): Let's test \(x=-3\).

\(f'(-3) = 6(-3+2)(-3+1) = 6(-1)(-2) = 12 > 0\). The function is increasing.
Interval \((-2, -1)\): Let's test \(x=-1.5\).

\(f'(-1.5) = 6(-1.5+2)(-1.5+1) = 6(0.5)(-0.5) = -1.5 < 0\). The function is decreasing.
Interval \((-1, \infty)\): Let's test \(x=0\).

\(f'(0) = 6(0+2)(0+1) = 12 > 0\). The function is increasing.

The function is decreasing only on the interval \((-2, -1)\).


Step 4: Final Answer:

The function is decreasing in the interval \((-2, -1)\). This corresponds to option (C).
Quick Tip: The sign of a quadratic \(a(x-r_1)(x-r_2)\) can be determined quickly by its graph. A parabola with a positive leading coefficient (\(a=6 > 0\)) is U-shaped. It is negative between its roots (\(-2\) and \(-1\)) and positive outside them.

Step 1: Understanding the Concept:

This is a related rates problem. We are given the rates of change of the radius and height of a cylinder and asked to find the rate of change of its volume at a specific instant. This involves implicit differentiation with respect to time.


Step 2: Key Formula or Approach:

1. Write down the formula for the volume of a cylinder: \(V = \pi r^2 h\).
2. Differentiate both sides of the equation with respect to time \(t\), using the product rule for the term \(r^2 h\).
3. Substitute the given values for the variables (\(r, h\)) and their rates of change (\(\frac{dr}{dt}, \frac{dh}{dt}\)) to find \(\frac{dV}{dt}\).


Step 3: Detailed Explanation:

The volume of a cylinder is \(V = \pi r^2 h\).

We are given:

Rate of change of radius: \(\frac{dr}{dt} = 2\) cm/s (increasing, so positive).
Rate of change of height: \(\frac{dh}{dt} = -3\) cm/s (decreasing, so negative).

We need to find \(\frac{dV}{dt}\) at the instant when \(r = 4\) cm and \(h = 6\) cm.

Differentiate the volume formula with respect to time \(t\). We must use the product rule on \(r^2h\). \[ \frac{dV}{dt} = \frac{d}{dt}(\pi r^2 h) = \pi \left[ \frac{d(r^2)}{dt} \cdot h + r^2 \cdot \frac{dh}{dt} \right] \]
Using the chain rule for \(\frac{d(r^2)}{dt}\), we get \(2r \frac{dr}{dt}\). \[ \frac{dV}{dt} = \pi \left[ (2r \frac{dr}{dt}) \cdot h + r^2 \cdot \frac{dh}{dt} \right] \]
Now, substitute the given values: \(r=4, h=6, \frac{dr}{dt}=2, \frac{dh}{dt}=-3\). \[ \frac{dV}{dt} = \pi \left[ (2 \cdot 4 \cdot 2) \cdot 6 + (4)^2 \cdot (-3) \right] \] \[ \frac{dV}{dt} = \pi [ (16) \cdot 6 + 16 \cdot (-3) ] \] \[ \frac{dV}{dt} = \pi [ 96 - 48 ] = 48\pi \]

Step 4: Final Answer:

The rate of change of the volume is \(48\pi\) cm\(^3\)/s. This corresponds to option (E).
Quick Tip: In related rates problems, be very careful with signs. An "increasing" rate is positive, while a "decreasing" rate is negative. Forgetting a negative sign is a common mistake.

Step 1: Understanding the Concept:

This is an optimization problem. We are asked to find two numbers that satisfy a given constraint (their sum is 12) and minimize a certain quantity (the sum of their squares). This can be solved using calculus by finding the critical points of the function to be minimized.


Step 2: Key Formula or Approach:

1. Define two variables for the numbers, say \(x\) and \(y\).
2. Write the constraint equation: \(x + y = 12\).
3. Write the objective function to be minimized: \(S = x^2 + y^2\).
4. Use the constraint to express \(S\) as a function of a single variable.
5. Find the derivative of \(S\) with respect to that variable and set it to zero to find the critical point.
6. Verify that this point corresponds to a minimum (e.g., using the second derivative test).


Step 3: Detailed Explanation:

Let the two positive numbers be \(x\) and \(y\).

Constraint: \(x + y = 12\). We can express \(y\) as \(y = 12 - x\).

Objective function: Minimize \(S = x^2 + y^2\).

Substitute the expression for \(y\) into \(S\): \[ S(x) = x^2 + (12 - x)^2 \]
Expand the expression: \[ S(x) = x^2 + (144 - 24x + x^2) = 2x^2 - 24x + 144 \]
To find the minimum, we take the first derivative and set it to zero: \[ S'(x) = \frac{d}{dx}(2x^2 - 24x + 144) = 4x - 24 \]
Set \(S'(x) = 0\): \[ 4x - 24 = 0 \implies 4x = 24 \implies x = 6 \]
Now, find the corresponding value of \(y\): \[ y = 12 - x = 12 - 6 = 6 \]
To confirm it is a minimum, we use the second derivative test: \[ S''(x) = \frac{d}{dx}(4x - 24) = 4 \]
Since \(S''(x) = 4 > 0\), the function has a minimum at this critical point.


Step 4: Final Answer:

The two numbers are 6 and 6. This corresponds to option (D).
Quick Tip: For many optimization problems of this type (fixed sum, minimize product of powers, or vice versa), the solution often occurs when the numbers are equal. This provides a good intuition to quickly check your answer.

Step 1: Understanding the Concept:

This integral involves a denominator with a sum of square roots. A standard technique to simplify such expressions is to multiply the numerator and denominator by the conjugate of the denominator.


Step 2: Key Formula or Approach:

1. Rationalize the denominator by multiplying by its conjugate, which is \(\sqrt{x} - \sqrt{x-1}\).
2. Simplify the resulting expression. The denominator will become a constant.
3. Integrate the simplified numerator term by term using the power rule for integration: \(\int u^n du = \frac{u^{n+1}}{n+1} + C\).


Step 3: Detailed Explanation:

The integral is \(\int \frac{1}{\sqrt{x} + \sqrt{x-1}} dx\).

Multiply the numerator and denominator by the conjugate \(\sqrt{x} - \sqrt{x-1}\): \[ \int \frac{1}{\sqrt{x} + \sqrt{x-1}} \cdot \frac{\sqrt{x} - \sqrt{x-1}}{\sqrt{x} - \sqrt{x-1}} dx \]
The denominator becomes \((\sqrt{x})^2 - (\sqrt{x-1})^2 = x - (x-1) = x - x + 1 = 1\).

The integral simplifies to: \[ \int (\sqrt{x} - \sqrt{x-1}) dx \]
This can be split into two separate integrals: \[ \int \sqrt{x} dx - \int \sqrt{x-1} dx = \int x^{1/2} dx - \int (x-1)^{1/2} dx \]
Now, apply the power rule for integration to each term: \[ \frac{x^{1/2 + 1}}{1/2 + 1} - \frac{(x-1)^{1/2 + 1}}{1/2 + 1} + C \] \[ \frac{x^{3/2}}{3/2} - \frac{(x-1)^{3/2}}{3/2} + C \] \[ \frac{2}{3}x^{3/2} - \frac{2}{3}(x-1)^{3/2} + C \]
Factor out the common term \(\frac{2}{3}\): \[ \frac{2}{3}\left(x^{3/2} - (x-1)^{3/2}\right) + C \]

Step 4: Final Answer:

The result of the integration is \(\frac{2}{3}\left(x^{3/2} - (x-1)^{3/2}\right) + C\). This corresponds to option (A).
Quick Tip: Rationalizing the denominator is a powerful technique for integrals involving sums or differences of square roots. It often simplifies the integrand dramatically, as seen here where the denominator becomes 1.

Note: The question in the provided image is hard to read. The version that correctly leads to the given answer is \(\int \frac{dx{2\cos^2 x \sqrt{\tan x}}\). The solution below is based on this corrected version.

Step 1: Understanding the Concept:

This is an integration problem that can be solved using a substitution after a trigonometric simplification. The key is to recognize that the derivative of \(\tan x\) is \(\sec^2 x\), which is related to \(1/\cos^2 x\).


Step 2: Key Formula or Approach:

1. Rewrite the integrand using the identity \(\sec x = \frac{1}{\cos x}\).
2. Use the method of u-substitution. Let \(u = \tan x\), which implies \(du = \sec^2 x dx\).
3. Integrate the resulting expression with respect to \(u\).
4. Substitute back to express the result in terms of \(x\).


Step 3: Detailed Explanation:

The integral is \(\int \frac{dx}{2\cos^2 x \sqrt{\tan x}}\).

We can rewrite \(\frac{1}{\cos^2 x}\) as \(\sec^2 x\). \[ \int \frac{\sec^2 x}{2\sqrt{\tan x}} dx \]
Let's make the substitution: \[ u = \tan x \]
Then, the differential is: \[ du = \sec^2 x dx \]
Substitute \(u\) and \(du\) into the integral: \[ \int \frac{1}{2\sqrt{u}} du = \frac{1}{2} \int u^{-1/2} du \]
Now, use the power rule for integration: \[ \frac{1}{2} \left( \frac{u^{-1/2 + 1}}{-1/2 + 1} \right) + C = \frac{1}{2} \left( \frac{u^{1/2}}{1/2} \right) + C = \frac{1}{2} (2\sqrt{u}) + C = \sqrt{u} + C \]
Finally, substitute back \(u = \tan x\): \[ \sqrt{\tan x} + C \]

Step 4: Final Answer:

The result of the integration is \(\sqrt{\tan x} + C\). This corresponds to option (B) (and A, which is identical).
Quick Tip: When you see an integral with both \(\tan x\) and \(\sec^2 x\) (or \(1/\cos^2 x\)), the substitution \(u = \tan x\) is almost always the correct approach.

Step 1: Understanding the Concept:

This is an initial value problem. We are given the derivative of a function and a specific point that the function passes through. We need to find the original function by performing antidifferentiation (integration) and then using the given point to solve for the constant of integration.


Step 2: Key Formula or Approach:

1. Find the general antiderivative \(f(x)\) by integrating \(f'(x)\): \(f(x) = \int f'(x) dx\).
2. Use the initial condition \(f(1)=0\) to find the value of the constant of integration, \(C\).
3. Write the specific function \(f(x)\) with the calculated value of \(C\).


Step 3: Detailed Explanation:

We are given \(f'(x) = 3x^2 - \frac{2}{x^3} = 3x^2 - 2x^{-3}\).

1. Integrate \(f'(x)\):
\[ f(x) = \int (3x^2 - 2x^{-3}) dx \]
Using the power rule for integration \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\): \[ f(x) = 3 \left(\frac{x^3}{3}\right) - 2 \left(\frac{x^{-2}}{-2}\right) + C \] \[ f(x) = x^3 - (-1)x^{-2} + C \] \[ f(x) = x^3 + x^{-2} + C = x^3 + \frac{1}{x^2} + C \]
2. Use the initial condition to find C:

We are given that \(f(1) = 0\). Substitute \(x=1\) into the expression for \(f(x)\): \[ 0 = (1)^3 + \frac{1}{(1)^2} + C \] \[ 0 = 1 + 1 + C \] \[ 0 = 2 + C \implies C = -2 \]
3. Write the final function:

Substitute \(C=-2\) back into the general form of \(f(x)\): \[ f(x) = x^3 + \frac{1}{x^2} - 2 \]

Step 4: Final Answer:

The function is \(f(x) = x^3 + \frac{1}{x^2} - 2\). This corresponds to option (D).
Quick Tip: Don't forget the constant of integration \(C\) when finding an indefinite integral. The initial condition is provided specifically to allow you to solve for this constant.

Step 1: Understanding the Concept:

This integral can be solved using integration by parts. The structure of the integrand, being a difference of two related terms, suggests that integrating one part might cancel out the other part.


Step 2: Key Formula or Approach:

The formula for integration by parts is \(\int u dv = uv - \int v du\). We will split the integral into two and apply integration by parts to the first term, \(\int \frac{1}{\log x} dx\).


Step 3: Detailed Explanation:

Let's evaluate the integral of the first term, \(\int \frac{1}{\log x} dx\), using integration by parts.
Let \(u = \frac{1}{\log x}\) and \(dv = dx\).
Then we find \(du\) and \(v\): \[ du = \frac{d}{dx}\left((\log x)^{-1}\right) = -1(\log x)^{-2} \cdot \frac{1}{x} dx = -\frac{1}{x(\log x)^2} dx \] \[ v = \int dx = x \]
Now apply the integration by parts formula: \[ \int \frac{1}{\log x} dx = uv - \int v du = \left(\frac{1}{\log x}\right)(x) - \int x \left(-\frac{1}{x(\log x)^2}\right) dx \] \[ \int \frac{1}{\log x} dx = \frac{x}{\log x} + \int \frac{1}{(\log x)^2} dx \]
Now, let's substitute this result back into the original problem: \[ \int \left(\frac{1}{\log x} - \frac{1}{(\log x)^2}\right) dx = \int \frac{1}{\log x} dx - \int \frac{1}{(\log x)^2} dx \] \[ = \left(\frac{x}{\log x} + \int \frac{1}{(\log x)^2} dx\right) - \int \frac{1}{(\log x)^2} dx \]
The two integral terms cancel each other out: \[ = \frac{x}{\log x} + C \]

Step 4: Final Answer:

The result of the integration is \(\frac{x}{\log x} + C\). This corresponds to option (D).
Quick Tip: This problem is an example of a common integration by parts pattern where integrating one part of a sum or difference leads to the cancellation of the other part. Another famous example is \(\int e^x(f(x)+f'(x))dx = e^x f(x)+C\).

Step 1: Understanding the Concept:

This integral involves the square root of a quadratic expression. The standard method for solving this is to first complete the square for the quadratic, and then use a standard integration formula.


Step 2: Key Formula or Approach:

1. Complete the square for \(x^2+2x+3\).
2. The integral will be in the form \(\int \sqrt{u^2 + a^2} du\).
3. Use the standard formula: \(\int \sqrt{u^2+a^2} du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\log|u+\sqrt{u^2+a^2}| + C\).


Step 3: Detailed Explanation:

1. Complete the square:

The quadratic expression is \(x^2 + 2x + 3\). \[ x^2 + 2x + 3 = (x^2 + 2x + 1) - 1 + 3 = (x+1)^2 + 2 \]
2. Rewrite the integral:
\[ \int \sqrt{(x+1)^2 + 2} \ dx \]
This is of the form \(\int \sqrt{u^2+a^2} du\) where \(u=x+1\) and \(a^2=2\), so \(a=\sqrt{2}\).
Since \(du = dx\), we can directly apply the formula.

3. Apply the standard formula:
\[ \int \sqrt{u^2+a^2} du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\log|u+\sqrt{u^2+a^2}| + C \]
Substitute \(u=x+1\) and \(a^2=2\): \[ \frac{x+1}{2}\sqrt{(x+1)^2+2} + \frac{2}{2}\log|(x+1)+\sqrt{(x+1)^2+2}| + C \]
Simplify the terms: \[ \frac{x+1}{2}\sqrt{x^2+2x+1+2} + \log|(x+1)+\sqrt{x^2+2x+1+2}| + C \] \[ \frac{x+1}{2}\sqrt{x^2+2x+3} + \log|(x+1)+\sqrt{x^2+2x+3}| + C \]

Step 4: Final Answer:

The result of the integration is \(\frac{x+1}{2}\sqrt{x^2+2x+3}+\log|(x+1)+\sqrt{x^2+2x+3}|+C\). This corresponds to option (D).
Quick Tip: Memorizing the three standard integral forms for \(\int \sqrt{x^2 \pm a^2} dx\) and \(\int \sqrt{a^2 - x^2} dx\) is crucial for solving these types of problems quickly in an exam setting.

Note: The question in the provided image is illegible. The version reconstructed here is a standard type that matches the given answer. The solution is based on this standard form.

Step 1: Understanding the Concept:

This problem uses a key property of definite integrals over an interval \([a, b]\). This property is particularly useful for integrands that have a certain symmetry with respect to the midpoint of the interval.


Step 2: Key Formula or Approach:

We will use the property: \[ \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \]
This property is sometimes called the "King Property" of definite integrals.


Step 3: Detailed Explanation:

Let the given integral be \(I\). \[ I = \int_{2^{4} \frac{e^{x^2}}{e^{x^2} + e^{(6-x)^2}} dx \quad \cdots (1) \]
The limits of integration are \(a=2\) and \(b=4\). So, \(a+b = 6\).

We apply the property by replacing \(x\) with \(a+b-x = 6-x\) everywhere in the integrand: \[ I = \int_{2}^{4} \frac{e^{(6-x)^2}}{e^{(6-x)^2} + e^{(6-(6-x))^2}} dx \]
Simplifying the exponent in the denominator: \(6-(6-x) = x\). \[ I = \int_{2}^{4} \frac{e^{(6-x)^2}}{e^{(6-x)^2} + e^{x^2}} dx \quad \cdots (2) \]
Now, we add the two equations for \(I\), equation (1) and equation (2): \[ I + I = \int_{2}^{4} \frac{e^{x^2}}{e^{x^2} + e^{(6-x)^2}} dx + \int_{2}^{4} \frac{e^{(6-x)^2}}{e^{(6-x)^2} + e^{x^2}} dx \]
Since the integrals have the same limits, we can combine the integrands: \[ 2I = \int_{2}^{4} \left( \frac{e^{x^2} + e^{(6-x)^2}}{e^{x^2} + e^{(6-x)^2}} \right) dx \]
The integrand simplifies to 1: \[ 2I = \int_{2}^{4} 1 \ dx \]
Now, we evaluate the simple integral: \[ 2I = [x]_2^4 = 4 - 2 = 2 \] \[ 2I = 2 \implies I = 1 \]

Step 4: Final Answer:

The value of the definite integral is 1. This corresponds to option (B).
Quick Tip: Look for integrals of the form \(\int_a^b \frac{f(x)}{f(x)+f(a+b-x)} dx\). They almost always evaluate to \(\frac{b-a}{2}\). In this case, \(a=2, b=4\), so the answer is \(\frac{4-2}{2} = 1\). This shortcut is extremely useful for competitive exams.

Step 1: Understanding the Concept:

This problem involves evaluating a definite integral over a symmetric interval \([-a, a]\). We can simplify the calculation by using the properties of even and odd functions.


Step 2: Key Formula or Approach:

For a definite integral over a symmetric interval \([-a, a]\):

If \(f(x)\) is an odd function (i.e., \(f(-x) = -f(x)\)), then \(\int_{-a}^{a} f(x) dx = 0\).
If \(f(x)\) is an even function (i.e., \(f(-x) = f(x)\)), then \(\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\).

We will check each term of the integrand for odd or even symmetry.


Step 3: Detailed Explanation:

Let's analyze each term in the integrand \(f(x) = x^3 + x\cos^2(2x) + \tan^5 x + 3\).

\(g(x) = x^3\): \(g(-x) = (-x)^3 = -x^3 = -g(x)\). This is an odd function.
\(h(x) = x\cos^2(2x)\): \(h(-x) = (-x)\cos^2(2(-x)) = -x\cos^2(-2x)\). Since \(\cos(-\theta) = \cos(\theta)\), we have \(h(-x) = -x\cos^2(2x) = -h(x)\). This is an odd function.
\(k(x) = \tan^5 x\): \(k(-x) = (\tan(-x))^5 = (-\tan x)^5 = -\tan^5 x = -k(x)\). This is an odd function.
\(p(x) = 3\): \(p(-x) = 3 = p(x)\). This is an even function.

The integral can be split into the sum of integrals of each term: \[ \int_{-a}^{a} x^3 dx + \int_{-a}^{a} x\cos^2(2x) dx + \int_{-a}^{a} \tan^5 x dx + \int_{-a}^{a} 3 dx \]
The integrals of the first three terms (the odd functions) are all zero. \[ 0 + 0 + 0 + \int_{-a}^{a} 3 dx \]
We only need to evaluate the integral of the even function part: \[ \int_{-a}^{a} 3 dx = [3x]_{-a}^{a} = 3(a) - 3(-a) = 3a + 3a = 6a \]
Alternatively, using the even function property: \[ \int_{-a}^{a} 3 dx = 2 \int_{0}^{a} 3 dx = 2[3x]_0^a = 2(3a - 0) = 6a \]

Step 4: Final Answer:

The value of the definite integral is \(6a\). This corresponds to option (D).
Quick Tip: Whenever you see a definite integral with symmetric limits like \([-a, a]\), your first thought should be to check for even and odd functions. This property can simplify complex-looking integrals to trivial ones.

Step 1: Understanding the Concept:

To find the area of the region bounded by a curve and the x-axis, we need to compute a definite integral of the function. The limits of integration are the x-coordinates of the points where the curve intersects the x-axis.


