Step 1: Understanding the Concept:

The given equation is a first-order linear differential equation of the form \(\frac{dy}{dx} + P(x)y = Q(x)\).

To solve this, we need to find the Integrating Factor (IF) and then multiply it by the dependent variable.


Step 2: Key Formula or Approach:

The Integrating Factor (IF) is given by:
\[ IF = e^{\int P(x) \, dx} \]

The general solution is then given by:
\[ y \cdot (IF) = \int Q(x) \cdot (IF) \, dx + C \]


Step 3: Detailed Explanation:

Identifying terms from the given equation \(\frac{dy}{dx} + \frac{1}{x}y = x^2\):
\(P(x) = \frac{1}{x}\) and \(Q(x) = x^2\).

Now, calculate the Integrating Factor:
\[ IF = e^{\int \frac{1}{x} \, dx} = e^{\ln x} = x \]

Substitute into the general solution formula:
\[ y \cdot x = \int (x^2 \cdot x) \, dx \]
\[ yx = \int x^3 \, dx \]

Integrating the right-hand side:
\[ yx = \frac{x^4}{4} + C \]

Dividing throughout by \(x\):
\[ y = \frac{x^3}{4} + \frac{C}{x} \]


Step 4: Final Answer:

The general solution of the differential equation is \(y = \frac{x^3}{4} + \frac{C}{x}\).
Quick Tip: Always ensure the coefficient of \(\frac{dy}{dx}\) is 1 before identifying \(P(x)\) and \(Q(x)\).
The Integrating Factor simplifies many linear equations instantly.

Step 1: Understanding the Concept:

The perpendicular distance from a point to a line represents the shortest distance between them.


Step 2: Key Formula or Approach:

The distance \(d\) of a point \((x_1, y_1)\) from the line \(ax + by + c = 0\) is:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]


Step 3: Detailed Explanation:

Here, the point is \((x_1, y_1) = (1, 2)\) and the line is \(3x - 4y + 10 = 0\).

Comparing with the standard form, we have \(a = 3, b = -4, c = 10\).

Substitute the values into the formula:
\[ d = \frac{|3(1) - 4(2) + 10|}{\sqrt{3^2 + (-4)^2}} \]
\[ d = \frac{|3 - 8 + 10|}{\sqrt{9 + 16}} \]
\[ d = \frac{|5|}{\sqrt{25}} \]
\[ d = \frac{5}{5} = 1 \]


Step 4: Final Answer:

The distance of the point from the line is \(1 unit\).
Quick Tip: Always take the absolute value of the numerator to ensure distance is positive.
Common Pythagorean triplets like \((3, 4, 5)\) help in calculating the denominator quickly.

Step 1: Understanding the Concept:

The feasible region is the area on a graph where all inequalities are satisfied simultaneously.


Step 2: Key Formula or Approach:

The region bounded by \(x \ge 0, y \ge 0\) and \(\frac{x}{a} + \frac{y}{b} \le 1\) is a right-angled triangle with area \(\frac{1}{2}ab\).


Step 3: Detailed Explanation:

The constraints are \(x \ge 0\) (y-axis and to the right), \(y \ge 0\) (x-axis and above).

The line \(x + y = 4\) intersects the x-axis at \((4, 0)\) and the y-axis at \((0, 4)\).

The region \(x + y \le 4\) includes the origin \((0, 0)\).

Therefore, the feasible region is a triangle with vertices at \((0, 0), (4, 0),\) and \((0, 4)\).

Area of triangle = \(\frac{1}{2} \times base \times height\)
\[ Area = \frac{1}{2} \times 4 \times 4 \]
\[ Area = 8 \]


Step 4: Final Answer:

The area of the solution region is \(8 sq. units\).
Quick Tip: For linear inequalities, the origin test (putting \(x=0, y=0\)) is the fastest way to check which side of the line to shade.

Step 1: Understanding the Concept:

The composite function \(fog(x)\) or \((f \circ g)(x)\) is defined as \(f(g(x))\).

It means we substitute the entire function \(g(x)\) as the input for \(f(x)\).


Step 2: Detailed Explanation:

Given \(f(x) = e^x\) and \(g(x) = \ln(x)\).

To find \(fog(x)\):
\[ fog(x) = f(g(x)) \]
\[ fog(x) = f(\ln(x)) \]

Substitute \(\ln(x)\) in place of \(x\) in \(e^x\):
\[ fog(x) = e^{\ln(x)} \]

Using the logarithmic property \(a^{\log_a(x)} = x\):
\[ fog(x) = x \]


Step 3: Final Answer:

The composite function \(fog(x)\) is equal to \(x\).
Quick Tip: Exponential and logarithmic functions with the same base are inverses of each other.
Therefore, \(f(f^{-1}(x)) = x\).

Step 1: Understanding the Concept:

Wall's formula provides a direct way to calculate integrals of the form \(\int_0^{\pi/2} \sin^n x \, dx\) or \(\int_0^{\pi/2} \cos^n x \, dx\).


Step 2: Key Formula or Approach:

For \(\int_0^{\pi/2} \sin^n x \, dx\):

If \(n\) is odd: \(I_n = \frac{(n-1)(n-3)...2}{n(n-2)...3} \times 1\)

If \(n\) is even: \(I_n = \frac{(n-1)(n-3)...1}{n(n-2)...2} \times \frac{\pi}{2}\)


Step 3: Detailed Explanation:

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

Using the formula for odd \(n\):
\[ I_5 = \frac{(5-1)(5-3)}{5(5-2)(5-4)} \times 1 \]
\[ I_5 = \frac{4 \times 2}{5 \times 3 \times 1} \]
\[ I_5 = \frac{8}{15} \]


Step 4: Final Answer:

The value of the integral is \(\frac{8}{15}\).
Quick Tip: Wall's formula only applies when the limits are exactly \(0\) to \(\pi/2\).
Remember to multiply by \(\pi/2\) only when the power \(n\) is even.

Step 1: Understanding the Concept:

For independent events, the probability of their intersection is the product of their individual probabilities.


Step 2: Key Formula or Approach:

Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

For independent events: \(P(A \cap B) = P(A) \cdot P(B)\)


Step 3: Detailed Explanation:

Substitute independent condition into addition rule:
\[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]

Substitute given values:
\[ 0.8 = 0.3 + P(B) - 0.3P(B) \]
\[ 0.8 - 0.3 = P(B)(1 - 0.3) \]
\[ 0.5 = 0.7P(B) \]
\[ P(B) = \frac{0.5}{0.7} = \frac{5}{7} \]


Step 4: Final Answer:

The probability \(P(B)\) is \(\frac{5}{7}\).
Quick Tip: Don't confuse independent events with mutually exclusive events.
For mutually exclusive events, \(P(A \cap B) = 0\).

Step 1: Understanding the Concept:

Events are mutually exclusive if they cannot happen at the same time. This means \(P(A \cap B) = 0\).


Step 2: Key Formula or Approach:

For mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\)


Step 3: Detailed Explanation:

Substitute the given values into the formula:
\[ 0.6 = 0.2 + P(B) \]
\[ P(B) = 0.6 - 0.2 \]
\[ P(B) = 0.4 \]


Step 4: Final Answer:

The probability \(P(B)\) is \(0.4\).
Quick Tip: For mutually exclusive events, the Venn diagrams of the two sets do not overlap.

Step 1: Understanding the Concept:

This problem uses the property of definite integrals: \(\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx\).


Step 2: Detailed Explanation:

Let \(I = \int_a^b \frac{f(x)}{f(x) + f(a+b-x)} \, dx\) \dots (i)

Applying the property \(\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx\):

Replace \(x\) with \(a+b-x\) in the integral:
\[ I = \int_a^b \frac{f(a+b-x)}{f(a+b-x) + f(a+b-(a+b-x))} \, dx \]
\[ I = \int_a^b \frac{f(a+b-x)}{f(a+b-x) + f(x)} \, dx \] \dots (ii)

Adding equations (i) and (ii):
\[ 2I = \int_a^b \frac{f(x) + f(a+b-x)}{f(x) + f(a+b-x)} \, dx \]

The integrand becomes 1:
\[ 2I = \int_a^b 1 \, dx \]
\[ 2I = [x]_a^b = b - a \]
\[ I = \frac{b - a}{2} \]


Step 3: Final Answer:

The value of the definite integral is \(\frac{b - a}{2}\).
Quick Tip: This is a standard property often used when the denominator remains invariant under the substitution \(x \rightarrow a+b-x\).

Step 1: Understanding the Concept:

The product of trigonometric functions can be converted into a sum using transformation formulas to facilitate integration.