Step 2: Key Formula or Approach:

1. Find the limits of integration by setting \(y=0\) and solving for \(x\).
2. The area \(A\) is given by the definite integral \(A = \int_a^b y \ dx\), where \(a\) and \(b\) are the intersection points. We need to ensure \(y \ge 0\) on the interval \([a, b]\).


Step 3: Detailed Explanation:

The curve is given by \(y = 3x - x^2\).

1. Find the limits of integration:

Set \(y=0\) to find where the curve intersects the x-axis: \[ 3x - x^2 = 0 \] \[ x(3 - x) = 0 \]
The intersection points are \(x=0\) and \(x=3\). These will be our limits of integration, \(a=0\) and \(b=3\).

The function \(y = -x^2 + 3x\) is a downward-opening parabola. Between its roots \(x=0\) and \(x=3\), the function value is positive, so the curve is above the x-axis.

2. Set up and evaluate the integral:

The area \(A\) is: \[ A = \int_{0}^{3} (3x - x^2) dx \]
Find the antiderivative: \[ A = \left[ \frac{3x^2}{2} - \frac{x^3}{3} \right]_0^3 \]
Evaluate at the limits: \[ A = \left( \frac{3(3)^2}{2} - \frac{(3)^3}{3} \right) - \left( \frac{3(0)^2}{2} - \frac{(0)^3}{3} \right) \] \[ A = \left( \frac{3 \cdot 9}{2} - \frac{27}{3} \right) - (0) \] \[ A = \frac{27}{2} - 9 \]
Find a common denominator: \[ A = \frac{27}{2} - \frac{18}{2} = \frac{9}{2} \]

Step 4: Final Answer:

The area of the region is \(\frac{9}{2}\) square units. This corresponds to option (E).
Quick Tip: For a parabolic segment bounded by the x-axis, defined by the quadratic \(y = k(x-a)(x-b)\), the area is given by the formula \(Area = \frac{1}{6}|k|(b-a)^3\). Here, \(y=-x(x-3)\), so \(k=-1, a=0, b=3\). Area = \(\frac{1}{6}|-1|(3-0)^3 = \frac{1}{6}(27) = \frac{9}{2}\).

Note: The problem statement is slightly ambiguous. The standard interpretation for this type of question is the area in the first quadrant bounded by \(y=|x|\), the vertical line \(x=4\), and the x-axis (\(y=0\)).

Step 1: Understanding the Concept:

We need to find the area of a geometric region defined by given boundaries. Sketching the region is the best way to understand its shape and determine how to calculate its area, either by using geometric formulas or by setting up a definite integral.


Step 2: Key Formula or Approach:

Method 1: Geometric Formula

The described region is a right-angled triangle. The area of a triangle is \(\frac{1{2} \times base \times height\).

Method 2: Integration

The area is given by the integral of the function defining the upper boundary from the starting x-value to the ending x-value.


Step 3: Detailed Explanation:

Let's sketch the boundaries:

\(y=|x|\) is a V-shaped graph. For \(x \ge 0\) (which includes the region up to \(x=4\)), this is simply the line \(y=x\).
\(x=4\) is a vertical line.
The x-axis is the line \(y=0\).

The region is a right-angled triangle in the first quadrant with vertices at:

The origin \((0,0)\).
The x-intercept of the line \(x=4\), which is \((4,0)\).
The intersection of \(y=x\) and \(x=4\), which is \((4,4)\).

Using Geometry:

The base of the triangle lies along the x-axis from \(x=0\) to \(x=4\), so the base length is 4.
The height of the triangle is the vertical distance from the base to the vertex \((4,4)\), so the height is 4. \[ Area = \frac{1}{2} \times base \times height = \frac{1}{2} \times 4 \times 4 = 8 \]
Using Integration:

The area is the integral of the upper curve (\(y=x\)) from \(x=0\) to \(x=4\). \[ Area = \int_{0}^{4} x \ dx = \left[\frac{x^2}{2}\right]_0^4 = \frac{4^2}{2} - \frac{0^2}{2} = \frac{16}{2} - 0 = 8 \]

Step 4: Final Answer:

The area of the region is 8 square units. This corresponds to option (C).
Quick Tip: For simple shapes like triangles, rectangles, and trapezoids, using basic geometry formulas is often much faster and less error-prone than setting up and evaluating an integral. Always sketch the region first.

Step 1: Understanding the Concept:

The order of a differential equation is the order of the highest derivative present in the equation. The degree of a differential equation is the highest power of the highest order derivative, after the equation has been cleared of any radicals or fractional exponents as far as the derivatives are concerned.


Step 2: Key Formula or Approach:

1. Identify the highest order derivative to find the order.
2. Rearrange the equation to eliminate any radicals or fractional powers involving the derivatives.
3. Identify the highest power of the highest order derivative to find the degree.


Step 3: Detailed Explanation:

The provided answer key states the order is 2 and the degree is 4. For this to be correct, the differential equation must have \(\frac{d^2y}{dx^2}\) as its highest derivative, and its highest power must be 4 after clearing radicals. The equation in the image is not clear, but let us assume the intended equation was: \[ \left(\frac{d^2y}{dx^2}\right)^2 + y = \sqrt{\frac{dy}{dx}} \]
1. Finding the Order: The highest order derivative in this assumed equation is \(\frac{d^2y}{dx^2}\). Therefore, the order is 2.
2. Finding the Degree: To find the degree, we must eliminate the square root. We rearrange the equation to isolate the radical:
\[ \left(\frac{d^2y}{dx^2}\right)^2 + y - \sqrt{\frac{dy}{dx}} = 0 \]
This is not a polynomial in derivatives. Let's try another plausible intended equation like:
\[ \sqrt{1+\left(\frac{dy}{dx}\right)^3} = \left(\frac{d^2y}{dx^2}\right)^2 \]
To clear the radical, we square both sides:
\[ 1+\left(\frac{dy}{dx}\right)^3 = \left(\left(\frac{d^2y}{dx^2}\right)^2\right)^2 = \left(\frac{d^2y}{dx^2}\right)^4 \]
The equation is now a polynomial in its derivatives. The highest order derivative is \(\frac{d^2y}{dx^2}\), and its highest power is 4.
Therefore, the order is 2 and the degree is 4.


Step 4: Final Answer:

Based on the assumption of the intended equation that matches the answer key, the order is 2 and the degree is 4. This corresponds to option (B).
Quick Tip: When finding the degree of a differential equation, it is essential to first make the equation a polynomial in the derivatives. This often involves squaring or raising the equation to a power to eliminate radicals and fractional exponents.

Step 1: Understanding the Concept:

The given differential equation is a first-order linear differential equation. It can be solved by recognizing the left side as the derivative of a product, or by using the integrating factor method.


Step 2: Key Formula or Approach:

The equation \(x\frac{dy}{dx} + y = e^x\) can be solved by observing that the left side is the result of the product rule for differentiation applied to \(xy\). \[ \frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + 1 \cdot y \]
So the equation can be rewritten as \(\frac{d}{dx}(xy) = e^x\).


Step 3: Detailed Explanation:

Starting with the equation: \[ x\frac{dy}{dx} + y = e^x \]
Recognize the left side as the derivative of \(xy\): \[ \frac{d}{dx}(xy) = e^x \]
Integrate both sides with respect to \(x\): \[ \int \frac{d}{dx}(xy) dx = \int e^x dx \] \[ xy = e^x + C \]
where C is the constant of integration.

Now, solve for \(y\) by dividing by \(x\): \[ y = \frac{e^x + C}{x} = \frac{e^x}{x} + \frac{C}{x} \]
This matches option (D). The question is mathematically sound and option (D) is the correct solution. The reason for its cancellation in an exam could be due to a printing error in the original paper or options that is not reflected in this version.


Step 4: Final Answer:

The correct solution to the differential equation is \(y = \frac{e^x}{x} + \frac{C}{x}\). While this matches option (D), we acknowledge the provided answer key states "Question Cancelled".
Quick Tip: Always check if a first-order linear differential equation of the form \(P(x)y' + P'(x)y = Q(x)\) can be simplified as \(\frac{d}{dx}(P(x)y) = Q(x)\). This is a quick way to solve it without calculating the integrating factor explicitly.

Step 1: Understanding the Concept:

In linear programming, the optimal value (minimum or maximum) of the objective function occurs at one or more of the corner points of the feasible region. If the optimal value occurs at two different corner points, it also occurs at every point on the line segment connecting them. This implies that the value of the objective function must be the same at these two points.


Step 2: Key Formula or Approach:

1. Evaluate the objective function \(Z = ax + by\) at the two specified points, (15,15) and (0,20).
2. Set the two resulting expressions for Z equal to each other.
3. Solve the resulting equation to find the relationship between a and b.
4. Note: The problem statement is flawed, as the value at these points is not the minimum for the feasible region. However, the intended method is to equate the values at the specified points.


Step 3: Detailed Explanation:

The objective function is \(Z = ax + by\).
We are told the minimum value of Z occurs at both (15,15) and (0,20). This means the value of Z must be the same at these two points.
Value of Z at (15,15): \[ Z_1 = a(15) + b(15) = 15a + 15b \]
Value of Z at (0,20): \[ Z_2 = a(0) + b(20) = 20b \]
Set \(Z_1 = Z_2\): \[ 15a + 15b = 20b \]
Subtract \(15b\) from both sides: \[ 15a = 20b - 15b \] \[ 15a = 5b \]
Divide both sides by 5: \[ 3a = b \]
This is the condition required.
(For completeness, let's check why the problem is flawed. Let \(b=3a\).
Z at (0,10) = \(10b = 30a\).
Z at (5,5) = \(5a+5b = 5a+15a = 20a\).
Z at (15,15) = \(15a+15b = 15a+45a = 60a\).
Z at (0,20) = \(20b = 20(3a) = 60a\).
The minimum value is actually \(20a\) at point (5,5), not \(60a\) at points (15,15) and (0,20). The question should have asked for the condition for the maximum to occur at these two points.)


Step 4: Final Answer:

Following the intended logic of the question, the condition on a and b is \(3a = b\). This corresponds to option (D).
Quick Tip: In LPP, if an objective function has the same optimal value at two corner points, its slope is the same as the slope of the line segment (boundary) connecting those two points. The slope of the line \(ax+by=Z\) is \(-a/b\). The slope of the segment from (15,15) to (0,20) is \((20-15)/(0-15) = 5/(-15) = -1/3\). So, \(-a/b = -1/3 \implies a/b=1/3 \implies 3a=b\).

Step 1: Understanding the Concept:

Percentage error is a measure of the uncertainty in a measurement relative to the size of the measurement itself. It is calculated by dividing the absolute error by the measured value and multiplying by 100. The absolute error is related to the least count of the measuring instrument.


Step 2: Key Formula or Approach:

1. Identify the measured value (\(L\)).
2. Identify the absolute error (\(\Delta L\)). For a single measurement, the maximum possible error is typically taken as the least count of the instrument.
3. Calculate the percentage error using the formula: \(Percentage Error = \frac{\Delta L}{L} \times 100%\).


Step 3: Detailed Explanation:

The measured value is \(L = 50\) cm.

The smallest division on the metre stick is the least count, which is given as 1 mm.

We need to convert the least count to the same units as the measurement.
Least Count (\(\Delta L\)) = 1 mm = 0.1 cm.

Now we calculate the percentage error: \[ Percentage Error = \frac{Least Count}{Measured Value} \times 100% \] \[ Percentage Error = \frac{0.1 cm}{50 cm} \times 100% \] \[ Percentage Error = \frac{0.1 \times 100}{50} % = \frac{10}{50} % = \frac{1}{5} % = 0.2% \]

Step 4: Final Answer:

The percentage error involved in the measurement is 0.2%. This corresponds to option (C).
Quick Tip: Ensure that the absolute error and the measured value are in the same units before calculating the percentage error. A common mistake is to divide values with different units.

Step 1: Understanding the Concept:

When adding or subtracting measurements, the result must be rounded to the same number of decimal places as the measurement with the fewest decimal places. The precision of the result is limited by the least precise measurement.


Step 2: Key Formula or Approach:

1. Convert all measurements to the same unit.
2. Identify the number of decimal places for each measurement.
3. Add the measurements.
4. Round the final result to the number of decimal places of the least precise measurement.


Step 3: Detailed Explanation:

The problem statement "200 m" is ambiguous regarding its precision. In scientific contexts, unless indicated otherwise (e.g., 200.), trailing zeros in a number without a decimal point are often ambiguous. However, to match the provided correct answer, we must assume that "200 m" is precise to one decimal place, i.e., it should be interpreted as 200.0 m.
1. Convert to meters:
\(200 mm = 200 \times 10^{-3} m = 0.200 m\)
2. Identify decimal places:
First measurement: 200.0 m (assumed precision) has 1 decimal place.
Second measurement: 0.200 m has 3 decimal places.
3. Add the values:
\[ 200.0 m + 0.200 m = 200.200 m \]
4. Round the result:
The least number of decimal places is 1. So, we must round the result to one decimal place.
\[ 200.200 m \approx 200.2 m \]

Step 4: Final Answer:

With the necessary assumption about the precision of "200 m", the value is 200.2 m. This corresponds to option (A).
Quick Tip: Remember the rules for significant figures: - For addition/subtraction, the result is limited by the number of decimal places. - For multiplication/division, the result is limited by the number of significant figures. - Ambiguity in trailing zeros is common; in exam questions, choose the interpretation that leads to one of the given options.

Step 1: Understanding the Concept:

The angle between a vector and a coordinate axis can be found using the dot product. The direction of the y-axis is represented by the unit vector \(\hat{j}\).


Step 2: Key Formula or Approach:

The angle \(\theta\) between two vectors \(\vec{A}\) and \(\vec{B}\) is given by the dot product formula: \[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]
The direction cosines of a vector \(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\) also give the cosines of the angles with the axes: \(\cos\beta = \frac{A_y}{|\vec{A}|}\), where \(\beta\) is the angle with the y-axis.


Step 3: Detailed Explanation:

Let \(\vec{A} = \hat{i} + \hat{j} + \hat{k}\). Let \(\beta\) be the angle it makes with the y-axis.
The vector representing the direction of the y-axis is \(\vec{B} = \hat{j} = 0\hat{i} + 1\hat{j} + 0\hat{k}\).

First, calculate the magnitude of \(\vec{A}\): \[ |\vec{A}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]
The magnitude of the unit vector \(\hat{j}\) is \(|\hat{j}| = 1\).

Next, calculate the dot product \(\vec{A} \cdot \hat{j}\): \[ \vec{A} \cdot \hat{j} = (\hat{i} + \hat{j} + \hat{k}) \cdot (0\hat{i} + 1\hat{j} + 0\hat{k}) = (1)(0) + (1)(1) + (1)(0) = 1 \]
Now, use the dot product formula to find \(\cos\beta\): \[ \cos \beta = \frac{\vec{A} \cdot \hat{j}}{|\vec{A}| |\hat{j}|} = \frac{1}{\sqrt{3} \cdot 1} = \frac{1}{\sqrt{3}} \]
Therefore, the angle is: \[ \beta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

Step 4: Final Answer:

The angle subtended by the vector with the y-axis is \(\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\). This corresponds to option (C).
Quick Tip: For any vector \(\vec{V} = V_x\hat{i} + V_y\hat{j} + V_z\hat{k}\), the cosine of the angle it makes with the y-axis is simply its y-component divided by its magnitude: \(\cos\beta = V_y / |\vec{V}|\). This is a direct application of the concept of direction cosines.

Step 1: Understanding the Concept:

This question relates motion parameters to their graphical representation. The key relationships for a velocity-time (v-t) graph are that its slope represents acceleration and its y-intercept represents the initial velocity.


Step 2: Detailed Explanation:

1. Constant Retardation: "Retardation" means the velocity is decreasing, which implies negative acceleration. "Constant" means the acceleration does not change over time.
2. Slope of v-t graph: Since the slope of a v-t graph represents acceleration, a constant acceleration corresponds to a constant slope. A graph with a constant slope is a straight line.
3. Sign of the slope: Since the acceleration is negative (retardation), the slope of the straight line must be negative. A straight line with a non-zero, non-infinite slope is called an oblique line.
4. Initial Velocity: The problem states the initial velocity is "non-zero". This means at time \(t=0\), the velocity \(v\) is not zero. Graphically, this means the line does not start from the origin but from some point on the velocity axis (y-axis).
Combining these points, the graph is an oblique (slanted) straight line with a negative slope, starting from a non-zero value on the velocity axis.


Step 4: Final Answer:

The velocity-time graph is an oblique straight line with a negative slope. This corresponds to option (D).
Quick Tip: Memorize the relationships for motion graphs: - Position-Time (x-t): Slope = velocity. - Velocity-Time (v-t): Slope = acceleration; Area under curve = displacement. - Acceleration-Time (a-t): Area under curve = change in velocity.

Step 1: Understanding the Concept:

This problem deals with the properties of projectile motion. Specifically, it involves the formula for the time of flight and how it depends on the angle of projection.


Step 2: Key Formula or Approach:

The time of flight (\(T\)) for a projectile launched with initial speed \(u\) at an angle \(\alpha\) with the horizontal is given by: \[ T = \frac{2u\sin\alpha}{g} \]
where \(g\) is the acceleration due to gravity.


Step 3: Detailed Explanation:

Let the initial speed for both bodies be \(u\).

For the first body:
The angle of projection is \(\alpha_1 = \theta\).
The time of flight \(T_1\) is: \[ T_1 = \frac{2u\sin\theta}{g} \]
For the second body:
The angle of projection is \(\alpha_2 = 90^\circ - \theta\).
The time of flight \(T_2\) is: \[ T_2 = \frac{2u\sin(90^\circ - \theta)}{g} \]
Using the trigonometric identity \(\sin(90^\circ - \theta) = \cos\theta\), we get: \[ T_2 = \frac{2u\cos\theta}{g} \]
Finding the ratio \(T_1/T_2\): \[ \frac{T_1}{T_2} = \frac{\frac{2u\sin\theta}{g}}{\frac{2u\cos\theta}{g}} \]
Cancel the common terms \(\frac{2u}{g}\) from the numerator and denominator: \[ \frac{T_1}{T_2} = \frac{\sin\theta}{\cos\theta} \]
Using the identity \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), we get: \[ \frac{T_1}{T_2} = \tan\theta \]

Step 4: Final Answer:

The ratio between their times of flight is \(\tan\theta\). This corresponds to option (E).
Quick Tip: Remember the properties for complementary projection angles (\(\theta\) and \(90^\circ - \theta\)) with the same initial speed: - Ranges are equal. - Ratio of times of flight is \(\tan\theta\). - Ratio of maximum heights is \(\tan^2\theta\).

Step 1: Understanding the Concept:

According to Newton's third law, the gun exerts a force on the bullets to accelerate them, and the bullets exert an equal and opposite recoil force on the gun. To keep the gun in position, an external force equal in magnitude to this recoil force must be applied. This force is equal to the rate of change of momentum of the bullets being fired.


Step 2: Key Formula or Approach:

The force is given by the rate of change of momentum: \(F = \frac{\Delta p}{\Delta t}\).
The total momentum imparted in a time \(\Delta t\) is the number of bullets fired in that time multiplied by the momentum of a single bullet. \[ F = (rate of firing) \times (momentum of one bullet) = (\frac{n}{t}) \times (mv) \]

Step 3: Detailed Explanation:

First, we list the given values and convert them to SI units.

Mass of one bullet, \(m = 40 g = 0.040 kg\).
Speed of bullet, \(v = 300 m/s\).
Rate of firing, \(R = 25 bullets/minute\). We need to convert this to bullets per second.
\[ R = \frac{25 bullets}{1 minute} \times \frac{1 minute}{60 seconds} = \frac{25}{60} bullets/s \]

The mass of the gun (5 kg) is not needed to calculate the force.

Now, calculate the force: \[ F = R \times m \times v \] \[ F = \left(\frac{25}{60}\right) \times (0.040) \times (300) \] \[ F = \frac{25}{60} \times (40 \times 10^{-3}) \times 300 \] \[ F = \frac{25}{60} \times (12000 \times 10^{-3}) = \frac{25}{60} \times 12 \] \[ F = \frac{25 \times 12}{60} = \frac{25}{5} = 5 N \]

Step 4: Final Answer:

The force required to keep the gun in position is 5 N. This corresponds to option (E).
Quick Tip: In problems involving continuous streams of particles (like bullets from a gun or water from a hose), the force is most easily calculated as the product of the rate of mass flow (\(dm/dt\)) and the velocity (\(v\)). Here, \(dm/dt = R \times m\).

Step 1: Understanding the Concept:

This problem is a direct application of Newton's second law of motion, \(F=ma\). Since the force is given as a vector, we first need to find its magnitude. The acceleration is given as a scalar, which represents its magnitude.


Step 2: Key Formula or Approach:

1. Find the magnitude of the force vector, \(|\vec{F}|\).
2. Use Newton's second law in scalar form: \(|\vec{F}| = m \cdot |\vec{a}|\).
3. Solve for the mass, \(m\).