Step 2: Key Formula or Approach:

Use the identity \(2 \sin A \cos B = \sin(A+B) + \sin(A-B)\).


Step 3: Detailed Explanation:

Rewrite the integrand:
\[ \sin 4x \cos 2x = \frac{1}{2} [2 \sin 4x \cos 2x] \]

Using the identity with \(A = 4x\) and \(B = 2x\):
\[ \sin 4x \cos 2x = \frac{1}{2} [\sin(4x + 2x) + \sin(4x - 2x)] \]
\[ \sin 4x \cos 2x = \frac{1}{2} [\sin 6x + \sin 2x] \]

Now integrate:
\[ I = \int_0^{\pi/2} \frac{1}{2} (\sin 6x + \sin 2x) \, dx \]
\[ I = \frac{1}{2} \left[ -\frac{\cos 6x}{6} - \frac{\cos 2x}{2} \right]_0^{\pi/2} \]

Applying limits:
\[ I = \frac{1}{2} \left[ \left( -\frac{\cos 3\pi}{6} - \frac{\cos \pi}{2} \right) - \left( -\frac{\cos 0}{6} - \frac{\cos 0}{2} \right) \right] \]

Note: \(\cos 3\pi = -1\), \(\cos \pi = -1\), \(\cos 0 = 1\).
\[ I = \frac{1}{2} \left[ \left( \frac{1}{6} + \frac{1}{2} \right) - \left( -\frac{1}{6} - \frac{1}{2} \right) \right] \]
\[ I = \frac{1}{2} \left[ \frac{1}{6} + \frac{1}{2} + \frac{1}{6} + \frac{1}{2} \right] \]
\[ I = \frac{1}{2} \left[ \frac{2}{6} + 1 \right] = \frac{1}{2} \left[ \frac{1}{3} + 1 \right] = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3} \]

Correction check: Let's re-evaluate limits carefully.

Limit at \(0\): \(-1/6 - 3/6 = -4/6 = -2/3\).

Limit at \(\pi/2\): \(1/6 + 3/6 = 4/6 = 2/3\).
\(I = \frac{1{2} [ 2/3 - (-2/3) ] = \frac{1}{2} [ 4/3 ] = \frac{2}{3}\).


Step 4: Final Answer:

The integral evaluates to \(\frac{2}{3}\).
Quick Tip: Always check the sign of trigonometric values at boundary points like \(\pi\), \(2\pi\), etc.
\(\cos(n\pi) = (-1)^n\).

Step 1: Understanding the Concept:

The binomial theorem describes the expansion of powers of a binomial.

The coefficients in this expansion are called binomial coefficients.


Step 2: Key Formula or Approach:

The binomial expansion of \((1+x)^n\) is:
\[ (1+x)^n = \binom{n}{0}x^0 + \binom{n}{1}x^1 + \binom{n}{2}x^2 + \dots + \binom{n}{n}x^n \]


Step 3: Detailed Explanation:

To find the sum of all coefficients, we set \(x = 1\) in the binomial expansion.

Substitute \(x = 1\) into the identity:
\[ (1+1)^n = \binom{n}{0}(1)^0 + \binom{n}{1}(1)^1 + \binom{n}{2}(1)^2 + \dots + \binom{n}{n}(1)^n \]
\[ 2^n = C_0 + C_1 + C_2 + \dots + C_n \]

Where \(C_r\) denotes \(\binom{n}{r}\).


Step 4: Final Answer:

The sum \(C_0 + C_1 + \dots + C_n\) is equal to \(2^n\).
Quick Tip: If you need the sum of even or odd coefficients, use \(x = 1\) and \(x = -1\) and then add or subtract the resulting equations.

Step 1: Understanding the Concept:

This is a problem of partitions and permutations. We need to find all combinations of 6 scores (each being 0, 1, 2, 3, 4, or 6) that sum to 30 and then calculate the number of unique sequences (arrangements) for each combination.


Step 2: Key Formula or Approach:

The sum of scores on 6 balls is \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 30\).

For a given set of scores with frequencies \(n_1, n_2, \dots, n_k\), the number of permutations is given by:
\[ P = \frac{n!}{n_1! n_2! \dots n_k!} \]


Step 3: Detailed Explanation:

Since the maximum score per ball is 6, the maximum total for 6 balls is \(6 \times 6 = 36\). To reach 30, the batsman must hit several high scores. Let's list the possible sets of scores:



Case 1: Five 6s and one 0

Scores: \{6, 6, 6, 6, 6, 0\

Sum: \(5(6) + 0 = 30\)

Permutations: \( \frac{6!}{5! \times 1!} = 6 \) ways.



Case 2: Four 6s, one 4, and one 2

Scores: \{6, 6, 6, 6, 4, 2\

Sum: \(4(6) + 4 + 2 = 30\)

Permutations: \( \frac{6!}{4! \times 1! \times 1!} = 30 \) ways.



Case 3: Four 6s and two 3s

Scores: \{6, 6, 6, 6, 3, 3\

Sum: \(4(6) + 2(3) = 30\)

Permutations: \( \frac{6!}{4! \times 2!} = 15 \) ways.



Case 4: Three 6s and three 4s

Scores: \{6, 6, 6, 4, 4, 4\

Sum: \(3(6) + 3(4) = 30\)

Permutations: \( \frac{6!}{3! \times 3!} = 20 \) ways.



Total cases = \(6 + 30 + 15 + 20 = 71\).


Step 4: Final Answer:

The total number of possible sequences to score 30 in 6 balls using the given scores is 71.
Quick Tip: To save time, start with the maximum possible value (6) and subtract from the total to see what remains for the other balls. Since 5 is missing from the options, only combinations of high numbers like 6, 4, 3, and 2 work.

Step 1: Understanding the Concept:

Three vectors are said to be coplanar if their scalar triple product (the determinant of their coefficients) is equal to zero. Although the question uses the term "collinear," in the context of three distinct 3D vectors provided for finding a missing variable, coplanarity is the standard geometric condition tested.


Step 2: Key Formula or Approach:
For vectors \(\vec{a}, \vec{b}, \vec{c}\) to be coplanar:
\[ \begin{vmatrix} a_x & a_y & a_z
b_x & b_y & b_z
c_x & c_y & c_z \end{vmatrix} = 0 \]


Step 3: Detailed Explanation:

Given vectors:
\(\vec{a} = 0\hat{i} + 3\hat{j} + 5\hat{k}\)
\(\vec{b} = 1\hat{i} - 4\hat{j} + 1\hat{k}\)
\(\vec{c} = 2\hat{i} + 5\hat{j} - k\hat{k}\)

Set the determinant to zero:
\[ \begin{vmatrix} 0 & 3 & 5
1 & -4 & 1
2 & 5 & -k \end{vmatrix} = 0 \]

Expanding along the first row:
\[ 0[(-4)(-k) - (5)(1)] - 3[(1)(-k) - (2)(1)] + 5[(1)(5) - (2)(-4)] = 0 \]
\[ -3[-k - 2] + 5[5 + 8] = 0 \]
\[ 3k + 6 + 5(13) = 0 \]
\[ 3k + 6 + 65 = 0 \]
\[ 3k + 71 = 0 \]
\[ k = -\frac{71}{3} \]


Step 4: Final Answer:

The value of \(k\) for which the vectors are coplanar is \(-\frac{71}{3}\).
Quick Tip: If you expand the determinant along a row or column containing a zero, you reduce the number of calculations needed significantly.

Step 1: Understanding the Concept:

The pH of a salt solution depends on the nature of the acid and base from which the salt is derived. Salts of strong acids and strong bases are neutral, while others undergo hydrolysis.


Step 2: Detailed Explanation:

1. \(NH_4Cl\): This is a salt of a weak base (\(NH_3\)) and a strong acid (\(HCl\)). It undergoes cationic hydrolysis, making the solution acidic. pH \(<\) 7.

2. \(NaCl\): This is a salt of a strong acid (\(HCl\)) and a strong base (\(NaOH\)). It does not undergo hydrolysis and remains neutral. pH \( \approx 7 \).

3. Sodium acetate (\(CH_3COONa\)): This is a salt of a weak acid (\(CH_3COOH\)) and a strong base (\(NaOH\)). It undergoes anionic hydrolysis, making the solution basic. pH \(>\) 7.

Therefore, the increasing order of pH is \(NH_4Cl < NaCl < Sodium acetate\).


Step 3: Final Answer:

The order is \(NH_4Cl < NaCl < Sodium acetate\).
Quick Tip: Remember: Strong Acid + Weak Base = Acidic solution; Weak Acid + Strong Base = Basic solution; Strong Acid + Strong Base = Neutral solution.