Step 3: Detailed Explanation:

The force vector is given as \(\vec{F} = \hat{i} + 2\hat{j} - 2\hat{k}\).

1. Calculate the magnitude of the force: \[ |\vec{F}| = \sqrt{(1)^2 + (2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 N \]
The magnitude of the acceleration is given as \(|\vec{a}| = a = 2 m/s^2\).

2. Use Newton's second law: \[ F = ma \] \[ 3 N = m \cdot (2 m/s^2) \]
3. Solve for mass \(m\): \[ m = \frac{3}{2} = 1.5 kg \]

Step 4: Final Answer:

The mass of the body is 1.5 kg. This corresponds to option (D).
Quick Tip: When force and acceleration are given in vector form (\(\vec{F}\) and \(\vec{a}\)), they must be parallel. The mass \(m\) is the scalar of proportionality: \(\vec{F} = m\vec{a}\). The problem simplifies here by providing the magnitude of the acceleration directly.

Step 1: Understanding the Concept:

This problem is an application of the Work-Energy Theorem. The theorem states that the net work done on an object is equal to the change in its kinetic energy.


Step 2: Key Formula or Approach:

1. The Work-Energy Theorem: \(W_{net} = \Delta K = K_{final} - K_{initial}\).
2. The work done by a constant force \(F\) over a distance \(d\) is \(W = Fd\cos\theta\).
3. We will equate the work done by the stopping force to the change in kinetic energy and solve for the distance \(d\).


Step 3: Detailed Explanation:

The initial kinetic energy of the body is \(K_{initial} = E\).

The body is stopped, so its final kinetic energy is \(K_{final} = 0\).

The change in kinetic energy is: \[ \Delta K = K_{final} - K_{initial} = 0 - E = -E \]
The stopping force \(F\) acts opposite to the direction of motion. If the stopping distance is \(d\), the angle \(\theta\) between the force and the displacement is 180°.
The work done by the stopping force is: \[ W = Fd\cos(180^\circ) = Fd(-1) = -Fd \]
According to the Work-Energy Theorem, \(W = \Delta K\): \[ -Fd = -E \]
Solving for the stopping distance \(d\): \[ d = \frac{E}{F} \]

Step 4: Final Answer:

The stopping distance is \(\frac{E}{F}\). This corresponds to option (C).
Quick Tip: The Work-Energy Theorem is a very powerful tool for problems involving force, distance, and changes in speed or kinetic energy. It often provides a more direct solution than using Newton's laws and kinematic equations.

Step 1: Understanding the Concept:

The work done by an external force to change the elongation of a spring is equal to the change in the elastic potential energy stored in the spring. The potential energy stored in a spring is a function of its elongation.


Step 2: Key Formula or Approach:

1. The elastic potential energy (\(U\)) stored in a spring with spring constant \(K\) and elongation \(x\) from its equilibrium position is \(U = \frac{1}{2}Kx^2\).
2. The work done (\(W\)) by an applied force is the change in potential energy: \(W = \Delta U = U_{final} - U_{initial}\).


Step 3: Detailed Explanation:

The initial elongation is \(x_1\). The initial potential energy stored in the spring is: \[ U_{initial} = \frac{1}{2}Kx_1^2 \]
The final elongation is \(x_2\). The final potential energy stored in the spring is: \[ U_{final} = \frac{1}{2}Kx_2^2 \]
The work done by the applied force to cause this change is equal to the change in the spring's potential energy: \[ W = U_{final} - U_{initial} = \frac{1}{2}Kx_2^2 - \frac{1}{2}Kx_1^2 \]
Factoring out the common term \(\frac{1}{2}K\): \[ W = \frac{1}{2}K(x_2^2 - x_1^2) \]

Step 4: Final Answer:

The work done is \(\frac{1}{2}K(x_2^2 - x_1^2)\). This corresponds to option (A).
Quick Tip: Be careful to distinguish between the work done by the spring and the work done by the applied force. The work done by the spring force is \(-\Delta U\), while the work done by the external applied force is \(+\Delta U\).

Step 1: Understanding the Concept:

The angular momentum of a particle about a point is conserved if and only if the net external torque about that same point is zero. We need to find the point about which the torque acting on a particle in uniform circular motion is zero.


Step 2: Key Formula or Approach:

1. Angular momentum is conserved when net torque \(\vec{\tau} = 0\).
2. Torque is defined as \(\vec{\tau} = \vec{r} \times \vec{F}\), where \(\vec{r}\) is the position vector from the reference point to the point of force application, and \(\vec{F}\) is the force.
3. In uniform circular motion, the net force is the centripetal force, which is always directed towards the center of the circle.


Step 3: Detailed Explanation:

Let a particle undergo uniform circular motion in a plane. The only force acting on the particle is the centripetal force, \(\vec{F}_c\). This force is always directed from the particle towards the center of the circle.

To check for conservation of angular momentum, we need to calculate the torque \(\vec{\tau} = \vec{r} \times \vec{F}_c\) about a chosen reference point.
Let's choose the center of the circle as the reference point.
- The position vector \(\vec{r}\) points from the center to the particle.
- The force vector \(\vec{F}_c\) points from the particle to the center.
This means that the vectors \(\vec{r}\) and \(\vec{F}_c\) are anti-parallel (the angle between them is 180°).

The cross product of two parallel or anti-parallel vectors is zero. \[ \vec{\tau}_{center} = \vec{r} \times \vec{F}_c = |\vec{r}||\vec{F}_c|\sin(180^\circ)\hat{n} = 0 \]
Since the torque about the center of the circle is zero, the angular momentum of the particle about the center is conserved. If any other point is chosen, \(\vec{r}\) and \(\vec{F}_c\) will not be anti-parallel, and the torque will generally be non-zero.


Step 4: Final Answer:

The angular momentum is conserved about the centre of the circle. This corresponds to option (D).
Quick Tip: Conservation of angular momentum is a key principle for central force motion (where the force is always directed towards a fixed central point). Circular motion is a special case of central force motion.

Step 1: Understanding the Concept:

This problem applies the rotational analogue of Newton's second law, which relates torque, moment of inertia, and angular acceleration. We need to first find the angular acceleration required to stop the wheel and then calculate the corresponding torque.


Step 2: Key Formula or Approach:

1. The rotational equation of motion is \(\tau = I\alpha\), where \(\tau\) is torque, \(I\) is moment of inertia, and \(\alpha\) is angular acceleration.
2. The rotational kinematic equation relating angular velocities, acceleration, and time is \(\omega_f = \omega_i + \alpha t\).


Step 3: Detailed Explanation:

First, we list the given values and convert them to SI units.

Moment of inertia, \(I = 4 \times 10^{-3} kgm^2\).
Initial angular speed, \(\omega_i = 25 rev/s\). We convert this to rad/s.
\[ \omega_i = 25 \frac{rev}{s} \times \frac{2\pi rad}{1 rev} = 50\pi rad/s \]
Final angular speed, \(\omega_f = 0\) (since it stops).
Time interval, \(t = 10 s\).

1. Calculate angular acceleration (\(\alpha\)):
Using \(\omega_f = \omega_i + \alpha t\): \[ 0 = 50\pi + \alpha(10) \] \[ -50\pi = 10\alpha \] \[ \alpha = -\frac{50\pi}{10} = -5\pi rad/s^2 \]
The negative sign indicates deceleration (retardation).

2. Calculate the torque (\(\tau\)):
We are interested in the magnitude of the torque required. \[ |\tau| = |I\alpha| = (4 \times 10^{-3} kgm^2) \times (5\pi rad/s^2) \] \[ |\tau| = 20\pi \times 10^{-3} Nm = 2\pi \times 10^1 \times 10^{-3} Nm = 2\pi \times 10^{-2} Nm \]

Step 4: Final Answer:

The required torque is \(2\pi \times 10^{-2}\) Nm. This corresponds to option (B).
Quick Tip: Always ensure your units are consistent before plugging values into physics formulas. The most common source of error in rotational dynamics problems is forgetting to convert revolutions per minute (rpm) or revolutions per second (rev/s) to radians per second (rad/s).

Step 1: Understanding the Concept:

This question tests the fundamental definition of torque as a vector (cross) product and the geometric properties of the cross product.


Step 2: Key Formula or Approach:

The torque \(\vec{\tau}\) about an origin due to a force \(\vec{F}\) acting at a position \(\vec{r}\) from the origin is defined by the cross product: \[ \vec{\tau} = \vec{r} \times \vec{F} \]
A key property of the cross product is that the resulting vector is perpendicular to the plane containing the two original vectors.


Step 3: Detailed Explanation:

By the mathematical definition of the cross product, the vector \(\vec{\tau}\) must be orthogonal (perpendicular) to both the position vector \(\vec{r}\) and the force vector \(\vec{F}\).

Since \(\vec{\tau}\) is perpendicular to \(\vec{r}\), the angle between them must be 90 degrees.
This is a direct geometric consequence of the definition of torque.


Step 4: Final Answer:

The angle between \(\vec{r}\) and \(\vec{\tau}\) is 90°. This corresponds to option (D).
Quick Tip: Remember the right-hand rule for cross products. If you point the fingers of your right hand in the direction of \(\vec{r}\) and curl them towards the direction of \(\vec{F}\), your thumb points in the direction of \(\vec{\tau}\). This clearly shows \(\vec{\tau}\) is perpendicular to the plane of \(\vec{r}\) and \(\vec{F}\).

Step 1: Understanding the Concept:

This question requires the direct application of the formula for gravitational potential energy between two point masses.


Step 2: Key Formula or Approach:

The gravitational potential energy \(U\) between two masses \(m_1\) and \(m_2\) separated by a distance \(r\) is given by the formula: \[ U = -\frac{Gm_1m_2}{r} \]
where G is the universal gravitational constant. The negative sign indicates that the gravitational force is attractive.


Step 3: Detailed Explanation:

We are given the following values:

Mass of the first body, \(m_1 = 1\) kg.
Mass of the second body, \(m_2 = 1\) kg.
Distance between them, \(r = 1\) m.

Substitute these values into the formula for gravitational potential energy: \[ U = -\frac{G(1)(1)}{1} \] \[ U = -G \]

Step 4: Final Answer:

The gravitational potential energy between the two bodies is -G. This corresponds to option (B).
Quick Tip: Gravitational potential energy is a scalar quantity. By convention, the potential energy is taken to be zero when the separation between the masses is infinite. For any finite separation, the potential energy is negative because work must be done against the attractive gravitational force to separate them to infinity.

Step 1: Understanding the Concept:

Escape velocity is the minimum speed an object must have at the surface of a planet to escape its gravitational field completely. This concept connects the gravitational potential energy and kinetic energy. We can derive the formula for escape velocity and then relate it to the surface gravity \(g\).


Step 2: Key Formula or Approach:

1. The escape velocity \(v_e\) is found by equating the initial kinetic energy of the object to the magnitude of its initial gravitational potential energy: \(\frac{1}{2}mv_e^2 = \frac{GMm}{r}\), which gives \(v_e = \sqrt{\frac{2GM}{r}}\). (Here, M is the planet's mass, not m as in the question prompt which uses m for planet mass).
2. The acceleration due to gravity \(g\) at the surface of a planet is given by \(g = \frac{GM}{r^2}\).
3. We will use the second formula to substitute for \(GM\) in the first formula.


Step 3: Detailed Explanation:

Let the mass of the planet be \(m\) and its radius be \(r\), as given in the question.
The standard formula for escape velocity (\(v_e\)) from the surface of this planet is: \[ v_e = \sqrt{\frac{2Gm}{r}} \]
The acceleration due to gravity (\(g\)) on the surface of this planet is: \[ g = \frac{Gm}{r^2} \]
From the equation for surface gravity, we can express the product \(Gm\) in terms of \(g\) and \(r\): \[ Gm = gr^2 \]
Now, substitute this expression for \(Gm\) into the escape velocity formula: \[ v_e = \sqrt{\frac{2(gr^2)}{r}} \]
Simplify the expression by canceling one \(r\): \[ v_e = \sqrt{2gr} \]

Step 4: Final Answer:

The escape velocity from the surface of the planet is \(\sqrt{2gr}\). This corresponds to option (A).
Quick Tip: It is useful to memorize both forms of the escape velocity formula: \(v_e = \sqrt{\frac{2GM}{R}}\) and \(v_e = \sqrt{2gR}\). The second form is often more convenient when the surface gravity \(g\) is known.

Step 1: Understanding the Concept:

This question asks for the specific terminology used in thermodynamics to describe different types of boundaries between systems, based on their ability to transfer heat.


Step 2: Detailed Explanation:

Let's define the terms given in the options:

Adiabatic wall: A boundary that does not allow the transfer of heat between the system and its surroundings. It is a perfect thermal insulator. "Insulated wall" and "non-conducting wall" are less technical synonyms.
Diathermic wall (or diathermal wall): A boundary that does allow the flow of heat. It is a thermally conducting wall. When two systems are separated by a diathermic wall, they can exchange heat until they reach the same temperature, a state known as thermal equilibrium.
Semiconducting wall: This term is primarily used in electronics and solid-state physics to describe materials with electrical conductivity between that of a conductor and an insulator. It is not a standard term for thermal boundaries in thermodynamics.

The question describes a wall that allows heat flow to reach thermal equilibrium, which is the definition of a diathermic wall.


Step 3: Final Answer:

The wall is called a diathermic wall. This corresponds to option (C).
Quick Tip: A simple way to remember the difference: - Diathermic sounds like "through thermal", meaning heat can go through. - Adiabatic contains "a-", a common prefix meaning "not" or "without". "A-dia-batic" means not allowing to pass through, in this context, heat.

Step 1: Understanding the Concept:

Pressure energy is a form of energy associated with a fluid due to its pressure. The question asks for this energy per unit mass. This is a fundamental concept in fluid dynamics, often appearing in Bernoulli's equation.


Step 2: Key Formula or Approach:

1. The work done by pressure P to move a fluid of volume V is considered its pressure energy, given by \(E = PV\).

2. Pressure energy per unit mass is this energy divided by the mass, \(m\). So, \(\frac{E}{m} = \frac{PV}{m}\).

3. The relationship between density (\(\rho\)), mass (\(m\)), and volume (\(V\)) is \(\rho = \frac{m}{V}\). This can be rearranged to express the term \(\frac{V}{m}\).


Step 3: Detailed Explanation:

The pressure energy of a certain amount of liquid with volume V at pressure P is given by the product PV.
\[ Pressure Energy = P \times V \]
The question asks for the pressure energy per unit mass. So we need to divide this by the mass \(m\) of the liquid.
\[ Pressure Energy per unit mass = \frac{PV}{m} = P \left(\frac{V}{m}\right) \]
We know that density (\(\rho\)) is defined as mass per unit volume:
\[ \rho = \frac{m}{V} \]
The reciprocal of density is the specific volume (volume per unit mass):
\[ \frac{1}{\rho} = \frac{V}{m} \]
Substituting this into our expression for pressure energy per unit mass:
\[ Pressure Energy per unit mass = P \left(\frac{1}{\rho}\right) = \frac{P}{\rho} \]
The term dV (change in volume) is related to work done during compression (\(W = P dV\)), but the intrinsic pressure energy per unit mass of the fluid is given by \(P/\rho\).


Step 4: Final Answer:

The pressure energy per unit mass of the liquid is \(\frac{P}{\rho}\). This corresponds to option (D).
Quick Tip: The terms in Bernoulli's equation, \(\frac{P}{\rho} + \frac{1}{2}v^2 + gh = constant\), all represent energy per unit mass. \(\frac{P}{\rho}\) is the pressure energy per unit mass, \(\frac{1}{2}v^2\) is the kinetic energy per unit mass, and \(gh\) is the potential energy per unit mass. Remembering this helps to identify these terms quickly.

Step 1: Understanding the Concept:

This problem is a direct application of Pascal's Principle, which is the fundamental principle behind hydraulic systems like a car lift. The principle states that a pressure change at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere.


Step 2: Key Formula or Approach:

1. According to Pascal's Principle, the pressure exerted on the first piston (\(P_1\)) is equal to the pressure on the second piston (\(P_2\)).
\[ P_1 = P_2 \]
2. Pressure is defined as force per unit area (\(P = F/A\)).
3. We can write \(P_1 = \frac{F_1}{A_1}\) and \(P_2 = \frac{F_2}{A_2}\).
4. By equating the pressures, we can solve for the unknown force \(F_2\).


Step 3: Detailed Explanation:

The pressure applied to the small piston is: \[ P_1 = \frac{F_1}{A_1} \]
According to Pascal's principle, this pressure is transmitted undiminished to the larger piston. So, the pressure on the second piston is: \[ P_2 = P_1 \]
The force \(F_2\) realised on the second piston of area \(A_2\) is related to this pressure by \(F_2 = P_2 A_2\).
Substituting \(P_2 = P_1 = \frac{F_1}{A_1}\): \[ F_2 = \left(\frac{F_1}{A_1}\right) A_2 \]
Rearranging the terms gives: \[ F_2 = F_1 \frac{A_2}{A_1} \]

Step 4: Final Answer:

The force realised on the second piston is \(F_1\frac{A_2}{A_1}\). This corresponds to option (B).
Quick Tip: Hydraulic systems are force multipliers. The mechanical advantage is the ratio of the areas \(\frac{A_2}{A_1}\). To lift a heavy object (large force \(F_2\)), a small force \(F_1\) is applied to a small area \(A_1\), which transmits pressure to a large area \(A_2\).

Step 1: Understanding the Concept:

This question requires knowledge of the definitions of various thermodynamic processes. We need to identify which pair incorrectly describes the process.


Step 2: Detailed Explanation:

Let's analyze each pair:

(A) Isothermal : Absorption or emission of heat

An isothermal process is one that occurs at a constant temperature (\(\Delta T = 0\)). According to the first law of thermodynamics, \(\Delta U = Q - W\). For an ideal gas, internal energy \(U\) depends only on temperature, so \(\Delta U = 0\). This means \(Q = W\). If work is done by the gas (\(W>0\)), it must absorb heat (\(Q>0\)) to keep its temperature constant. If work is done on the gas (\(W<0\)), it must emit heat (\(Q<0\)). So, heat is indeed exchanged. This pair is a correct match.
(B) Isobaric : Pressure constant

This is the definition of an isobaric process. This pair is correct.
(C) Isochoric : Volume constant

This is the definition of an isochoric (or isovolumetric) process. This pair is correct.
(D) Irreversible : Loss of heat

Irreversible processes are those that cannot be reversed to restore both the system and surroundings to their original states. They often involve dissipative effects like friction, which convert mechanical energy into heat. This "loss" of useful energy as heat is a characteristic feature. This pair is a plausible match.
(E) Adiabatic : Heat exchange

An adiabatic process is defined as one in which there is no heat exchange between the system and its surroundings (\(Q=0\)). The description "Heat exchange" is the exact opposite of the definition of an adiabatic process. This is a clear mismatch.


Step 4: Final Answer:

The mismatched pair is (E) Adiabatic : Heat exchange.
Quick Tip: Remember the key definitions: \textbf{Isothermal}: \(\Delta T = 0\) (constant temperature) \textbf{Isobaric}: \(\Delta P = 0\) (constant pressure) \textbf{Isochoric}: \(\Delta V = 0\) (constant volume) \textbf{Adiabatic}: \(Q = 0\) (no heat exchange)

Step 1: Understanding the Concept:

The thermal efficiency of a heat engine is defined as the ratio of the net work done by the engine to the heat absorbed from the high-temperature source. It represents how effectively the engine converts heat into work.


Step 2: Key Formula or Approach:

1. The efficiency \(\eta\) of a heat engine is given by \(\eta = \frac{W}{Q_H}\), where \(W\) is the work done and \(Q_H\) is the heat absorbed from the source.
2. From the first law of thermodynamics applied to a cycle, the work done is the difference between the heat absorbed and the heat rejected: \(W = Q_H - Q_L\), where \(Q_L\) is the heat rejected to the sink.
3. Substituting for \(W\), the efficiency formula becomes \(\eta = \frac{Q_H - Q_L}{Q_H} = 1 - \frac{Q_L}{Q_H}\).


Step 3: Detailed Explanation:

We are given the ratio of the heat rejected to the sink (\(Q_L\)) to the heat absorbed from the source (\(Q_H\)). \[ \frac{Q_L}{Q_H} = \frac{1}{4} \]
Now, we use the formula for efficiency: \[ \eta = 1 - \frac{Q_L}{Q_H} \]
Substitute the given ratio into the formula: \[ \eta = 1 - \frac{1}{4} = \frac{3}{4} \]
To express the efficiency as a percentage, we multiply by 100: \[ \eta = \frac{3}{4} \times 100% = 75% \]

Step 4: Final Answer:

The efficiency of the engine is 75%. This corresponds to option (A).
Quick Tip: Efficiency is always a fraction representing (useful output) / (total input). For a heat engine, the useful output is work (\(W\)) and the input is the heat from the source (\(Q_H\)). The heat rejected (\(Q_L\)) is the waste. The efficiency is simply \(1 - (waste/input)\).

Step 1: Understanding the Concept:

The mean free path (\(\lambda\)) is the average distance a gas molecule travels between successive collisions with other molecules. This concept is central to the kinetic theory of gases and helps explain transport phenomena like diffusion and viscosity.


Step 2: Key Formula or Approach:

The formula for the mean free path is derived from kinetic theory as: \[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \]
where \(d\) is the molecular diameter and \(n\) is the number density (number of molecules per unit volume).

The ideal gas law can be written as \(PV = NkT\), where \(N\) is the total number of molecules, \(k\) is Boltzmann's constant, and \(T\) is the absolute temperature. The number density is \(n = N/V\). We can express \(n\) in terms of pressure and temperature.


Step 3: Detailed Explanation:

From the ideal gas law, \(PV=NkT\), we can find the number density \(n\): \[ n = \frac{N}{V} = \frac{P}{kT} \]
Now, substitute this expression for \(n\) into the mean free path formula: \[ \lambda = \frac{1}{\sqrt{2} \pi d^2 \left(\frac{P}{kT}\right)} = \frac{kT}{\sqrt{2} \pi d^2 P} \]
From this final expression, we can see the proportionalities:

\(\lambda \propto T\) (directly proportional to absolute temperature)
\(\lambda \propto \frac{1}{P}\) (inversely proportional to pressure)
\(\lambda \propto \frac{1}{d^2}\) (inversely proportional to the square of the molecular diameter)

The question asks what the mean free path is directly proportional to. Based on our analysis, it is directly proportional to the absolute temperature.


Step 4: Final Answer:

The mean free path is directly proportional to the absolute temperature. This corresponds to option (D).
Quick Tip: Intuitively, at a higher temperature (for a fixed pressure), molecules move faster, covering more distance between collisions. Also, at a fixed temperature, higher pressure means molecules are closer together (higher number density), leading to more frequent collisions and a shorter mean free path.

Step 1: Understanding the Concept:

The ideal gas law is based on a simplified model of a gas that makes two key assumptions: (1) the volume of the gas molecules is negligible, and (2) the intermolecular forces between gas molecules are negligible. Real gases deviate from this ideal behavior, especially when these assumptions are no longer valid. The question asks for the conditions under which a real gas behaves most like an ideal gas.


Step 2: Detailed Explanation:

We need to find the conditions where the two main assumptions of the ideal gas model are most closely met for a real gas.

Negligible Molecular Volume: The volume occupied by the molecules themselves should be insignificant compared to the total volume of the container. This condition is best met at low pressure, where the gas expands to fill a large volume, making the molecules very far apart.
Negligible Intermolecular Forces: The attractive or repulsive forces between molecules should be minimal. The effect of these forces is small when the molecules are far apart (i.e., at low pressure) and when their kinetic energy is much larger than the potential energy of interaction. High kinetic energy corresponds to high temperature.

Therefore, a real gas most closely approximates an ideal gas at the conditions of low pressure and high temperature.


Step 4: Final Answer:

The required conditions are low pressure and high temperature. This corresponds to option (E).
Quick Tip: Think of the opposite conditions: high pressure and low temperature. These conditions force the molecules close together and slow them down, making both their size and the forces between them very significant. This is when real gases deviate most from ideal behavior and can even liquefy or solidify.

Step 1: Understanding the Concept:

This question tests the fundamental characteristics of velocity and acceleration at different points in Simple Harmonic Motion (SHM). In SHM, the restoring force, and thus the acceleration, is directly proportional to the displacement from the mean position and is always directed towards it.


Step 2: Detailed Explanation:

Let's analyze the velocity and acceleration at the specified positions:

Mean Position (O): This is the equilibrium point where the net force on the particle is zero.

Acceleration (a): Since force is zero (\(F = -kx = 0\)), the acceleration is also zero (\(a = F/m = 0\)).
Velocity (V): The particle has its maximum speed as it passes through the mean position.

Extreme Positions (A and B): These are the points of maximum displacement.

Acceleration (a): The displacement is maximum, so the restoring force is maximum. Therefore, the magnitude of acceleration is maximum at these points.
Velocity (V): The particle momentarily stops at the extreme positions before reversing its direction, so the velocity is zero.


Now let's evaluate the given options based on this analysis:

(A) at A (extreme position), a is maximum, not 0. Incorrect.
(B) at B (extreme position), a is maximum, not 0. Incorrect.
(C) at O (mean position), a is zero, not maximum. Incorrect.
(D) at O, V is maximum, but a is zero. Incorrect.
(E) at O (mean position), a is zero. Correct.


Step 4: Final Answer:

The correct statement is that at the mean position O, the acceleration a = 0. This corresponds to option (E).
Quick Tip: A useful mnemonic for SHM: \textbf{Center (Mean):} Max speed, zero acceleration. \textbf{Ends (Extremes):} Zero speed, max acceleration. Velocity and acceleration are \(\pi/2\) (90 degrees) out of phase. When one is at its maximum, the other is zero.

Step 1: Understanding the Concept:

This problem requires evaluating the given position function of a particle in SHM at a specific time.


Step 2: Key Formula or Approach:

The displacement \(x\) at any time \(t\) is given by the equation \(x(t) = 5\cos(4\pi t)\). We need to substitute \(t=3\) s into this equation and calculate the value of \(x\).


Step 3: Detailed Explanation:

The displacement equation is: \[ x(t) = 5\cos(4\pi t) \]
We need to find the displacement at \(t = 3\) s. \[ x(3) = 5\cos(4\pi \times 3) \] \[ x(3) = 5\cos(12\pi) \]
The cosine function is periodic with a period of \(2\pi\). This means that \(\cos(\theta + 2n\pi) = \cos(\theta)\) for any integer \(n\). In this case, \(12\pi = 6 \times 2\pi\).
So, \[ \cos(12\pi) = \cos(0) = 1 \]
Now, substitute this value back into the displacement equation: \[ x(3) = 5 \times 1 = 5 m \]

Step 4: Final Answer:

The displacement after 3 s is 5 m. This corresponds to option (B).
Quick Tip: For arguments of trigonometric functions in the form of \(n\pi\), where \(n\) is an integer, remember these key values: \(\cos(n\pi) = (-1)^n\) (i.e., 1 for even n, -1 for odd n) \(\sin(n\pi) = 0\) Here, n=12 is even, so \(\cos(12\pi) = 1\).

Step 1: Understanding the Concept:

Beats are the periodic variation in loudness (amplitude) of a sound wave that results from the superposition of two sound waves of slightly different frequencies. The beat frequency is the number of these loudness variations heard per second.


Step 2: Key Formula or Approach:

1. The beat frequency (\(f_{beat}\)) is defined as the number of beats per unit time.
\[ f_{beat} = \frac{Number of beats}{Time interval} \]
2. The beat frequency is also equal to the absolute difference between the frequencies of the two sources, \(f_1\) and \(f_2\).
\[ f_{beat} = |f_1 - f_2| \]

Step 3: Detailed Explanation:

We are given that 24 beats are produced in a time interval of 3 seconds.

First, we calculate the beat frequency in Hertz (beats per second): \[ f_{beat} = \frac{24 beats}{3 s} = 8 beats/s = 8 Hz \]
The difference between the two source frequencies is equal to this beat frequency. \[ |f_1 - f_2| = f_{beat} = 8 Hz \]

Step 4: Final Answer:

The difference between the two frequencies is 8 Hz. This corresponds to option (C).
Quick Tip: Beat frequency is always expressed in Hertz (Hz), which means "per second". If the time is given in minutes or any other unit, make sure to convert it to seconds to find the frequency in Hz.

Step 1: Understanding the Concept:

This question asks about the relationship between the electric potential (\(V\)) of a dipole and the distance (\(r\)) from its center along its axis. This is a standard result from electrostatics.


Step 2: Key Formula or Approach:

The electric potential \(V\) at a point on the axis of a dipole with dipole moment \(p\) at a distance \(r\) from its center is given by: \[ V_{axial} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^2} \]
(This formula is valid for \(r\) much larger than the separation of the charges in the dipole).


Step 3: Detailed Explanation:

From the formula \(V_{axial} = \frac{kp}{r^2}\) (where \(k = \frac{1}{4\pi\epsilon_0}\)), we can see that the potential \(V\) is proportional to \(\frac{1}{r^2}\). \[ V \propto \frac{1}{r^2} \]
The question asks what the potential is inversely proportional to.
If a quantity \(Y\) is proportional to \(\frac{1}{X}\), we say that \(Y\) is inversely proportional to \(X\).
In our case, \(V\) is proportional to \(\frac{1}{r^2}\).
Therefore, \(V\) is inversely proportional to \(r^2\).


Step 4: Final Answer:

The electric potential is inversely proportional to \(r^2\). This corresponds to option (C).
Quick Tip: Pay close attention to the wording. "Proportional to \(r^{-2}\)" is the same as "inversely proportional to \(r^2\)". The question phrasing determines which form is the correct answer. For a dipole: Potential \(V \propto 1/r^2\) Field \(E \propto 1/r^3\) For a point charge: Potential \(V \propto 1/r\) Field \(E \propto 1/r^2\)

Step 1: Understanding the Concept:

This problem relates the potential difference (\(\Delta V\)) between two points to the electric field (\(E\)) and the distance (\(d\)) between them. For a uniform electric field, this relationship is linear.


Step 2: Key Formula or Approach:

Assuming the electric field between the conductors is uniform (as is typical for such problems, e.g., between parallel plates), the magnitude of the electric field is given by: \[ E = \frac{\Delta V}{d} \]
where \(\Delta V\) is the potential difference and \(d\) is the distance between the conductors.


Step 3: Detailed Explanation:

First, we list the given values and convert them to SI units.

Potential difference, \(\Delta V = 4 \times 10^3\) V.
Distance, \(d = 2 cm\). We must convert this to meters: \(d = 2 \times 10^{-2}\) m.

Now, we apply the formula: \[ E = \frac{4 \times 10^3 V}{2 \times 10^{-2} m} \] \[ E = \frac{4}{2} \times 10^{3 - (-2)} V/m \] \[ E = 2 \times 10^{3+2} V/m = 2 \times 10^5 V/m \]
The unit Vm\(^{-1}\) is the same as V/m.


Step 4: Final Answer:

The electric field between the conductors is \(2 \times 10^5\) Vm\(^{-1}\). This corresponds to option (E).
Quick Tip: Unit conversion is a critical step in physics problems. A common mistake is forgetting to convert quantities like centimeters to meters before performing calculations. Always double-check that all your values are in a consistent set of units (usually SI).

Step 1: Understanding the Concept:

Electrostatic energy density (\(u_E\)) is the energy stored in an electric field per unit volume. It represents the concentration of energy in the space where the electric field exists, such as between the plates of a capacitor.


Step 2: Key Formula or Approach:

The formula for the energy density in a region of free space (or in a dielectric with permittivity \(\epsilon\)) with an electric field of magnitude \(E\) is: \[ u_E = \frac{1}{2}\epsilon E^2 \]
From this formula, we can determine the proportionality between \(u_E\) and \(E\).


Step 3: Detailed Explanation:

The formula for electrostatic energy density is \(u_E = \frac{1}{2}\epsilon E^2\).
In this expression, \(\frac{1}{2}\) is a constant and \(\epsilon\) (the permittivity of the medium) is also a constant for a given medium.
Therefore, the energy density \(u_E\) is directly proportional to the square of the magnitude of the electric field, \(E\). \[ u_E \propto E^2 \]

Step 4: Final Answer:

The electrostatic energy density is directly proportional to \(E^2\). This corresponds to option (A).
Quick Tip: Energy densities in both electric and magnetic fields are proportional to the square of the respective field strengths. Electric energy density: \(u_E \propto E^2\) Magnetic energy density: \(u_B \propto B^2\) This is a general feature of energy stored in fields.

Step 1: Understanding the Concept:

This question asks to identify the particles responsible for carrying electric current in an electrolytic solution. Different types of materials have different charge carriers.


Step 2: Detailed Explanation:

An electrolyte is a substance (typically an ionic compound like a salt, acid, or base) that produces an electrically conducting solution when dissolved in a polar solvent, such as water.

The dissolution process causes the ionic compound to dissociate into its constituent ions: positively charged ions (cations) and negatively charged ions (anions).
When an external electric field is applied (by placing electrodes in the solution), these ions start to move.
Cations (positive ions) are attracted to and move towards the negative electrode (cathode).
Anions (negative ions) are attracted to and move towards the positive electrode (anode).
This directed motion of both positive and negative ions in opposite directions constitutes the flow of electric current through the electrolyte.

Free electrons are the charge carriers in metallic conductors, but they are not the primary mobile charge carriers within the bulk of an electrolytic solution.


Step 4: Final Answer:

In an electrolyte, the mobile charge carriers are negative and positive ions. This corresponds to option (D).
Quick Tip: It's helpful to remember the charge carriers in different media: \textbf{Solids (Metals):} Free electrons. \textbf{Solids (Semiconductors):} Electrons and holes. \textbf{Liquids (Electrolytes):} Positive and negative ions. \textbf{Gases (Ionized):} Positive ions and free electrons.

Step 1: Understanding the Concept:

This problem deals with how the electrical resistance of a conductor depends on its geometric properties, specifically its length and cross-sectional area.


Step 2: Key Formula or Approach:

The resistance \(R\) of a conductor is given by the formula: \[ R = \rho \frac{L}{A} \]
where \(\rho\) is the resistivity of the material (a constant for a given material), \(L\) is the length of the conductor, and \(A\) is its cross-sectional area. We need to see how \(R\) changes when \(L\) and \(A\) are changed.


Step 3: Detailed Explanation:

Let the initial resistance be \(R_{initial}\). \[ R_{initial} = \rho \frac{L}{A} \]
Now, we are given the new conditions:

The new length, \(L_{new} = \frac{L}{2}\).
The new area, \(A_{new} = \frac{A}{2}\).

Let's calculate the new resistance, \(R_{new}\), using these new values: \[ R_{new} = \rho \frac{L_{new}}{A_{new}} = \rho \frac{(L/2)}{(A/2)} \]
The factors of \(1/2\) in the numerator and denominator cancel out: \[ R_{new} = \rho \frac{L}{A} \]
Comparing the new resistance with the initial resistance: \[ R_{new} = R_{initial} \]
The resistance remains unchanged.


Step 4: Final Answer:

The resistance would remain unchanged. This corresponds to option (B).
Quick Tip: Resistance is directly proportional to length (\(R \propto L\)) and inversely proportional to area (\(R \propto 1/A\)). If both \(L\) and \(A\) are changed by the same factor (e.g., both are halved, or both are doubled), their effects on the resistance will cancel each other out, leaving the resistance unchanged.

Step 1: Understanding the Concept:

This problem involves calculating the power loss (dissipation) in transmission lines. The power is dissipated as heat due to the resistance of the cables. This loss is often called Joule heating or \(I^2R\) loss. To calculate it, we first need to find the current flowing through the cables.


Step 2: Key Formula or Approach:

1. The formula for electrical power transmitted is \(P_{trans} = V \times I\), where V is the transmission voltage and I is the current.
2. The formula for power dissipated in a resistor is \(P_{diss} = I^2 R\), where I is the current flowing through the resistor and R is its resistance.
3. We will first use the transmission power and voltage to find the current, and then use the current and cable resistance to find the dissipated power.


Step 3: Detailed Explanation:

1. Calculate the current (I):

We are given:

Power transmitted, \(P_{trans} = 11 kW = 11000 W\).
Transmission voltage, \(V = 220 V\).

Using the formula \(P_{trans} = VI\), we can solve for I: \[ I = \frac{P_{trans}}{V} = \frac{11000 W}{220 V} = \frac{1100}{22} A = 50 A \]
So, the current flowing through the transmission cables is 50 A.


2. Calculate the power dissipated (\(P_{diss}\)):

We are given:

Resistance of the cables, \(R = 0.03 \, \Omega\).

Using the formula \(P_{diss} = I^2 R\): \[ P_{diss} = (50 A)^2 \times (0.03 \, \Omega) = 2500 \times 0.03 W = 75 W \]
The options are given in kilowatts (kW), so we convert our answer: \[ P_{diss} = 75 W = \frac{75}{1000} kW = 0.075 kW \]

Step 4: Final Answer:

The power dissipated in the transmission cables is 0.075 kW. This corresponds to option (C).
Quick Tip: This problem highlights why power is transmitted at very high voltages. For the same amount of power transmitted, a higher voltage results in a lower current (\(I=P/V\)). Since power loss is proportional to the square of the current (\(I^2R\)), a lower current drastically reduces the energy wasted in the transmission lines.

Step 1: Understanding the Concept:

The Earth's magnetic field at any point can be resolved into a horizontal component (\(B_H\)) and a vertical component (\(B_V\)). The dip angle (or angle of inclination, \(\delta\)) is the angle that the total magnetic field vector makes with the horizontal direction.


Step 2: Key Formula or Approach:

The horizontal and vertical components are related to the total magnetic field \(B_E\) and the dip angle \(\delta\) by:

\(B_H = B_E \cos\delta\)
\(B_V = B_E \sin\delta\)

By dividing the second equation by the first, we get a direct relationship between the components and the dip angle: \[ \frac{B_V}{B_H} = \frac{B_E \sin\delta}{B_E \cos\delta} = \tan\delta \]
We can use this formula to find the dip angle.


Step 3: Detailed Explanation:

We are given the components of the Earth's magnetic field:

Horizontal component, \(B_H = 0.26\) G.
Vertical component, \(B_V = 0.26\sqrt{3}\) G.

Using the formula \(\tan\delta = \frac{B_V}{B_H}\): \[ \tan\delta = \frac{0.26\sqrt{3}}{0.26} \]
The factor 0.26 cancels out: \[ \tan\delta = \sqrt{3} \]
We need to find the angle \(\delta\) for which the tangent is \(\sqrt{3}\). From standard trigonometric values, we know that: \[ \delta = \tan^{-1}(\sqrt{3}) = 60^\circ \]

Step 4: Final Answer:

The dip angle is 60\(^\circ\). This corresponds to option (D).
Quick Tip: Remember the special triangle for 30-60-90 degrees. If the side opposite 30° is 1, the side opposite 60° is \(\sqrt{3}\), and the hypotenuse is 2. This helps you recall that \(\tan(60^\circ) = \frac{opposite}{adjacent} = \frac{\sqrt{3}}{1} = \sqrt{3}\).

Step 1: Understanding the Concept:

A current-carrying loop of wire placed in a magnetic field experiences a torque. This torque depends on the current, the magnetic field, the area of the loop, and the orientation of the loop relative to the field. The question asks for the maximum possible torque.


Step 2: Key Formula or Approach:

The torque \(\vec{\tau}\) on a current loop with \(N\) turns, area \(A\), and current \(I\) in a uniform magnetic field \(\vec{B}\) is given by: \[ \vec{\tau} = \vec{\mu} \times \vec{B} \]
where \(\vec{\mu} = NIA\hat{n}\) is the magnetic dipole moment of the loop (\(\hat{n}\) is the normal vector to the plane of the loop).
The magnitude of the torque is: \[ \tau = |\vec{\mu}| |\vec{B}| \sin\theta = NIAB\sin\theta \]
where \(\theta\) is the angle between the magnetic moment (the normal to the loop's area) and the magnetic field. For a single loop, \(N=1\).


Step 3: Detailed Explanation:

The magnitude of the torque on the rectangular coil (assuming a single turn, \(N=1\)) is: \[ \tau = IAB\sin\theta \]
The torque is maximum when the sine function is maximum. The maximum value of \(\sin\theta\) is 1, which occurs when \(\theta = 90^\circ\). This means the magnetic moment vector is perpendicular to the magnetic field, or equivalently, the plane of the coil is parallel to the magnetic field.
The maximum torque is therefore: \[ \tau_{max} = IAB(1) = IBA \]
The length \(l\) is a property of the coil, but the torque is directly expressed in terms of the coil's area \(A\), not its length alone.


Step 4: Final Answer:

The maximum torque is IBA. This corresponds to option (A).
Quick Tip: The torque on a current loop is maximum when the plane of the loop is aligned with the magnetic field lines (\(\theta = 90^\circ\)). The torque is zero when the plane of the loop is perpendicular to the magnetic field lines (\(\theta = 0^\circ\)). This is a key concept for understanding electric motors.

Step 1: Understanding the Concept:

This question asks about the technology used in older Cathode Ray Tube (CRT) televisions to control the electron beam. The electron beam is deflected by magnetic fields to scan across the screen and create an image.