Step 1: Understanding the Concept:

According to the Brønsted-Lowry theory, the stronger the acid, the weaker its conjugate base.


Step 2: Detailed Explanation:

Let's compare the acid strengths of the given compounds:

- \(HBr\) and \(HCl\) are mineral acids (strong acids). Between them, \(HBr\) is stronger than \(HCl\) because the \(H-Br\) bond is longer and weaker than the \(H-Cl\) bond.

- Oxalic acid and acetic acid are organic acids and are significantly weaker than mineral acids.

Since \(HBr\) is the strongest acid among the choices, its conjugate base (\(Br^-\)) is the weakest (least basic) base.


Step 3: Final Answer:
\(HBr\) has the least basic conjugate base.
Quick Tip: For hydrohalic acids, acid strength increases down the group: \(HF < HCl < HBr < HI\). Consequently, the basicity of their conjugate bases decreases: \(F^- > Cl^- > Br^- > I^-\).

Step 1: Understanding the Concept:

Molecular Orbital Theory (MOT) describes the distribution of electrons in molecules. For homonuclear diatomic molecules with \(Z \le 7\) (like \(N_2\)), the \(\pi 2p\) orbitals are lower in energy than the \(\sigma 2p_z\) orbital due to \(s-p\) mixing.


Step 2: Detailed Explanation:

The nitrogen atom has 7 electrons, so the \(N_2\) molecule has a total of 14 electrons.

The filling order for \(N_2\) is:

1. \(\sigma 1s\) (2e)

2. \(\sigma^* 1s\) (2e)

3. \(\sigma 2s\) (2e)

4. \(\sigma^* 2s\) (2e)

5. \(\pi 2p_x\) and \(\pi 2p_y\) (4e total, as they are degenerate)

6. \(\sigma 2p_z\) (2e)

Total = \(2+2+2+2+4+2 = 14\) electrons.

The configuration is: \(\sigma 1s^2 \sigma^* 1s^2 \sigma 2s^2 \sigma^* 2s^2 (\pi 2p_x^2 = \pi 2p_y^2) \sigma 2p_z^2\).


Step 3: Final Answer:

The configuration is \(\sigma 1s^2 \sigma^* 1s^2 \sigma 2s^2 \sigma^* 2s^2 (\pi 2p_x^2 = \pi 2p_y^2) \sigma 2p_z^2\).
Quick Tip: For \(O_2\) and \(F_2\) (\(Z > 7\)), the order changes: \(\sigma 2p_z\) becomes lower in energy than \(\pi 2p_x\) and \(\pi 2p_y\).

Step 1: Understanding the Concept:

A primary standard is a highly purified compound that serves as a reference material in titrations. It must be stable, non-hygroscopic, and have high purity.


Step 2: Detailed Explanation:
\(NaOH\) is not a primary standard because:

1. It is highly hygroscopic (it absorbs moisture from the atmosphere).

2. It reacts with atmospheric carbon dioxide (\(CO_2\)) to form sodium carbonate (\(Na_2CO_3\)).

3. It is difficult to obtain in a very high state of purity.

As a result, \(NaOH\) solutions must be standardized against a primary standard (like Potassium Hydrogen Phthalate or Oxalic Acid) before use, making it a secondary standard.


Step 3: Final Answer:
\(NaOH\) is a secondary standard, not a primary standard.
Quick Tip: Common primary standards for acid-base titrations include anhydrous sodium carbonate (\(Na_2CO_3\)) for acids and oxalic acid (\(H_2C_2O_4 \cdot 2H_2O\)) for bases.

Step 1: Understanding the Concept:

Miller indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice.


Step 2: Key Formula or Approach:

The indices \((hkl)\) are calculated as follows:

1. Determine the intercepts of the plane on the three axes (\(x, y, z\)) in terms of lattice constants (\(a, b, c\)).

2. Take the reciprocals of these intercepts.

3. Clear fractions to obtain the smallest set of integers.


Step 3: Detailed Explanation:

For example, if a plane intersects the axes at \(1a\), \(2b\), and \(\infty c\):

- Intercepts: 1, 2, \(\infty\)

- Reciprocals: \(1/1, 1/2, 1/\infty \rightarrow 1, 0.5, 0\)

- Smallest integers: Multiply by 2 \(\rightarrow (210)\).

These indices uniquely define the orientation of the plane within the crystal system.