Step 2: Detailed Explanation:

In a CRT television, an electron gun produces a beam of electrons. This beam is directed towards a fluorescent screen. To form an image, the beam must be deflected both horizontally and vertically to "paint" the picture line by line.
This deflection is achieved by passing the electron beam through a region of controlled magnetic fields. These fields are generated by coils of wire placed around the neck of the CRT. These coils are known as deflection coils or deflection yokes.
When current is passed through these coils, they produce a magnetic field. By varying the current's magnitude and direction in a systematic way (using a sawtooth waveform), the magnetic field can be controlled to steer the electron beam across the entire screen.
A coil of wire that produces a magnetic field when current flows through it is the definition of an electromagnet. Specifically, deflection yokes are a form of electromagnet. A solenoid is a specific type of electromagnet, as is a toroid. A circular coil is also an electromagnet.
Given the options, "electromagnet" is the most general and correct term. A "solenoid" could also be considered, as the deflection coils are wound around the tube.
The question is marked as "Cancelled". This is likely because multiple options could be considered correct. "Electromagnet" (B) is the general principle. "Solenoid" (E) and "circular coil" (D) describe the form of the electromagnet used. A toroid (A) is also a type of electromagnet but not typically used for this purpose. The ambiguity makes the question problematic.


Step 4: Final Answer:

The magnetic field in a CRT is produced by deflection coils, which are a type of electromagnet. Option (B) is the best description. However, the question is marked as cancelled, likely due to the ambiguity with other options like (D) and (E).
Quick Tip: Modern flat-screen TVs (LCD, LED, OLED, Plasma) do not use electron beams or magnetic deflection. They use grids of pixels that emit or filter light directly. This question is specific to the older CRT technology.

Step 1: Understanding the Concept:

Magnetic flux (\(\Phi_B\)) through a surface is a measure of the total number of magnetic field lines passing through that surface. It depends on the strength of the magnetic field, the area of the surface, and the orientation of the surface relative to the field.


Step 2: Key Formula or Approach:

The magnetic flux through a flat coil of area \(A\) in a uniform magnetic field \(B\) is given by the dot product of the magnetic field vector and the area vector: \[ \Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta \]
where \(\theta\) is the angle between the magnetic field vector \(\vec{B}\) and the area vector \(\vec{A}\). The area vector is a vector perpendicular (normal) to the plane of the coil. We need to solve this equation for \(\theta\).


Step 3: Detailed Explanation:

We are given the following values:

Magnetic flux, \(\Phi_B = 4\) Wb.
Area of the coil, \(A = 0.5\) m\(^2\).
Magnetic field strength, \(B = 16\) T.

We use the formula \(\Phi_B = BA\cos\theta\) and solve for \(\cos\theta\). \[ \cos\theta = \frac{\Phi_B}{BA} \]
Substitute the given values: \[ \cos\theta = \frac{4}{16 \times 0.5} = \frac{4}{8} = \frac{1}{2} \]
We need to find the angle \(\theta\) for which the cosine is \(\frac{1}{2}\). From standard trigonometric values, we know that: \[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

Step 4: Final Answer:

The angle between the magnetic field and the area vector is 60\(^\circ\). This corresponds to option (D).
Quick Tip: Be careful to distinguish between the "angle with the area vector" and the "angle with the plane of the coil". The area vector is normal to the plane. If the angle with the plane is \(\alpha\), then the angle with the normal is \(\theta = 90^\circ - \alpha\). The formula \(\Phi_B = BA\cos\theta\) always uses the angle with the normal (area vector).

Step 1: Understanding the Concept:

Self-inductance (\(L\)) is a property of an electrical circuit (like a coil) that describes its opposition to a change in current flowing through it. It is an intrinsic property determined by the geometry of the coil and the material properties of its core.


Step 2: Key Formula or Approach:

The self-inductance \(L\) is defined by the relationship between the magnetic flux (\(\Phi_B\)) through the coil and the current (\(I\)) producing it: \(N\Phi_B = LI\), where \(N\) is the number of turns.
For a long solenoid, a common example, the self-inductance is given by the formula: \[ L = \frac{\mu N^2 A}{l} \]
where \(\mu\) is the permeability of the medium, \(N\) is the total number of turns, \(A\) is the cross-sectional area, and \(l\) is the length of the solenoid. We can analyze this formula to see the dependencies.


Step 3: Detailed Explanation:

Let's analyze the factors affecting self-inductance based on the formula for a solenoid, \(L = \frac{\mu N^2 A}{l}\):

Permeability of the medium (\(\mu\)): \(L\) is directly proportional to \(\mu\). So, inductance depends on the medium. (Option E is a dependency).
Number of turns (N): \(L\) is proportional to \(N^2\). So, inductance depends on the number of turns. (Option B is a dependency).
Area of cross-section (A): \(L\) is directly proportional to \(A\). So, inductance depends on the area. (Option C is a dependency).
Radius (r): The area \(A\) is related to the radius by \(A = \pi r^2\). Since \(L\) depends on \(A\), it also depends on the radius. (Option A is a dependency).
Current (I): The self-inductance \(L\) is defined as the ratio \(L = N\Phi_B/I\). While the flux \(\Phi_B\) is proportional to the current \(I\), their ratio \(L\) is a constant of proportionality. It is a geometric and material property of the coil itself and does not depend on the amount of current flowing through it (as long as the material's permeability remains constant).


Step 4: Final Answer:

The self-inductance of a coil does not depend on the current through it. This corresponds to option (D).
Quick Tip: Think of the analogy with other circuit components. Resistance (\(R\)) is a property of a resistor; it doesn't depend on the voltage across it or the current through it (\(R=V/I\)). Similarly, capacitance (\(C\)) is a property of a capacitor; it doesn't depend on the charge on it or the voltage across it (\(C=Q/V\)). Inductance (\(L\)) is a property of an inductor; it doesn't depend on the flux or the current.

Step 1: Understanding the Concept:

This question asks to identify the physical phenomenon that uniquely demonstrates that electromagnetic waves (like light) are transverse. A transverse wave is one in which the oscillations are perpendicular to the direction of wave propagation. A longitudinal wave has oscillations parallel to the direction of propagation.


Step 2: Detailed Explanation:

Let's analyze the phenomena listed:

Interference, Diffraction, Dispersion: These are all wave phenomena. Interference (superposition of waves), diffraction (bending of waves around obstacles), and dispersion (splitting of light into colors) can be exhibited by both transverse and longitudinal waves. For example, sound waves, which are longitudinal, also show interference and diffraction. Therefore, these phenomena prove the wave nature of light but not its transverse nature.
Photoelectric effect: This phenomenon, where electrons are emitted from a material when light shines on it, demonstrates the particle nature of light (photons), not its wave nature.
Polarization: This is the phenomenon where the oscillations of a wave are restricted to a single plane. Unpolarized light has electric field oscillations in all directions perpendicular to the direction of travel. A polarizer can filter this light so that only oscillations in one specific direction pass through. This filtering process is only possible if the oscillations are transverse to the direction of motion. A longitudinal wave (like sound) cannot be polarized because its oscillations have only one possible direction—along the direction of propagation.

Thus, polarization is the only phenomenon in the list that is exclusive to transverse waves and therefore proves the transverse nature of electromagnetic waves.


Step 4: Final Answer:

Polarization of light proves the transverse nature of electromagnetic waves. This corresponds to option (C).
Quick Tip: Remember this key distinction: Interference and diffraction prove the \textbf{wave nature} of light. The photoelectric effect proves the \textbf{particle nature} of light. Polarization proves the \textbf{transverse wave nature} of light.

Step 1: Understanding the Concept:

This question involves the prism formula, which relates the refractive index (\(n\)) of a prism's material to its prism angle (\(A\)) and the angle of minimum deviation (\(D_m\)).


Step 2: Key Formula or Approach:

The prism formula is: \[ n = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
We are given the special condition that the angle of the prism is equal to the angle of minimum deviation, i.e., \(A = D_m\). We need to substitute this condition into the formula and simplify.


Step 3: Detailed Explanation:

Substitute \(D_m = A\) into the prism formula: \[ n = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{2A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]
Now, we use the double-angle identity for sine: \(\sin(A) = 2\sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)\).
Substitute this into the expression for \(n\): \[ n = \frac{2\sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Cancel the common term \(\sin\left(\frac{A}{2}\right)\) from the numerator and denominator: \[ n = 2\cos\left(\frac{A}{2}\right) \]

Step 4: Final Answer:

The refractive index of the material is \(2\cos\left(\frac{A}{2}\right)\). This corresponds to option (A).
Quick Tip: Knowing trigonometric identities, especially double-angle and half-angle formulas, is crucial for simplifying expressions in optics and wave problems. \(\sin(2\theta)=2\sin\theta\cos\theta\) is one of the most frequently used identities.

Step 1: Understanding the Concept:

This question asks for the definition of a wavefront according to Huygens' principle, which is a fundamental concept in the wave theory of light.


Step 2: Detailed Explanation:

Huygens' Principle describes how waves propagate. Let's analyze the core ideas:

A wavefront is defined as the locus of all points in a medium that are in the same state of vibration, or have the same phase. For example, if you drop a pebble in a pond, the circular ripples are wavefronts—all points on a given circle are at a crest at the same time. This makes a wavefront a surface of constant phase.
Huygens' principle states that every point on a given wavefront can be considered as a source of new secondary spherical wavelets. The new wavefront at a later time is the surface tangent to all these secondary wavelets (their envelope).

Now let's look at the options:

(A) a single ray of light: A ray is a line that indicates the direction of energy propagation. It is perpendicular to the wavefront, not the wavefront itself.
(B) a surface of constant phase: This is the precise definition of a wavefront. All points on the surface are oscillating in unison.
(C) a surface of varying phase: This is the opposite of the definition.
(D) a random arrangement of waves: A wavefront is a highly ordered surface, not a random arrangement.
(E) a region where crests and troughs overlap: This describes interference, not a single wavefront.


Step 4: Final Answer:

According to Huygens Principle, a wavefront is a surface of constant phase. This corresponds to option (B).
Quick Tip: Visualize wavefronts and rays: For a point source: Wavefronts are concentric spheres, rays are radial lines pointing outwards. For a line source: Wavefronts are coaxial cylinders, rays are radial lines in planes perpendicular to the source. For a plane wave (source at infinity): Wavefronts are parallel planes, rays are parallel lines perpendicular to the planes. In all cases, rays are perpendicular to wavefronts.

Step 1: Understanding the Concept:

This question deals with Young's double-slit experiment and the factors affecting the fringe width of the resulting interference pattern.


Step 2: Key Formula or Approach:

The fringe width (\(\beta\)) in a Young's double-slit experiment is given by the formula: \[ \beta = \frac{\lambda D}{d} \]
where:

\(\lambda\) is the wavelength of the light.
\(D\) is the distance from the slits to the screen.
\(d\) is the separation between the two slits.

We need to analyze how \(\beta\) changes when \(\lambda\) is changed, keeping \(D\) and \(d\) constant.


Step 3: Detailed Explanation:

From the formula \(\beta = \frac{\lambda D}{d}\), we can see that the fringe width \(\beta\) is directly proportional to the wavelength \(\lambda\), assuming \(D\) and \(d\) are kept constant ("with the same set up"). \[ \beta \propto \lambda \]
This means that if we want to change the fringe width by a certain factor, the wavelength must be changed by the same factor.
The problem states that the new fringe width (\(\beta_{new}\)) should be 3 times the original fringe width (\(\beta_{original}\)). \[ \beta_{new} = 3 \times \beta_{original} \]
Since \(\beta \propto \lambda\), this implies that the new wavelength (\(\lambda_{new}\)) must be 3 times the original wavelength (\(\lambda_{original}\)). \[ \lambda_{new} = 3 \times \lambda_{original} \]
Therefore, the wavelength of the light used must be tripled. The specific numerical values (600 nm, 0.5 mm, 2 m) are not needed to answer the question about the relationship, but they could be used to calculate the initial fringe width if required.


Step 4: Final Answer:

The wavelength of light used is tripled. This corresponds to option (A).
Quick Tip: Proportionality is a key concept in physics. For formulas like \(\beta = \frac{\lambda D}{d}\), ask yourself how the result changes if one variable is doubled, halved, etc., while others are constant. This is a very common type of exam question that tests conceptual understanding rather than just calculation.

Step 1: Understanding the Concept:

This question relates to the photoelectric effect and Einstein's photoelectric equation. The stopping potential (\(V_s\)) is the minimum negative voltage applied to the collector plate that stops even the most energetic photoelectrons from reaching it. It is a measure of the maximum kinetic energy of the emitted electrons.


Step 2: Key Formula or Approach:

Einstein's photoelectric equation is: \[ K_{max} = hf - \phi_0 \]
where:

\(K_{max}\) is the maximum kinetic energy of the photoelectrons.
\(h\) is Planck's constant.
\(f\) is the frequency of the incident light.
\(\phi_0\) is the work function of the material (\(\phi_0 = hf_0\), where \(f_0\) is the threshold frequency).

The stopping potential \(V_s\) is related to the maximum kinetic energy by: \[ K_{max} = eV_s \]
where \(e\) is the elementary charge.
Combining these two equations gives the relationship between stopping potential and frequency.


Step 3: Detailed Explanation:

By equating the two expressions for \(K_{max}\), we get: \[ eV_s = hf - \phi_0 \]
Solving for the stopping potential \(V_s\): \[ V_s = \frac{h}{e}f - \frac{\phi_0}{e} \]
This equation is in the form of a linear equation \(y = mx + c\), where:

\(y = V_s\) (the stopping potential)
\(x = f\) (the frequency)
\(m = \frac{h}{e}\) (the slope, which is a positive constant)
\(c = -\frac{\phi_0}{e}\) (the y-intercept, which is a negative constant)

Since the equation for \(V_s\) is a linear function of \(f\) with a positive slope, the stopping potential increases linearly with the frequency \(f\).
The intensity of light affects the number of photoelectrons emitted per second (the photoelectric current), but not their maximum kinetic energy or the stopping potential.


Step 4: Final Answer:

The stopping potential increases linearly with \(f\). This corresponds to option (A).
Quick Tip: Remember the key dependencies in the photoelectric effect: \textbf{Maximum KE / Stopping Potential} depends on \textbf{frequency} (linearly). \textbf{Photoelectric Current} depends on \textbf{intensity}. These are the core experimental results that classical wave theory could not explain.

Step 1: Understanding the Concept:

This question asks for the name of a specific process of electron emission. There are several ways to provide electrons in a metal with enough energy to overcome the work function and escape the surface.


Step 2: Detailed Explanation:

Let's define the different types of emission listed:

Photoelectric emission: Emission of electrons when light (photons) of suitable frequency strikes a metal surface.
Field emission (or cold cathode emission): Emission of electrons induced by a very strong external electric field (of the order of \(10^8\) V/m). The strong field drastically thins the potential barrier at the metal surface, allowing electrons to "tunnel" through and escape, even without having high thermal or photon-induced energy.
Thermionic emission: Emission of electrons when a metal is heated to a high temperature. The thermal energy gives the electrons enough kinetic energy to overcome the work function.
Beta emission: This is a type of radioactive decay where a beta particle (an electron or a positron) is emitted from an atomic nucleus. It is a nuclear process, not a process of electron emission from a metal surface due to external stimuli.
Gamma emission: This is also a type of radioactive decay where a gamma ray (a high-energy photon) is emitted from an excited nucleus. It does not involve the emission of electrons from a metal.

The question specifically describes electron emission due to a "very strong electric field", which matches the definition of field emission.


Step 4: Final Answer:

The process is called field emission. This corresponds to option (B).
Quick Tip: The names of the emission processes are quite descriptive: \textbf{Photo-} (light) \(\implies\) light causes emission. \textbf{Thermionic-} (thermal/heat) \(\implies\) heat causes emission. \textbf{Field-} (electric field) \(\implies\) electric field causes emission.

Step 1: Understanding the Concept:

This question asks for the typical length scale associated with the size (radius or diameter) of an atomic nucleus.


Step 2: Key Formula or Approach:

The empirical formula for the radius \(R\) of a nucleus is: \[ R = R_0 A^{1/3} \]
where \(A\) is the mass number of the nucleus and \(R_0\) is a constant approximately equal to \(1.2 \times 10^{-15}\) m. The unit \(10^{-15}\) m is also known as a femtometer (fm) or a fermi.


Step 3: Detailed Explanation:

Let's calculate the radius for a few nuclei:

For Hydrogen (\(A=1\)): \(R \approx 1.2 \times 10^{-15}\) m.
For Carbon (\(A=12\)): \(R \approx (1.2 \times 10^{-15}) \times (12)^{1/3} \approx (1.2 \times 2.29) \times 10^{-15} \approx 2.7 \times 10^{-15}\) m.
For Uranium (\(A=238\)): \(R \approx (1.2 \times 10^{-15}) \times (238)^{1/3} \approx (1.2 \times 6.2) \times 10^{-15} \approx 7.4 \times 10^{-15}\) m.

As seen from these examples, the size of atomic nuclei ranges from about 1 to 10 femtometers. Therefore, the characteristic order of magnitude for the size of a nucleus is \(10^{-15}\) m.
Option (A) is \(10^{-15}\) m, which is the correct order of magnitude.
The reason for the question being marked as "Cancelled" is unclear, as option (A) is scientifically correct. It could be due to a printing error in the exam paper that is not visible here.


Step 4: Final Answer:

The size of a nucleus is of the order of \(10^{-15}\) m. This corresponds to option (A). We acknowledge the question was marked as cancelled.
Quick Tip: It's useful to have a sense of scale for different physical objects: Nucleus: \(\sim 10^{-15}\) m (femtometer) Atom: \(\sim 10^{-10}\) m (angstrom) Visible light wavelength: \(\sim 10^{-7}\) m (hundreds of nanometers) The size of an atom is about 100,000 times larger than the size of its nucleus.

Step 1: Understanding the Concept:

This question asks to identify which type of radiation from the given list has the shortest wavelength. This requires knowledge of the electromagnetic spectrum and the nature of different types of radiation from nuclear decay.


Step 2: Detailed Explanation:

Let's order the given radiations by their typical wavelengths, from longest to shortest.

Ultra-violet (UV) rays: Part of the electromagnetic spectrum with wavelengths shorter than visible light, typically 10 nm to 400 nm.
X-rays: Electromagnetic radiation with wavelengths shorter than UV rays, typically 0.01 nm to 10 nm.
Gamma (\(\gamma\)) rays: Electromagnetic radiation emitted from the nucleus during radioactive decay. They have the shortest wavelengths in the electromagnetic spectrum, typically less than 0.01 nm (or 10 pm).

Now let's consider the other options, which are not electromagnetic waves:

Alpha (\(\alpha\)) rays: These are streams of alpha particles (Helium nuclei). They are particles, not electromagnetic radiation. We can associate a de Broglie wavelength with them, but they are not part of the electromagnetic spectrum.
Beta (\(\beta\)) rays: These are streams of beta particles (electrons or positrons). Like alpha rays, they are particles, not electromagnetic radiation.

Comparing the electromagnetic radiations (UV, X-rays, gamma rays), gamma rays have the shortest wavelengths.


Step 4: Final Answer:

The radiations of extremely short wavelength are gamma rays. This corresponds to option (C).
Quick Tip: A useful mnemonic for the electromagnetic spectrum in order of increasing frequency (and decreasing wavelength) is: \textbf{R}oman \textbf{M}en \textbf{I}nvented \textbf{V}ery \textbf{U}nusual \textbf{X}-ray \textbf{G}uns. (Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma ray)

Step 1: Understanding the Concept:

This question asks to identify a specific naturally occurring crystal that was historically important in the early days of radio technology for its use as a detector.


Step 2: Detailed Explanation:

In early radio receivers, particularly in crystal radios, a detector was needed to demodulate the amplitude-modulated (AM) radio signal. This process, also known as rectification, converts the high-frequency alternating current signal from the antenna into a direct current signal that varies with the audio information.

Galena: This is the natural mineral form of lead(II) sulfide (PbS). A crystal of galena, when touched by a fine wire (called a "cat's whisker"), forms a crude point-contact semiconductor diode. This diode allows current to flow more easily in one direction than the other, thereby rectifying the radio signal. Galena was one of the most common and effective materials used in crystal detectors for early radios.
Ruby: This is a gemstone, a variety of the mineral corundum (aluminum oxide) colored by chromium. It is famous for its use in early lasers (ruby laser) but not as a radio detector.
Silicon and Germanium: These are elemental semiconductors that became the basis for modern solid-state electronics, including diodes and transistors, starting in the mid-20th century. While highly purified, man-made silicon and germanium are used for modern detectors, the question asks for a naturally occurring crystal used historically.
Zinc selenide: This is a semiconductor material used in modern applications like LEDs and infrared windows, but it was not the crystal used in early radio detectors.


Step 3: Final Answer:

The naturally occurring crystal used as a detector of radio waves was Galena. This corresponds to option (B).
Quick Tip: The "crystal" in "crystal radio" refers to the galena crystal (or sometimes other minerals like pyrite) that formed the rectifying diode, which was the key component for detecting the audio signal from the radio wave.

Step 1: Understanding the Concept:

This question is about the charge carrier concentrations in extrinsic semiconductors, specifically a p-type semiconductor. Semiconductors can be intrinsic (pure) or extrinsic (doped with impurities).


Step 2: Detailed Explanation:


Intrinsic Semiconductor: In a pure semiconductor (like silicon or germanium), thermal energy creates electron-hole pairs. The concentration of electrons (\(n_e\)) is equal to the concentration of holes (\(n_h\)), and this is known as the intrinsic carrier concentration (\(n_i\)). So, \(n_e = n_h = n_i\).
Extrinsic Semiconductor: To increase conductivity, pure semiconductors are doped with impurities.

n-type semiconductor: Doped with a pentavalent impurity (e.g., Phosphorus in Silicon). These 'donor' atoms provide extra free electrons. In an n-type semiconductor, electrons are the majority charge carriers, and holes are the minority charge carriers. Thus, \(n_e \gg n_h\).
p-type semiconductor: Doped with a trivalent impurity (e.g., Boron in Silicon). These 'acceptor' atoms create vacancies for electrons, which are known as holes. In a p-type semiconductor, holes are the majority charge carriers, and electrons are the minority charge carriers. Thus, \(n_h \gg n_e\).


The question asks about a p-type semiconductor, where holes are the majority carriers. Therefore, the concentration of holes is much greater than the concentration of electrons.


Step 3: Final Answer:

In a p-type semiconductor, \(n_h \gg n_e\). This corresponds to option (C).
Quick Tip: A simple mnemonic: \textbf{n}-type has a majority of \textbf{n}egative charge carriers (electrons). \textbf{p}-type has a majority of \textbf{p}ositive charge carriers (holes).

Step 1: Understanding the Concept:

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution.


Step 2: Key Formula or Approach:

1. Calculate the number of moles of the solute (KCl) using the formula:
\[ Moles = \frac{Mass of solute}{Molar mass of solute} \]
2. Calculate the molarity (M) using the formula:
\[ Molarity (M) = \frac{Moles of solute}{Volume of solution in litres} \]

Step 3: Detailed Explanation:

1. Calculate the moles of KCl:

Given:

Mass of KCl = 149 g
Molar mass of KCl = 74.5 g/mol
\[ Moles of KCl = \frac{149 g}{74.5 g/mol} = 2 mol \]
2. Calculate the molarity:

Given:

Volume of solution = 10 litres
\[ Molarity (M) = \frac{2 mol}{10 L} = 0.2 mol/L = 0.2 M \]

Step 4: Final Answer:

The molarity of the solution is 0.2 M. This corresponds to option (D).
Quick Tip: Always ensure the volume is in litres when calculating molarity. If the volume is given in milliliters (mL) or cubic centimeters (cm\(^3\)), convert it to litres by dividing by 1000.

Step 1: Understanding the Concept:

This question tests fundamental concepts of atomic structure, including orbital energies, quantum numbers, nodes, and electron capacity of shells. We need to identify the incorrect statement.


Step 2: Detailed Explanation:

Let's analyze each statement:

(A) For multi-electron atoms, the energy of orbitals depends on both n and l. As the atomic number (Z) increases, the nuclear charge increases. This increased attraction lowers the energy of all orbitals (makes them more negative). The statement says the energy *increases*, which is incorrect. The energy levels become lower (more stable). However, let's re-read it. It could mean the splitting between subshells increases. But the most direct interpretation is that the energy values themselves increase, which is false. There might be an ambiguity here, but let's check other options which might be more definitively wrong.
(B) For a p-orbital (or d, f), the lobes represent regions of high electron probability. The point or plane where the lobes meet is called a nodal plane or nodal point, where the wavefunction, and thus the probability density (\(|\psi|^2\)), is zero. This statement is true.
(C) This is the Aufbau principle, specifically the (n+l) rule. Orbitals are filled in order of increasing (n+l) value. For orbitals with the same (n+l) value, the one with the lower n value has lower energy. So, a lower (n+l) value corresponds to lower energy. This statement is true.
(D) The total number of nodes for an orbital with principal quantum number n is indeed (n-1). These are divided into (n-l-1) radial nodes and l angular nodes. This statement is true.
(E) A shell with principal quantum number 'n' contains 'n' subshells (l=0 to n-1). The total number of orbitals in the shell is \(n^2\). Since each orbital can hold a maximum of 2 electrons (Pauli Exclusion Principle), the maximum number of electrons in the shell is \(2 \times (number of orbitals) = 2n^2\). The statement says the maximum number of electrons is \(n^2\), which is incorrect. It gives the number of orbitals, not electrons.

Comparing the statements, statement (E) is definitively and unambiguously incorrect based on a fundamental formula. Statement (A) is also technically incorrect but might be considered true under a specific interpretation about relative energies. However, (E) is a direct contradiction of a well-known rule.


Step 3: Final Answer:

The statement that is NOT true is (E).
Quick Tip: Remember these key formulas for a shell with principal quantum number \(n\): Number of subshells = \(n\) Number of orbitals = \(n^2\) Maximum number of electrons = \(2n^2\)

Step 1: Understanding the Concept:

This question asks about the physical significance of the different quantum numbers that describe an electron in an atom. Each quantum number specifies a particular property of the electron's state.


Step 2: Detailed Explanation:

Let's review the four main quantum numbers:

Principal Quantum Number (n): This determines the principal electron shell, and it is the primary factor in determining the energy of the electron. It also relates to the average distance of the electron from the nucleus. It can have integer values n = 1, 2, 3, ...
Azimuthal or Angular Momentum Quantum Number (l): This determines the shape of the orbital and the subshell it belongs to (s, p, d, f). It can have integer values from 0 to n-1. For example, l=0 is an s-orbital (spherical), l=1 is a p-orbital (dumbbell-shaped), etc.
Magnetic Quantum Number (\(m_l\)): This determines the orientation of the orbital in space relative to an external magnetic field. For a given value of l, \(m_l\) can take integer values from -l to +l, including 0. For example, for a p-orbital (l=1), \(m_l\) can be -1, 0, or +1, corresponding to the \(p_x\), \(p_y\), and \(p_z\) orbitals, which are oriented along the x, y, and z axes, respectively.
Spin Quantum Number (\(m_s\)): This describes the intrinsic angular momentum of the electron, which is quantized and is often visualized as the electron's "spin". It can have one of two values: +1/2 or -1/2.

The question specifically asks which quantum number determines the orientation of the orbital. As explained above, this is the role of the magnetic quantum number, \(m_l\).


Step 3: Final Answer:

The magnetic quantum number, \(m_l\), determines the orientation of the orbital. This corresponds to option (C).
Quick Tip: A simple summary of quantum numbers: \textbf{n}: Shell (Energy/Size) \textbf{l}: Subshell (Shape) \textbf{\(m_l\)}: Orbital (Orientation) \textbf{\(m_s\)}: Electron (Spin)

Step 1: Understanding the Concept:

This question tests knowledge about the f-block elements (Lanthanoids and Actinoids) in the periodic table, including their definition, position, properties, and electronic configuration. We need to identify the incorrect statement.


Step 2: Detailed Explanation:

Let's analyze each statement:

(A) This is the definition of f-block elements. The differentiating (last) electron enters the antepenultimate f-subshell. This statement is correct.
(B) The f-block consists of two series:

The Lanthanoid series (4f-series) goes from Cerium (Ce, Z=58) to Lutetium (Lu, Z=71).
The Actinoid series (5f-series) goes from Thorium (Th, Z=90) to Lawrencium (Lr, Z=103).

This statement correctly identifies the atomic number ranges for the two series. This statement is correct.
(C) All actinoid elements are radioactive. Some, like Uranium and Thorium, are naturally occurring, while the heavier ones are synthetically produced. This statement is correct.
(D) The f-block consists of the Lanthanoid series (14 elements) and the Actinoid series (14 elements). In total, there are \(14 + 14 = 28\) f-block elements. This statement is correct.
(E) The general outer electronic configuration for Actinoids involves filling the 5f subshell. The principal quantum number of the outermost shell (ns\(^2\)) for the Actinoid series is n=7. The f-orbital being filled is the antepenultimate one, which is \((n-2)f\). The d-orbital is the penultimate one, \((n-1)d\).
Therefore, the general configuration should be written as \([Rn] (n-2)f^{1-14} (n-1)d^{0-1} ns^2\), which for Actinoids is \([Rn] 5f^{1-14} 6d^{0-1} 7s^2\). The statement gives the f-orbital as \((n-1)f\), which would be the 6f orbital. This is incorrect. The f-subshell being filled is always two shells inside the outermost shell.


Step 3: Final Answer:

The incorrect statement is (E) because the f-orbital being filled should be designated as \((n-2)f\), not \((n-1)f\).
Quick Tip: For electron configurations: \textbf{s-block and p-block} (main group): Electrons fill the outermost shell (\(n\)). \textbf{d-block} (transition metals): Electrons fill the penultimate shell (\(n-1\)). \textbf{f-block} (inner transition metals): Electrons fill the antepenultimate shell (\(n-2\)).

Step 1: Understanding the Concept:

This question asks for a specific bond angle in the ethene molecule (C\(_2\)H\(_4\)). This is determined by the hybridization of the carbon atoms and VSEPR theory.


Step 2: Detailed Explanation:

The structure of ethene is H\(_2\)C=CH\(_2\).

Each carbon atom is bonded to two hydrogen atoms and one other carbon atom via a double bond.
To form these bonds (one sigma C-C, two sigma C-H), each carbon atom uses three hybrid orbitals. This corresponds to sp\(^2\) hybridization.
According to VSEPR theory, for a central atom with three electron domains (one C=C double bond and two C-H single bonds count as three domains), the ideal geometry is trigonal planar, with bond angles of 120\(^\circ\).
However, in the actual ethene molecule, the electron domains are not identical. The C=C double bond contains more electron density than the C-H single bonds. This greater electron density in the double bond causes it to exert a stronger repulsion on the single bonds.
This stronger repulsion pushes the C-H bonds slightly closer together, reducing the H-C-H bond angle to a value slightly less than the ideal 120\(^\circ\).
Conversely, the H-C=C bond angle is slightly larger than 120\(^\circ\) (experimentally around 121.3\(^\circ\)).
The experimentally determined H-C-H bond angle in ethene is approximately 117.6\(^\circ\).

Among the given options, 117.6\(^\circ\) is the most accurate value.


Step 3: Final Answer:

The H-C-H bond angle in ethene is 117.6\(^\circ\). This corresponds to option (A).
Quick Tip: Remember VSEPR theory's refinement: lone pairs and multiple bonds are more repulsive than single bonds. They tend to compress the angles between the single bonds. In ethene, the C=C double bond compresses the H-C-H angle from the ideal 120\(^\circ\) to about 117\(^\circ\).

Step 1: Understanding the Concept:

This question asks for the defining condition of an adiabatic process in thermodynamics.


Step 2: Detailed Explanation:

Let's define the conditions listed in the options:

(A) \(\Delta T = 0\): This is the condition for an isothermal process (constant temperature).
(B) \(\Delta P = 0\): This is the condition for an isobaric process (constant pressure).
(C) q = 0: This is the condition for an adiabatic process. An adiabatic process is one in which no heat (q) is transferred into or out of the system. This can be achieved by carrying out the process in a perfectly insulated container or by carrying it out so rapidly that there is no time for significant heat transfer.
(D) w = 0: This is the condition for an isochoric process (constant volume), because work done by/on a gas is \(w = -\int P_{ext} dV\), and if volume is constant (\(dV=0\)), then \(w=0\).
(E) \(\Delta U = 0\): This is the condition for constant internal energy. For an ideal gas, this is equivalent to an isothermal process (\(\Delta T = 0\)). For a cyclic process, the net change in internal energy is also zero.

The defining characteristic of an adiabatic process is that there is no heat exchange with the surroundings.


Step 3: Final Answer:

The correct condition for an adiabatic process is q = 0. This corresponds to option (C).
Quick Tip: Associate the thermodynamic process names with their defining constant variable: Iso\textbf{thermal} \(\to\) Temperature Iso\textbf{baric} \(\to\) Pressure (bar is a unit of pressure) Iso\textbf{choric} \(\to\) Volume \textbf{Adia}batic \(\to\) No Heat (from Greek 'adiabatos' - impassable)

Step 1: Understanding the Concept:

We need to determine the signs of the enthalpy change (\(\Delta H\)) and entropy change (\(\Delta S\)) for the decomposition of phosphorus pentachloride gas.


Step 2: Detailed Explanation:

1. Enthalpy Change (\(\Delta H\)):

The reaction is a decomposition reaction: \(PCl_5(g) \to PCl_3(g) + Cl_2(g)\).
Decomposition reactions involve the breaking of chemical bonds. The breaking of bonds requires an input of energy. In this case, we are breaking two P-Cl bonds in the PCl\(_5\) molecule. Since bond breaking is an endothermic process, the reaction absorbs heat from the surroundings.
Therefore, the enthalpy change is positive. \[ \Delta H > 0 \]
2. Entropy Change (\(\Delta S\)):

Entropy (\(\Delta S\)) is a measure of the disorder or randomness of a system.
In the given reaction, we start with 1 mole of gaseous reactant (\(PCl_5\)) and produce 2 moles of gaseous products (\(1 mole of PCl_3 + 1 mole of Cl_2\)).
The number of moles of gas increases (\(\Delta n_g = 2 - 1 = 1 > 0\)). An increase in the number of gas molecules leads to a significant increase in the randomness and disorder of the system.
Therefore, the entropy change is positive. \[ \Delta S > 0 \]

Step 3: Final Answer:

For this decomposition, both \(\Delta H\) and \(\Delta S\) are positive. This corresponds to option (B).
Quick Tip: Quick rules for predicting signs of \(\Delta S\): If the number of moles of gas increases in a reaction, \(\Delta S\) is usually positive. If the number of moles of gas decreases, \(\Delta S\) is usually negative. Phase changes: \(S_{solid} < S_{liquid} < S_{gas}\). Melting, boiling, or sublimation lead to \(\Delta S > 0\).

Step 1: Understanding the Concept:

This problem involves the relationship between the equilibrium constant expressed in terms of partial pressures (K\(_p\)) and the equilibrium constant expressed in terms of molar concentrations (K\(_c\)).


Step 2: Key Formula or Approach:

The relationship between K\(_p\) and K\(_c\) is given by the equation: \[ K_p = K_c(RT)^{\Delta n_g} \]
where:

R is the ideal gas constant.
T is the absolute temperature in Kelvin.
\(\Delta n_g\) is the change in the number of moles of gas in the balanced chemical equation (\(\Delta n_g = moles of gaseous products - moles of gaseous reactants\)).


Step 3: Detailed Explanation:

The given reaction is: \(2NOCl(g) \rightleftharpoons 2NO(g) + Cl_2(g)\).

1. Calculate \(\Delta n_g\):

Moles of gaseous products = (moles of NO) + (moles of Cl\(_2\)) = 2 + 1 = 3.
Moles of gaseous reactants = (moles of NOCl) = 2. \[ \Delta n_g = 3 - 2 = 1 \]
2. Use the relationship formula:

We are given:

\(K_p = 8.21 \times 10^{-2}\)
\(R = 0.0821\) L atm mol\(^{-1}\) K\(^{-1}\)
\(T = 1000\) K

Substitute these values into the formula \(K_p = K_c(RT)^{\Delta n_g}\): \[ 8.21 \times 10^{-2} = K_c (0.0821 \times 1000)^1 \] \[ 8.21 \times 10^{-2} = K_c (82.1) \]
Now, solve for K\(_c\): \[ K_c = \frac{8.21 \times 10^{-2}}{82.1} \]
We can write \(82.1\) as \(8.21 \times 10^1\). \[ K_c = \frac{8.21 \times 10^{-2}}{8.21 \times 10^1} = \frac{10^{-2}}{10^1} = 10^{-2-1} = 10^{-3} \]

Step 4: Final Answer:

The value of K\(_c\) is \(10^{-3}\). This corresponds to option (A).
Quick Tip: The value of R can be approximated or manipulated for quick calculations. Notice that \(8.21 \times 10^{-2}\) and \(0.0821\) are related by a factor of 10. Recognizing these patterns can speed up the arithmetic.

Step 1: Understanding the Concept:

This question asks about the role of a catalyst in a reversible reaction at equilibrium. A catalyst is a substance that increases the rate of a chemical reaction without being consumed in the process.


Step 2: Detailed Explanation:

A catalyst works by providing an alternative reaction pathway with a lower activation energy (\(E_a\)). Let's analyze its effect on an equilibrium: \( Reactants \rightleftharpoons Products \).

A catalyst lowers the activation energy for the forward reaction, thereby increasing the rate of the forward reaction.
Crucially, it also lowers the activation energy for the reverse reaction by the exact same amount. This increases the rate of the reverse reaction as well.
Because the rates of both the forward and reverse reactions are increased equally, the catalyst does not change the position of the equilibrium. The equilibrium composition (the amounts of reactants and products at equilibrium) and the value of the equilibrium constant (K) remain unchanged.
The only effect of the catalyst is to allow the system to reach equilibrium faster.

Now let's evaluate the options:

(A) and (B) are incorrect because the catalyst affects both forward and reverse reactions.
(C) is incorrect. The effectiveness of a catalyst is not related to the magnitude of the equilibrium constant K.
(D) is incorrect. A catalyst does not affect the equilibrium composition.
(E) is correct. It correctly states that the catalyst lowers the activation energy for both the forward and reverse reactions by the same amount.


Step 3: Final Answer:

The correct statement is that a catalyst lowers the activation energy for the forward and reverse reaction by the same amount. This corresponds to option (E).
Quick Tip: Think of a catalyst as a "matchmaker" for a reaction. It helps reactants get together to form products more easily, and it also helps products break apart to form reactants more easily. It facilitates the process in both directions but doesn't favor one side over the other at equilibrium.

Step 1: Understanding the Concept:

This question asks for the incorrect definition of reduction. Reduction is one half of a redox (reduction-oxidation) reaction.


Step 2: Detailed Explanation:

Let's define reduction from different perspectives:

In terms of Oxygen/Hydrogen (Classical definition):

Reduction is the removal of oxygen from a substance or the addition of hydrogen to a substance.
Oxidation is the addition of oxygen or the removal of hydrogen.

In terms of Electrons (Modern definition):

Reduction Is Gain of electrons. (Mnemonic: RIG - Reduction Is Gain). This is the same as the addition of an electron.
Oxidation Is Loss of electrons. (Mnemonic: OIL - Oxidation Is Loss). This is the same as the removal of an electron.

In terms of Oxidation Number:

Reduction is a decrease in oxidation number.
Oxidation is an increase in oxidation number.


Now let's check the options against these definitions for reduction:

(A) Removal of oxygen - Correct definition of reduction.
(B) Addition of hydrogen - Correct definition of reduction.
(C) Addition of electron - Correct definition of reduction.
(D) Decrease in oxidation number - Correct definition of reduction.
(E) Removal of an electron - This is the definition of oxidation, not reduction.


Step 4: Final Answer:

The incorrect statement for the concept of reduction is "Removal of an electron". This corresponds to option (E).
Quick Tip: Use the mnemonic "OIL RIG" to remember the electron transfer definitions: \textbf{O}xidation \textbf{I}s \textbf{L}oss (of electrons) \textbf{R}eduction \textbf{I}s \textbf{G}ain (of electrons)

Step 1: Understanding the Concept:

This problem relates three important quantities in electrochemistry: conductivity (\(\kappa\)), resistance (\(R\)), and the cell constant (\(G^*\)). The cell constant is a geometric factor specific to the conductivity cell used.


Step 2: Key Formula or Approach:

The relationship between these quantities is given by the formula: \[ \kappa = \frac{1}{R} \times G^* \]
or \[ R = \rho \frac{l}{A} \] where conductivity \(\kappa = 1/\rho\) and cell constant \(G^* = l/A\).
So, \(R = \frac{1}{\kappa} \frac{l}{A} \implies R = \frac{G^*}{\kappa}\).
We can rearrange this to solve for the cell constant: \[ G^* = \kappa \times R \]

Step 3: Detailed Explanation:

We are given the following values:

Conductivity, \(\kappa = 0.012\) ohm\(^{-1}\) cm\(^{-1}\) (also written as S cm\(^{-1}\)).
Resistance, \(R = 50\) ohm (\(\Omega\)).

Note: A "decinormal" solution means 0.1 N, but this information is not needed for the calculation as the conductivity is directly provided.