Step 4: Final Answer:

Miller indices represent the orientation of crystal planes.
Quick Tip: Planes that are parallel to an axis have an intercept of infinity, which results in a Miller index of 0 for that axis.

Step 1: Understanding the Concept:

EDTA (Ethylenediaminetetraacetic acid) is a hexadentate ligand used in complexometric titrations. The stability of the metal-EDTA complex is highly dependent on the pH of the solution.


Step 2: Detailed Explanation:

For the titration of alkaline earth metals like \(Ca^{2+}\) and \(Mg^{2+}\) with EDTA:

- At low pH, the EDTA exists in protonated forms and cannot effectively bind to the metal ions.

- An ammonia-ammonium chloride (\(NH_3/NH_4Cl\)) buffer is used to maintain a pH of approximately 10.

- This pH ensures that the EDTA is sufficiently deprotonated (\(Y^{4-}\) form) to form a stable complex with Calcium and that the Eriochrome Black T (EBT) indicator functions correctly to show a sharp color change at the endpoint.


Step 3: Final Answer:

The typical pH for Calcium titration with EDTA is 10.
Quick Tip: For specific determination of Calcium in the presence of Magnesium, the pH is sometimes raised to 12-13 to precipitate Magnesium as \(Mg(OH)_2\), using hydroxynaphthol blue as an indicator.

Step 1: Understanding the Concept:

When an object rolls down an inclined plane without slipping, its potential energy is converted into both translational and rotational kinetic energy.

The object with the higher acceleration (or lower moment of inertia relative to its mass) will reach the bottom in less time.


Step 2: Key Formula or Approach:

The acceleration \(a\) of an object rolling down an inclined plane is given by:
\[ a = \frac{g \sin \theta}{1 + \frac{I}{MR^2}} = \frac{g \sin \theta}{1 + \frac{K^2}{R^2}} \]

Where \(I\) is the moment of inertia, \(M\) is mass, \(R\) is radius, and \(K\) is the radius of gyration.


Step 3: Detailed Explanation:

For a Ring:

The moment of inertia \(I_{ring} = MR^2\).

Thus, \(\frac{I}{MR^2} = 1\).

Acceleration \(a_{ring} = \frac{g \sin \theta}{1 + 1} = 0.5 g \sin \theta\).



For a Disc:

The moment of inertia \(I_{disc} = \frac{1}{2}MR^2\).

Thus, \(\frac{I}{MR^2} = 0.5\).

Acceleration \(a_{disc} = \frac{g \sin \theta}{1 + 0.5} = \frac{g \sin \theta}{1.5} \approx 0.67 g \sin \theta\).



Since \(a_{disc} > a_{ring}\), the disc has a higher acceleration and will cover the same distance in a shorter time.


Step 4: Final Answer:

The disc will reach the bottom of the inclined plane first.
Quick Tip: Remember the order of "reaching first" for standard shapes: Sphere \(>\) Disc \(>\) Ring.
Smaller the value of \(\frac{I}{MR^2}\), faster the object reaches the bottom.

Step 1: Understanding the Concept:

Work done by a constant force is defined as the dot product of the force vector and the displacement vector.


Step 2: Key Formula or Approach:

Work done \(W = \vec{F} \cdot \vec{d}\)

Displacement vector \(\vec{d} = \vec{r}_B - \vec{r}_A\)


Step 3: Detailed Explanation:

First, find the displacement vector \(\vec{d}\):
\[ \vec{d} = (2-1)\hat{i} + (4-1)\hat{j} + (5-2)\hat{k} \]
\[ \vec{d} = (1\hat{i} + 3\hat{j} + 3\hat{k}) \, m \]

Now, calculate the work done using the dot product:
\[ W = \vec{F} \cdot \vec{d} = (3\hat{i} + 2\hat{j} + 4\hat{k}) \cdot (1\hat{i} + 3\hat{j} + 3\hat{k}) \]
\[ W = (3 \times 1) + (2 \times 3) + (4 \times 3) \]
\[ W = 3 + 6 + 12 \]
\[ W = 21 \, J \]


Step 4: Final Answer:

The work done by the force is 21 J.
Quick Tip: For the dot product, only multiply corresponding components (\(i\) with \(i\), \(j\) with \(j\), \(k\) with \(k\)) and sum them up.
Always ensure coordinates are subtracted in the order (Final - Initial).

Step 1: Understanding the Concept:

The electric field produced by an infinite sheet of charge is uniform and independent of the distance from the sheet.

The direction of the field is away from a positive charge and towards a negative charge.


Step 2: Key Formula or Approach:

Electric field due to a single infinite sheet with charge density \(\sigma\) is:
\[ E = \frac{\sigma}{2\epsilon_0} \]


Step 3: Detailed Explanation:
Consider two sheets, Sheet 1 and Sheet 2, both with \(+\sigma\).

In the region outside (to the left of both or right of both):

The electric field vectors from both sheets point in the same direction.

Net field \(E_{net} = E_1 + E_2\)
\[ E_{net} = \frac{\sigma}{2\epsilon_0} + \frac{\sigma}{2\epsilon_0} \]
\[ E_{net} = \frac{2\sigma}{2\epsilon_0} = \frac{\sigma}{\epsilon_0} \]

Note: In the region between the sheets, the fields oppose each other, resulting in a net field of \(0\).


Step 4: Final Answer:

The net electric field outside the common region is \( \frac{\sigma}{\epsilon_0} \).
Quick Tip: If the sheets had opposite charges (\(+\sigma\) and \(-\sigma\)), the field would be \(\frac{\sigma}{\epsilon_0}\) \textbf{inside} and \(0\) \textbf{outside}.

Step 1: Understanding the Concept:

According to the Principle of Flotation, for a floating body, the weight of the body is equal to the weight of the liquid displaced (Buoyant force).


Step 2: Key Formula or Approach:
Weight of body = Buoyant force
\[ M \cdot g = V_{submerged} \cdot \rho_{liquid} \cdot g \]

Since the weight of the object remains constant regardless of the liquid:
\[ V_1 \cdot \rho_1 = V_2 \cdot \rho_2 \]


Step 3: Detailed Explanation:
Given the ratio of submerged volumes:
\[ \frac{V_1}{V_2} = \frac{2}{3} \]

From the flotation condition, we have:
\[ \rho_1 V_1 = \rho_2 V_2 \]

Rearranging for the ratio of densities:
\[ \frac{\rho_1}{\rho_2} = \frac{V_2}{V_1} \]

Substitute the given ratio:
\[ \frac{\rho_1}{\rho_2} = \frac{3}{2} \]


Step 4: Final Answer:

The ratio of the densities of the liquids is \( 3 : 2 \).
Quick Tip: The density of the liquid is inversely proportional to the volume submerged.
If less volume is submerged, it means the liquid is denser.

Step 1: Understanding the Concept:

The given sequence is a number series where the terms (starting from the second one) follow a specific growth pattern. In competitive exams, such patterns often involve powers of 2, squares, or cubes, sometimes with a constant subtracted or added.


Step 2: Key Formula or Approach:

Let's analyze the relationship between the terms by observing their proximity to powers of 2:

- \(7\) is close to \(2^3 = 8\)

- \(31\) is close to \(2^5 = 32\)

- \(127\) is close to \(2^7 = 128\)

- \(511\) is close to \(2^9 = 512\)


Step 3: Detailed Explanation:

By expressing the terms as powers of 2 minus a constant:
\[ a_2 = 2^3 - 1 = 7 \]
\[ a_3 = 2^5 - 1 = 31 \]
\[ a_4 = 2^7 - 1 = 127 \]
\[ a_5 = 2^9 - 1 = 511 \]

The pattern for the powers of 2 is a sequence of consecutive odd integers: \(3, 5, 7, 9, \dots\).



Alternatively, looking at the differences between terms:
\(7 - (-3) = 10\)
\(31 - 7 = 24\)
\(127 - 31 = 96\)
\(511 - 127 = 384\)



The ratio of consecutive differences from the second difference onwards is:
\(\frac{96}{24} = 4\)
\(\frac{384}{96} = 4\)



Following this ratio, the next difference should be:
\[ Next Difference = 384 \times 4 = 1536 \]
\[ Next Term = 511 + 1536 = 2047 \]



Using the power pattern, the next odd power after 9 is 11:
\[ Next Term = 2^{11} - 1 = 2048 - 1 = 2047 \]


Step 4: Final Answer:

Both the difference method and the power-of-2 method lead to the same result. The next term in the series is 2047.
Quick Tip: Familiarize yourself with the powers of 2 (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048). If you see numbers like 31, 63, 127, 255, or 511 in a series, there is a very high probability that the pattern involves \(2^n - 1\).