Now, we use the formula to find the cell constant, \(G^*\): \[ G^* = \kappa \times R \] \[ G^* = (0.012 ohm^{-1} cm^{-1}) \times (50 ohm) \]
The "ohm" and "ohm\(^{-1}\)" units cancel out, leaving the unit cm\(^{-1}\). \[ G^* = 0.012 \times 50 cm^{-1} \] \[ G^* = 0.6 cm^{-1} \]

Step 4: Final Answer:

The cell constant value is 0.6 cm\(^{-1}\). This corresponds to option (E).
Quick Tip: Remember the fundamental relationships: Resistance \(R\) depends on material and geometry. Resistivity \(\rho\) depends only on the material. Conductance \(G = 1/R\). Conductivity \(\kappa = 1/\rho\). The key formula connecting them via the cell geometry (\(l/A\)) is \(R = \rho \frac{l}{A}\) or \(\kappa = G \frac{l}{A}\). The cell constant is \(G^* = l/A\).

Step 1: Understanding the Concept:

This problem involves the colligative property of freezing point depression. The addition of a non-volatile solute to a solvent lowers the freezing point of the solvent. The extent of this depression is proportional to the molality of the solution.


Step 2: Key Formula or Approach:

1. The formula for freezing point depression is: \(\Delta T_f = K_f \times m\), where \(\Delta T_f\) is the depression in freezing point, \(K_f\) is the freezing point depression constant, and \(m\) is the molality of the solution.
2. Molality (\(m\)) is defined as: \(m = \frac{moles of solute}{mass of solvent in kg}\).
3. Moles of solute = \(\frac{mass of solute}{molar mass of solute}\).
We can combine these to solve for the molar mass of the solute.


Step 3: Detailed Explanation:

First, let's write the combined formula. Let \(w_2\) be the mass of the solute, \(M_2\) be the molar mass of the solute, and \(w_1\) be the mass of the solvent. \[ m = \frac{w_2 / M_2}{w_1 (in kg)} \]
So, the freezing point depression formula becomes: \[ \Delta T_f = K_f \times \frac{w_2}{M_2 \times w_1(in kg)} \]
We need to solve for the molar mass, \(M_2\): \[ M_2 = K_f \times \frac{w_2}{\Delta T_f \times w_1(in kg)} \]
Now, substitute the given values, making sure the units are correct:

\(\Delta T_f = 0.20\) K
\(K_f = 5\) K kg mol\(^{-1}\)
Mass of solute, \(w_2 = 1\) g
Mass of solvent (benzene), \(w_1 = 50\) g. We must convert this to kg: \(w_1 = \frac{50}{1000} = 0.05\) kg.
\[ M_2 = 5 \times \frac{1}{0.20 \times 0.05} \] \[ M_2 = \frac{5}{0.01} = 500 g/mol \]

Step 3: Final Answer:

The molar mass of the solute is 500 g/mol. This corresponds to option (A).
Quick Tip: A common mistake in colligative property problems is confusing molarity and molality. Remember: Mola\textbf{r}ity (M) = moles of solute / \textbf{L}iters of solution. Mola\textbf{l}ity (m) = moles of solute / \textbf{k}ilograms of solvent. Freezing point depression and boiling point elevation depend on molality.

Step 1: Understanding the Concept:

This question asks for the name of a specific term in the Arrhenius equation, which describes the temperature dependence of reaction rates.


Step 2: Key Formula or Approach:

The Arrhenius equation is: \[ k = A e^{-E_a / RT} \]
where:

\(k\) is the rate constant.
\(A\) is the pre-exponential factor.
\(E_a\) is the activation energy.
\(R\) is the ideal gas constant.
\(T\) is the absolute temperature.

We need to identify the correct name for the factor \(A\).


Step 3: Detailed Explanation:

The factor \(A\) in the Arrhenius equation is known by several names:

Pre-exponential factor: This is its most literal name, as it comes before the exponential term.
Frequency factor: This is its most common name. It relates to the frequency of collisions between reactant molecules that have the correct orientation to react.
Arrhenius factor: Another name for it.

Collision theory provides a more detailed interpretation of \(A\) as \(A = pZ\), where \(Z\) is the collision frequency (the total number of collisions per unit time per unit volume) and \(p\) is the steric or probability factor (the fraction of collisions that have the correct orientation).
Let's look at the options:

(A) probability factor: This is part of A (\(p\)), but not A itself.
(B) activation energy: This is \(E_a\), the term in the exponent.
(C) collision frequency: This is part of A (\(Z\)), but not A itself.
(D) reaction coordinate: This is the x-axis on a reaction energy profile diagram.
(E) frequency factor: This is the correct and most common name for the entire pre-exponential factor \(A\).


Step 4: Final Answer:

The pre-exponential factor is called the frequency factor. This corresponds to option (E).
Quick Tip: The Arrhenius equation breaks down the rate constant into two parts: the frequency factor \(A\) (representing the frequency of properly oriented collisions) and the exponential factor \(e^{-E_a / RT}\) (representing the fraction of collisions with sufficient energy to overcome the activation barrier).

Step 1: Understanding the Concept:

This problem involves the kinetics of a first-order reaction. We need to relate the half-life period to the rate constant and then use the rate law to find the reaction rate at a given concentration.


Step 2: Key Formula or Approach:

1. For a first-order reaction, the rate constant (\(k\)) is related to the half-life (\(t_{1/2}\)) by the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
2. The rate law for a first-order reaction is:
\[ Rate = k[A] \]
where [A] is the concentration of the reactant.


Step 3: Detailed Explanation:

Step 3.1: Calculate the rate constant (k):

We are given the half-life period, \(t_{1/2} = 10\) minutes. \[ k = \frac{0.693}{10 min} = 0.0693 min^{-1} \]
Step 3.2: Calculate the reaction rate:

We are given the concentration of A, [A] = 0.1 mol lit\(^{-1}\).
Using the rate law: \[ Rate = k[A] = (0.0693 min^{-1}) \times (0.1 mol lit^{-1}) \] \[ Rate = 0.00693 mol lit^{-1} min^{-1} \]
To match the format of the options, we can write this in scientific notation: \[ Rate = 6.93 \times 10^{-3} mol lit^{-1} min^{-1} \]

Step 4: Final Answer:

The rate of the reaction is \(6.93 \times 10^{-3}\) mol lit\(^{-1}\) min\(^{-1}\). This corresponds to option (B).
Quick Tip: For first-order reactions, the half-life is independent of the initial concentration. This is a unique characteristic that is often tested. Remember that 0.693 is the approximate value of \(\ln(2)\).

Step 1: Understanding the Concept:

This question tests the properties of transition metal ions, specifically Cr\(^{2+}\) and Mn\(^{3+}\). We need to analyze their electronic configuration, redox behavior, and magnetic properties.


Step 2: Detailed Explanation:

Let's evaluate each statement:

(i) Cr\(^{2+}\) is a reducing agent: A reducing agent gets oxidized itself. The electronic configuration of Cr (Z=24) is \([Ar] 3d^5 4s^1\). The configuration of Cr\(^{2+}\) is \([Ar] 3d^4\). Cr\(^{2+}\) can be easily oxidized to Cr\(^{3+}\) (\([Ar] 3d^3\)). The Cr\(^{3+}\) ion is particularly stable in aqueous solution because it has a half-filled \(t_{2g}\) level in an octahedral crystal field. Because Cr\(^{2+}\) is readily oxidized, it acts as a strong reducing agent. This statement is correct.
(ii) Mn\(^{3+}\) is an oxidising agent in acidic medium: An oxidizing agent gets reduced itself. The electronic configuration of Mn (Z=25) is \([Ar] 3d^5 4s^2\). The configuration of Mn\(^{3+}\) is \([Ar] 3d^4\). Mn\(^{3+}\) can be readily reduced to Mn\(^{2+}\) (\([Ar] 3d^5\)), which has a very stable half-filled d-subshell. Because Mn\(^{3+}\) is easily reduced, it acts as a strong oxidizing agent. This statement is correct.
(iii) Both Cr\(^{2+}\) and Mn\(^{3+}\) exhibit d\(^4\) electronic configuration: As determined above, the configuration of Cr\(^{2+}\) is \([Ar] 3d^4\) and the configuration of Mn\(^{3+}\) is also \([Ar] 3d^4\). This statement is correct.
(iv) The highest oxide of Mn is Mn\(_3\)O\(_4\): The highest possible oxidation state for Manganese is +7. This is found in the permanganate ion (MnO\(_4^-\)) and in the oxide Mn\(_2\)O\(_7\). Mn\(_3\)O\(_4\) is a mixed oxide with an average oxidation state of +8/3. Therefore, the highest oxide is Mn\(_2\)O\(_7\), not Mn\(_3\)O\(_4\). This statement is incorrect.
(v) Cr\(^{2+}\) and Mn\(^{3+}\) have the same magnetic moment...: Both ions have a d\(^4\) configuration. In a high-spin state (which is common for these ions), they will both have 4 unpaired electrons. The spin-only magnetic moment is calculated as \(\mu = \sqrt{n(n+2)}\) Bohr Magnetons, where n is the number of unpaired electrons. Since both have n=4, they will have the same magnetic moment. This statement is correct.

The correct statements are (i), (ii), (iii), and (v).


Step 3: Final Answer:

The combination of correct statements is (i), (ii), (iii) and (v). This corresponds to option (E).
Quick Tip: The stability of half-filled (\(d^5\)) and completely filled (\(d^{10}\)) d-orbitals, as well as the stability of half-filled \(t_{2g}\) orbitals in crystal field theory (\(t_{2g}^3\)), are key factors that determine the redox properties of transition metal ions.

Step 1: Understanding the Concept:

A substance is diamagnetic if it contains no unpaired electrons. All electrons are paired up in orbitals. Diamagnetic substances are weakly repelled by a magnetic field. We need to determine the electronic configuration of each given ion and check for unpaired electrons.


Step 2: Detailed Explanation:

Let's write the electronic configuration for each ion:

(A) Zn\(^{2+}\): The atomic number of Zn is 30. Its electronic configuration is \([Ar] 3d^{10} 4s^2\). To form the Zn\(^{2+}\) ion, it loses the two 4s electrons. The configuration is \([Ar] 3d^{10}\). The 3d subshell is completely filled, meaning all 10 electrons are paired. There are no unpaired electrons. Thus, Zn\(^{2+}\) is diamagnetic.
(B) Ni\(^{2+}\): The atomic number of Ni is 28. Its configuration is \([Ar] 3d^8 4s^2\). Ni\(^{2+}\) has the configuration \([Ar] 3d^8\). According to Hund's rule, the 3d\(^8\) configuration will have 2 unpaired electrons (3 filled orbitals, 2 half-filled orbitals). It is paramagnetic.
(C) Co\(^{2+}\): The atomic number of Co is 27. Its configuration is \([Ar] 3d^7 4s^2\). Co\(^{2+}\) has the configuration \([Ar] 3d^7\). It will have 3 unpaired electrons. It is paramagnetic.
(D) Cu\(^{2+}\): The atomic number of Cu is 29. Its configuration is \([Ar] 3d^{10} 4s^1\). Cu\(^{2+}\) has the configuration \([Ar] 3d^9\). It will have 1 unpaired electron. It is paramagnetic.
(E) Mn\(^{2+}\): The atomic number of Mn is 25. Its configuration is \([Ar] 3d^5 4s^2\). Mn\(^{2+}\) has the configuration \([Ar] 3d^5\). It has 5 unpaired electrons (a stable half-filled subshell). It is strongly paramagnetic.


Step 3: Final Answer:

The only ion with no unpaired electrons is Zn\(^{2+}\), making it diamagnetic. This corresponds to option (A).
Quick Tip: Paramagnetism is caused by unpaired electrons, while diamagnetism is a property of all substances and is caused by paired electrons. Paramagnetism is much stronger, so if a substance has any unpaired electrons, it will be paramagnetic. Only substances with all electrons paired are diamagnetic.

Step 1: Understanding the Concept:

This is a matching question that tests knowledge of important industrial catalysts and the processes they are used in.


Step 2: Detailed Explanation:

Let's match each catalyst in Column-I with its corresponding process in Column-II.

(a) TiCl\(_4\) + Al(CH\(_3\))\(_3\): This combination is known as a Ziegler-Natta catalyst, which is famously used for the polymerization of alkenes like ethene to produce polyethene. So, (a) matches with (iii).
(b) PdCl\(_2\): Palladium(II) chloride is the key catalyst in the Wacker process, which is the industrial method for oxidizing ethylene to acetaldehyde (ethanal). So, (b) matches with (iv).
(c) Fe: Finely divided iron, promoted with K\(_2\)O and Al\(_2\)O\(_3\), is the catalyst used in the Haber process (or Haber-Bosch process) for the synthesis of ammonia from nitrogen and hydrogen. So, (c) matches with (v).
(d) Ni: Finely divided Nickel (or Pt, Pd) is a common catalyst for the hydrogenation of fats and oils. This process converts unsaturated fats (containing C=C double bonds) into saturated fats, turning liquid oils into solid or semi-solid fats (like margarine). So, (d) matches with (ii).
(e) V\(_2\)O\(_5\): Vanadium pentoxide is the catalyst used in the Contact process, specifically for the key step of oxidizing sulfur dioxide (SO\(_2\)) to sulfur trioxide (SO\(_3\)), which is then used to manufacture sulfuric acid (H\(_2\)SO\(_4\)). So, (e) matches with (i).

Putting it all together: (a)-(iii), (b)-(iv), (c)-(v), (d)-(ii), (e)-(i).


Step 3: Final Answer:

The correct matching is (a)-(iii), (b)-(iv), (c)-(v), (d)-(ii), (e)-(i). This corresponds to option (A).
Quick Tip: Certain industrial processes and their catalysts are very frequently asked in exams. It's highly beneficial to memorize the following pairs: Haber Process (Fe), Contact Process (V\(_2\)O\(_5\)), Ostwald Process (Pt-Rh gauze), Hydrogenation (Ni/Pt/Pd), and Ziegler-Natta polymerization (TiCl\(_4\)/AlR\(_3\)).

Step 1: Understanding the Concept:

This question asks about the characteristic oxidation state of the lanthanoids, which are the first series of f-block elements.


Step 2: Detailed Explanation:

The lanthanoid elements have the general valence shell electronic configuration \([Xe] 4f^{1-14} 5d^{0-1} 6s^2\).

The most characteristic and stable oxidation state for all the lanthanoids is +3.
This is because they readily lose the two outermost 6s electrons and one more electron (usually from the 5d or sometimes the 4f subshell) to form the Ln\(^{3+}\) ion.
The energy difference between the 4f, 5d, and 6s orbitals is relatively small, which allows for this consistent loss of three electrons.
While some lanthanoids also show other oxidation states like +2 (e.g., Eu\(^{2+}\), Yb\(^{2+}\)) or +4 (e.g., Ce\(^{4+}\)) due to the extra stability of empty, half-filled, or completely filled f-orbitals, the +3 state is the only one common to all of them and is generally the most stable in aqueous solutions.


Step 3: Final Answer:

The common oxidation state of the elements of the lanthanoid series is +3. This corresponds to option (C).
Quick Tip: For f-block elements (both lanthanoids and actinoids), the most common oxidation state is +3. This is a very important general rule to remember for these elements.

Step 1: Understanding the Concept:

This question requires a comparison of two coordination complexes of Nickel(II) with different ligands. We need to use Valence Bond Theory or Crystal Field Theory to determine their properties like geometry, hybridization, and magnetic moment.


Step 2: Detailed Explanation:

First, let's analyze each complex individually. In both complexes, let the oxidation state of Ni be x.
For [NiCl\(_4\)]\(^{2-}\): \(x + 4(-1) = -2 \implies x = +2\).
For [Ni(CN)\(_4\)]\(^{2-}\): \(x + 4(-1) = -2 \implies x = +2\).
So, (iv) Oxidation state of nickel is the same for both. This cannot be a point of difference. This eliminates options A, C, and E.

Now let's analyze the Ni\(^{2+}\) ion and the effect of the ligands.
The configuration of Ni (Z=28) is \([Ar] 3d^8 4s^2\). The configuration of Ni\(^{2+}\) is \([Ar] 3d^8\).

For [NiCl\(_4\)]\(^{2-}\):

Ligand: Cl\(^-\) is a weak-field ligand.
It does not cause pairing of the 3d electrons. The 3d\(^8\) configuration has two unpaired electrons.
For coordination number 4, it uses the outer 4s and 4p orbitals for bonding.
(iii) Hybridisation: sp\(^3\).
(ii) Geometry: Tetrahedral.
(i) Magnetic moment: Since there are 2 unpaired electrons, it is paramagnetic. \(\mu = \sqrt{2(2+2)} = \sqrt{8}\) BM.

For [Ni(CN)\(_4\)]\(^{2-}\):

Ligand: CN\(^-\) is a strong-field ligand.
It forces the 3d electrons to pair up. The two unpaired electrons in the 3d\(^8\) configuration are paired, leaving one 3d orbital empty.
For coordination number 4, it uses this empty 3d orbital, the 4s orbital, and two 4p orbitals.
(iii) Hybridisation: dsp\(^2\).
(ii) Geometry: Square planar.
(i) Magnetic moment: Since all electrons are paired, there are 0 unpaired electrons. It is diamagnetic. \(\mu = 0\) BM.

Comparison:

(i) Magnetic moment: Different (paramagnetic vs. diamagnetic).
(ii) Geometry: Different (tetrahedral vs. square planar).
(iii) Hybridisation: Different (sp\(^3\) vs. dsp\(^2\)).
(iv) Oxidation state: Same (+2).

The two complexes differ in magnetic moment, geometry, and hybridization.


Step 3: Final Answer:

The properties by which the complexes differ are (i), (ii), and (iii). This corresponds to option (B).
Quick Tip: For coordination number 4, the geometry is usually tetrahedral. However, for d\(^8\) ions (like Ni\(^{2+}\), Pd\(^{2+}\), Pt\(^{2+}\)) with strong-field ligands, square planar geometry with dsp\(^2\) hybridization is common.

Step 1: Understanding the Concept:

The color of a transition metal complex is due to the absorption of light of a specific wavelength (color) to promote a d-electron to a higher energy d-orbital (d-d transition). The color we observe is the complementary color of the light that was absorbed.


Step 2: Detailed Explanation:

This is a knowledge-based matching question based on the colors and absorption spectra of common coordination complexes.

(a) [Ti(H\(_2\)O)\(_6\)]\(^{3+}\): Ti\(^{3+}\) is a d\(^1\) ion. This complex is known for its violet/purple color. To appear violet, it must absorb light in the yellow-green region of the spectrum (around 500-560 nm). "Blue-green" (iii) fits this description. So, (a) matches with (iii).
(b) [Cu(H\(_2\)O)\(_4\)]\(^{2+}\) / [Cu(H\(_2\)O)\(_6\)]\(^{2+}\): Cu\(^{2+}\) is a d\(^9\) ion. This complex is famously light blue. To appear light blue, it must absorb light in the orange-red region of the spectrum. "Red" (iv) is the closest match. So, (b) matches with (iv).
(c) [CoCl(NH\(_3\))\(_5\)]\(^{2+}\): This Co\(^{3+}\) complex is purplish-red or violet. It absorbs in the yellow-green part of the spectrum. The option given is "Yellow" (ii). This is a reasonable match as absorbing yellow light would lead to an observed violet color. So, (c) matches with (ii).
(d) [Co(NH\(_3\))\(_6\)]\(^{3+}\): This Co\(^{3+}\) complex is yellow-orange. To appear yellow-orange, it must absorb light in the blue-violet region of the spectrum. The option given is "Blue" (i). So, (d) matches with (i).

Combining the matches: (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i).


Step 3: Final Answer:

The correct matching is (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i). This corresponds to option (A).
Quick Tip: Remembering the color wheel helps predict observed colors. The color you see is complementary (opposite) to the color absorbed. For example: Absorbs Violet \(\to\) Appears Yellow-Green Absorbs Blue \(\to\) Appears Orange Absorbs Green \(\to\) Appears Purple Absorbs Yellow \(\to\) Appears Violet Absorbs Orange \(\to\) Appears Blue Absorbs Red \(\to\) Appears Green

Step 1: Understanding the Concept:

In organic chemistry, the alpha (\(\alpha\)) carbon is the carbon atom directly attached to a functional group. Alpha-hydrogens are the hydrogen atoms attached to the alpha-carbon.


Step 2: Detailed Explanation:

Let's identify the \(\alpha\)-carbon and count the \(\alpha\)-hydrogens for each molecule. The functional group here is the chlorine atom (-Cl).

Tertiary butyl chloride: The structure is (CH\(_3\))\(_3\)C-Cl.
The \(\alpha\)-carbon is the central carbon atom bonded to the Cl. This carbon is also bonded to three other carbon atoms (the methyl groups) and has zero hydrogen atoms attached to it. So, it has 0 \(\alpha\)-hydrogens.
Isopropyl chloride: The structure is (CH\(_3\))\(_2\)CH-Cl.
The \(\alpha\)-carbon is the CH group bonded to the Cl. This carbon has one hydrogen atom attached to it. So, it has 1 \(\alpha\)-hydrogen.
Ethyl chloride: The structure is CH\(_3\)CH\(_2\)-Cl.
The \(\alpha\)-carbon is the CH\(_2\) group bonded to the Cl. This carbon has two hydrogen atoms attached to it. So, it has 2 \(\alpha\)-hydrogens.
Methyl chloride: The structure is CH\(_3\)-Cl.
The \(\alpha\)-carbon is the CH\(_3\) group bonded to the Cl. This carbon has three hydrogen atoms attached to it. So, it has 3 \(\alpha\)-hydrogens.

The respective numbers of \(\alpha\)-hydrogens are 0, 1, 2, and 3.


Step 3: Final Answer:

The number of \(\alpha\)-hydrogens are 0, 1, 2 and 3, respectively. This corresponds to option (A).
Quick Tip: Be careful not to confuse \(\alpha\)-hydrogens with \(\beta\)-hydrogens. Beta-hydrogens are attached to the beta-carbon, which is the carbon atom adjacent to the alpha-carbon. The number of \(\beta\)-hydrogens is important for elimination reactions.

Step 1: Understanding the Concept:

This question asks for the order of reactivity of different types of alkyl halides (primary, secondary, tertiary) towards \(\beta\)-elimination reactions. The most common \(\beta\)-elimination mechanism is the E2 reaction. The rate of this reaction depends on factors that stabilize the transition state.


Step 2: Detailed Explanation:

A \(\beta\)-elimination reaction involves the removal of a leaving group from the \(\alpha\)-carbon and a proton from the \(\beta\)-carbon to form an alkene.
The rate of elimination reactions (particularly E2, which is the most common for strong bases) is determined by the stability of the alkene that is formed. The transition state of the reaction has partial double-bond character and resembles the final alkene product.

Alkene Stability: The stability of alkenes increases with the number of alkyl groups attached to the double-bonded carbons. This is known as Zaitsev's rule. The order of stability is:
Tetrasubstituted \(>\) Trisubstituted \(>\) Disubstituted \(>\) Monosubstituted.
Alkyl Halide Type and Product Stability:

Tertiary (3\(^\circ\)) halides typically form the most highly substituted (and therefore most stable) alkenes upon elimination. This leads to a more stable transition state and a faster reaction rate.
Secondary (2\(^\circ\)) halides form less substituted alkenes than tertiary halides, leading to an intermediate reaction rate.
Primary (1\(^\circ\)) halides form the least substituted (and least stable) alkenes, resulting in the slowest reaction rate.


Therefore, the general order of reactivity for alkyl halides in E2 elimination reactions is: \[ Tertiary (3\(^\circ\)) > Secondary (2\(^\circ\)) > Primary (1\(^\circ\)) \]

Step 3: Final Answer:

The correct order of the rate of \(\beta\)-elimination is Tertiary \(>\) Secondary \(>\) Primary. This corresponds to option (C).
Quick Tip: The reactivity order for elimination (E2) is the opposite of the reactivity order for S\(_N\)2 substitution reactions. \textbf{S\(_N\)2:} Primary \(>\) Secondary \(>\) Tertiary (due to steric hindrance). \textbf{E2:} Tertiary \(>\) Secondary \(>\) Primary (due to alkene product stability). Keeping this contrast in mind helps to remember both trends.

Step 1: Understanding the Concept:

This question asks to identify a specific named reaction in organic chemistry used for the synthesis of alkyl halides. The reaction shown is a halide exchange reaction, where a chloroalkane is converted into an iodoalkane.


Step 2: Detailed Explanation:

Let's analyze the given reaction and the options:

The Reaction: CH\(_3\)CH\(_2\)Cl + NaI \(\to\) CH\(_3\)CH\(_2\)I + NaCl. This reaction involves the exchange of a halogen atom (Cl is replaced by I). It is typically carried out in dry acetone. Sodium iodide (NaI) is soluble in acetone, while the sodium chloride (NaCl) formed as a byproduct is not. According to Le Chatelier's principle, the precipitation of NaCl drives the equilibrium to the right, favoring the formation of the alkyl iodide.
(D) Finkelstein reaction: This is the name given to this specific halide exchange reaction for the preparation of alkyl iodides. This is the correct answer.
(A) Wurtz reaction: This reaction involves the coupling of two alkyl halides in the presence of sodium metal to form a higher alkane (2R-X + 2Na \(\to\) R-R + 2NaX).
(B) Wurtz-Fittig reaction: This is a variation of the Wurtz reaction where an aryl halide and an alkyl halide are coupled to form an alkylbenzene (Ar-X + R-X + 2Na \(\to\) Ar-R + 2NaX).
(C) Williamson synthesis: This is a method for preparing ethers by reacting a sodium alkoxide with an alkyl halide (R-ONa + R'-X \(\to\) R-O-R' + NaX).
(E) Etard reaction: This reaction involves the oxidation of a methyl group on an aromatic ring to an aldehyde group using chromyl chloride (CrO\(_2\)Cl\(_2\)).


Step 3: Final Answer:

The given reaction is the Finkelstein reaction. This corresponds to option (D).
Quick Tip: The Finkelstein reaction is an S\(_N\)2 reaction. The choice of acetone as a solvent is crucial because it selectively precipitates the NaCl or NaBr byproduct, driving the reaction forward. A similar reaction using heavy metal fluorides (like AgF, Hg\(_2\)F\(_2\)) to prepare alkyl fluorides is called the Swarts reaction.

Step 1: Understanding the Concept:

Nucleophilic Aromatic Substitution (S\(_N\)Ar) is a reaction where a nucleophile displaces a leaving group on an aromatic ring. The rate of this reaction is highly dependent on the presence of electron-withdrawing groups (EWGs) on the ring.


Step 2: Detailed Explanation:

The mechanism of S\(_N\)Ar involves two steps:
1. Attack of the nucleophile on the carbon bearing the leaving group, forming a resonance-stabilized carbanion intermediate called a Meisenheimer complex. This is the rate-determining step.
2. Loss of the leaving group from the intermediate to restore the aromaticity of the ring.

The reactivity is determined by the stability of the Meisenheimer complex. Strong electron-withdrawing groups, especially when located at the \textit{ortho and \textit{para positions relative to the leaving group, can delocalize the negative charge of the intermediate through resonance, thereby stabilizing it and increasing the reaction rate.

Let's analyze the given compounds (the leaving group is Cl in all cases):

(D) Chlorobenzene: Has no electron-withdrawing groups. It is very unreactive.
(A) p-Nitrochlorobenzene: Has one -NO\(_2\) group (a strong EWG) at the \textit{para position. This position is ideal for resonance stabilization of the negative charge. This compound is much more reactive than chlorobenzene.
(B) m-Nitrochlorobenzene: Has one -NO\(_2\) group at the \textit{meta position. From the meta position, the -NO\(_2\) group can only exert its electron-withdrawing inductive effect (-I). It cannot participate in resonance to delocalize the negative charge of the intermediate. It is less reactive than the ortho or para isomers.
(C) 2,5-Dinitrochlorobenzene: Has two -NO\(_2\) groups. One is at the \textit{ortho position (activating), but the other is at the \textit{meta position (less activating).
(E) 2,4-Dinitrochlorobenzene: Has two -NO\(_2\) groups, one at the \textit{ortho position and one at the \textit{para position. Both groups are in the optimal positions to stabilize the Meisenheimer complex through both resonance and inductive effects. The combined effect of two such groups makes this compound extremely reactive towards S\(_N\)Ar.

Comparing the compounds, the reactivity increases with the number of strongly activating EWGs at ortho/para positions. Therefore, 2,4-dinitrochlorobenzene is the most reactive.


Step 3: Final Answer:

The compound with nitro groups at both the ortho and para positions is the most reactive. This corresponds to option (E).
Quick Tip: For S\(_N\)Ar reactions, remember the rule: More electron-withdrawing groups at ortho/para positions = faster reaction. The order of reactivity is generally: 2,4,6-trinitro > 2,4-dinitro > 4-nitro > 2-nitro > 3-nitro > unsubstitued.

Step 1: Understanding the Concept:

This reaction is an electrophilic aromatic substitution (EAS), specifically the bromination of an activated benzene ring. The starting material is 4-methylphenol (p-cresol), which has two activating substituents: a hydroxyl group (-OH) and a methyl group (-CH\(_3\)). We need to determine the position where the electrophile (Br\(^+\)) will attack.


Step 2: Detailed Explanation:

1. Identifying the Directing Groups:

The hydroxyl group (-OH) is a very strong activating group and an \textit{ortho, para-director.
The methyl group (-CH\(_3\)) is a weakly activating group and also an \textit{ortho, para-director.

2. Determining the Dominant Director: When multiple activating groups are present, the strongest activator controls the position of substitution. The -OH group is a much stronger activator than the -CH\(_3\) group. Therefore, the position of bromination will be determined by the -OH group.
3. Finding Available Positions: The -OH group directs incoming electrophiles to its ortho and para positions.

The \textit{para position relative to the -OH group is already occupied by the methyl group.
The two \textit{ortho positions (C2 and C6) are available. These two positions are equivalent due to the symmetry of the molecule.

4. Predicting the Product: The electrophile, Br\(^+\) (generated from Br\(_2\) and the Lewis acid catalyst FeBr\(_3\)), will attack one of the ortho positions to the -OH group. This results in the formation of 2-bromo-4-methylphenol.

Let's examine the options:

(A) shows 3-bromo-4-methylphenol. Bromination occurs meta to the -OH group, which is incorrect as -OH is an o,p-director.
(B) and (C) both show 2-bromo-4-methylphenol, which is the correct product. The image quality or OCR might be causing confusion, but the structure with bromine ortho to the -OH group is the correct one. Assuming option (C) represents this structure correctly.
(D) shows bromination on the methyl group (benzylic bromination). This occurs under radical conditions (e.g., using NBS and light), not under EAS conditions with FeBr\(_3\).
(E) shows dibromination, which could occur with excess bromine, but the question asks for the major (mono-substituted) product.


Step 3: Final Answer:

The major product is 2-bromo-4-methylphenol, where the bromine atom is ortho to the strongly activating -OH group. This is represented by option (C).
Quick Tip: When a benzene ring has multiple substituents, the position of electrophilic attack is determined by a hierarchy: 1. The strongest activating group wins. 2. If directing effects oppose each other, a mixture may form, but the product from the stronger activator usually predominates. 3. Steric hindrance can also play a role, sometimes favoring the para position over the more crowded ortho position.

Step 1: Understanding the Concept:

The question asks for a chemical test that can distinguish between two ketones: benzophenone and acetophenone. This requires identifying a structural difference between them that leads to a selective reaction.


Step 2: Detailed Explanation:

Let's look at the structures of the two compounds:

Acetophenone: C\(_6\)H\(_5\)COCH\(_3\). This is a methyl ketone because it has a methyl group (-CH\(_3\)) directly attached to the carbonyl carbon.
Benzophenone: C\(_6\)H\(_5\)COC\(_6\)H\(_5\). The carbonyl carbon is attached to two phenyl groups. It is not a methyl ketone.

Now let's analyze the reagents:

(A) Fehling's reagent and (E) Tollen's reagent are mild oxidizing agents used to test for aldehydes. They do not react with ketones.
(B) Lucas reagent (conc. HCl + anhyd. ZnCl\(_2\)) is used to distinguish between primary, secondary, and tertiary alcohols. It does not react with ketones.
(D) Aqueous CrO\(_3\) (chromic acid) is a strong oxidizing agent, but it would not typically provide a simple visual test to distinguish between these two relatively stable ketones under normal conditions.
(C) Iodine and alkali (I\(_2\)/NaOH or I\(_2\)/KOH): This is the reagent for the iodoform test. This test is a specific and reliable method for identifying the presence of a methyl ketone group (CH\(_3\)CO-) or a secondary alcohol group (CH\(_3\)CH(OH)-) that can be oxidized to a methyl ketone.

Acetophenone, being a methyl ketone, will give a positive iodoform test. It reacts to form iodoform (CHI\(_3\)), which is a yellow solid precipitate with a characteristic antiseptic smell.
Benzophenone does not have a methyl ketone group, so it will give a negative iodoform test (no yellow precipitate).


This clear difference in reactivity allows the two compounds to be distinguished.


Step 3: Final Answer:

Treating with iodine and alkali (the iodoform test) can distinguish between benzophenone and acetophenone. This corresponds to option (C).
Quick Tip: The iodoform test is a very important qualitative test in organic chemistry. Any compound containing the CH\(_3\)CO- group or a group that can be oxidized to it (like CH\(_3\)CH(OH)-) will give a positive result. Ethanol and acetaldehyde are exceptions that also give a positive test.

Step 1: Understanding the Concept:

This is an example of a base-catalyzed crossed aldol condensation, specifically the Claisen-Schmidt condensation. This reaction occurs between an aldehyde (with no \(\alpha\)-hydrogens) and a ketone (with \(\alpha\)-hydrogens). The ketone forms the enolate, which then attacks the aldehyde.


Step 2: Detailed Explanation:

Reactants: Benzaldehyde (C\(_6\)H\(_5\)CHO) and Acetophenone (C\(_6\)H\(_5\)COCH\(_3\)).
Catalyst: NaOH (a strong base).

1. Enolate Formation: The base (OH\(^-\)) removes an acidic \(\alpha\)-hydrogen from the methyl group of acetophenone. Benzaldehyde has no \(\alpha\)-hydrogens and cannot form an enolate.
\[ C_6H_5COCH_3 + OH^- \rightleftharpoons C_6H_5CO\overline{C}H_2 + H_2O \]
2. Nucleophilic Attack: The enolate carbanion acts as a nucleophile and attacks the electrophilic carbonyl carbon of benzaldehyde.
\[ C_6H_5CHO + C_6H_5CO\overline{C}H_2 \to C_6H_5CH(O^-)CH_2COC_6H_5 \]
3. Protonation: The intermediate alkoxide is protonated by water to give the aldol addition product, a \(\beta\)-hydroxy ketone.
\[ C_6H_5CH(O^-)CH_2COC_6H_5 + H_2O \to C_6H_5CH(OH)CH_2COC_6H_5 + OH^- \]
4. Dehydration (Condensation): The aldol addition product readily dehydrates upon gentle warming (or even at room temperature in this case) because the resulting double bond is conjugated with both the benzene ring and the carbonyl group, which makes the product very stable. A base removes a proton from the \(\alpha\)-carbon, and the -OH group leaves.
\[ C_6H_5CH(OH)CH_2COC_6H_5 \xrightarrow{-H_2O} C_6H_5CH=CHCOC_6H_5 \]
The final product is 1,3-diphenylprop-2-en-1-one, commonly known as chalcone.


Step 3: Final Answer:

The major product of the reaction is C\(_6\)H\(_5\)CH=CHCOC\(_6\)H\(_5\). This corresponds to option (A).
Quick Tip: Crossed aldol condensations are most effective when one of the carbonyl compounds has no \(\alpha\)-hydrogens (like benzaldehyde or formaldehyde). This prevents it from self-condensing and ensures it acts only as the electrophile (the "enolate acceptor").

Step 1: Understanding the Concept:

The acidity of a carboxylic acid (R-COOH) is determined by the stability of its conjugate base, the carboxylate anion (R-COO\(^-\)). Any factor that stabilizes this anion will increase the acidity of the parent acid. Electron-withdrawing groups (EWGs) attached to the R group stabilize the anion by delocalizing or withdrawing its negative charge, primarily through the inductive effect (-I effect).


Step 2: Detailed Explanation:

We need to compare the electron-withdrawing strength of the groups attached to the -COOH group in each option.

(E) CH\(_3\)COOH (Acetic acid): The methyl group (-CH\(_3\)) is an electron-donating group (+I effect), which destabilizes the carboxylate anion. This is our reference weak acid.
(D) Br-CH\(_2\)COOH (Bromoacetic acid): The bromine atom is electronegative and exerts an electron-withdrawing inductive effect (-I effect), which stabilizes the anion. This makes it a stronger acid than acetic acid.
(C) NC-CH\(_2\)COOH (Cyanoacetic acid): The cyano group (-CN) is a strong electron-withdrawing group, stronger than halogens. It makes the acid stronger than bromoacetic acid.
(A) FCH\(_2\)COOH (Fluoroacetic acid): Fluorine is the most electronegative element, so its -I effect is stronger than that of Br or CN. This is a very strong acid.
(B) CF\(_3\)COOH (Trifluoroacetic acid): This molecule has three fluorine atoms attached to the \(\alpha\)-carbon. The inductive effects of the three highly electronegative fluorine atoms are cumulative, creating an extremely powerful electron-withdrawing effect. This effect strongly stabilizes the trifluoroacetate anion by pulling electron density away from the -COO\(^-\) group.

Comparing the strengths:
The strength of the -I effect of the substituents is in the order: \[ -CF_3 > -F > -CN > -Br > -CH_3 (+I effect) \]
Therefore, the order of acidity is: \[ CF_3COOH > FCH_2COOH > NC-CH_2COOH > Br-CH_2COOH > CH_3COOH \]
The strongest acid in the list is trifluoroacetic acid.


Step 3: Final Answer:

The strongest acid is CF\(_3\)COOH. This corresponds to option (B).
Quick Tip: When comparing acid strengths based on the inductive effect, remember these rules: 1. \textbf{Number:} More EWGs have a greater effect (e.g., -CF\(_3\) vs -F). 2. \textbf{Strength (Electronegativity):} A more electronegative atom has a stronger effect (F > Cl > Br > I). 3. \textbf{Distance:} The inductive effect weakens rapidly with distance. An EWG on the \(\alpha\)-carbon has a much larger effect than one on the \(\beta\)-carbon.

Step 1: Understanding the Concept:

This is a matching question that requires identifying common reagents, reaction types, and compound classes related to the chemistry of amines.


Step 2: Detailed Explanation:

Let's analyze each item in Column-I and find its correct match in Column-II.

(a) Benzenesulphonyl chloride (C\(_6\)H\(_5\)SO\(_2\)Cl): This compound is famously known as (iii) Hinsberg's reagent. It is used to distinguish between primary, secondary, and tertiary amines based on the solubility of the resulting sulfonamide products in alkali.
(b) Conversion of amide to amine: This refers to a reaction that converts an amide (RCONH\(_2\)) into a primary amine (RNH\(_2\)) with one less carbon atom. This specific transformation is the (iv) Hofmann's bromamide reaction (or degradation), which uses Br\(_2\) and NaOH.
(c) Conversion of primary amine to isocyanide: This describes the (i) Carbylamine reaction (also known as the isocyanide test). When a primary amine is heated with chloroform (CHCl\(_3\)) and alcoholic KOH, it produces a foul-smelling isocyanide (R-NC). This is a characteristic test for primary amines.
(d) Diethylamine ((CH\(_3\)CH\(_2\))\(_2\)NH): In this molecule, the nitrogen atom is bonded to two ethyl (alkyl) groups and one hydrogen atom. An amine where the nitrogen is bonded to two alkyl/aryl groups is classified as a (ii) Secondary amine.

Combining the correct pairs gives: (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii).


Step 3: Final Answer:

The correct set of combinations is (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii). This corresponds to option (C).
Quick Tip: Named reactions and reagents are high-yield topics for chemistry exams. Creating flashcards with the reagent, reaction name, substrate, product, and key features (like the smell in the carbylamine test) is an effective way to memorize them.

Step 1: Understanding the Concept:

This is a fundamental question in biochemistry, asking for the monomeric units or building blocks that make up peptides.


Step 2: Detailed Explanation:


Peptides are biological molecules formed by linking amino acids together in a chain.
The bond that connects two amino acids is called a peptide bond, which is an amide bond formed between the carboxyl group (-COOH) of one amino acid and the amino group (-NH\(_2\)) of another.
Hydrolysis is a chemical reaction that breaks bonds by adding water. When a peptide is hydrolyzed (either by acid, base, or enzymes), the peptide bonds are broken.
Breaking the peptide bonds releases the individual amino acids that were originally linked to form the peptide chain.

Let's look at the other options:

(A) Glucose is the monomer of carbohydrates like starch and cellulose.
(B) Fatty acids (along with glycerol) are the components of lipids like triglycerides.
(D) Ribose sugar, phosphate, and a nitrogenous base are the components of ribonucleotides, the monomers of RNA.
(E) A heterocyclic base and a sugar form a nucleoside, a component of nucleic acids.

Therefore, the only correct answer is amino acids.


Step 3: Final Answer:

Peptide on hydrolysis gives amino acids. This corresponds to option (C).
Quick Tip: Remember the basic monomer-polymer relationships for the main classes of biomolecules: \textbf{Monomer:} Amino Acid \(\to\) \textbf{Polymer:} Protein/Peptide \textbf{Monomer:} Monosaccharide (e.g., glucose) \(\to\) \textbf{Polymer:} Polysaccharide (e.g., starch) \textbf{Monomer:} Nucleotide \(\to\) \textbf{Polymer:} Nucleic Acid (DNA/RNA)