Step 1: Understanding the Concept:

The Highest Common Factor (HCF) of a set of numbers expressed in their prime factorization is the product of the lowest powers of all common prime factors.


Step 2: Key Formula or Approach:

Given two numbers:
\[ N_1 = 2^2 \cdot 3^3 \]
\[ N_2 = 2^3 \cdot 3^2 \]


Step 3: Detailed Explanation:

Identify the common prime factors in both expressions, which are 2 and 3.

Compare the exponents for prime factor 2: The powers are 2 and 3. The minimum is 2. So, we take \(2^2\).

Compare the exponents for prime factor 3: The powers are 3 and 2. The minimum is 2. So, we take \(3^2\).

The HCF is the product of these terms:
\[ HCF = 2^2 \cdot 3^2 \]


Step 4: Final Answer:

The HCF is \(2^2 \cdot 3^2\).
Quick Tip: For HCF, think "Highest" common factor but use the "Lowest" powers. For LCM, think "Lowest" common multiple but use the "Highest" powers.

Step 1: Understanding the Concept:

Probability measures the likelihood of an event occurring and is calculated as the ratio of favorable outcomes to total possible outcomes.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of favorable outcomes}{Total number of outcomes} \]


Step 3: Detailed Explanation:

The letters in the word "FEBRUARY" are F, E, B, R, U, A, R, Y.

Total number of letters = 8.

The vowels in the word are E, U, and A.

Number of vowels (favorable outcomes) = 3.

Applying the formula:
\[ P(vowel) = \frac{3}{8} \]


Step 4: Final Answer:

The probability is \(\frac{3}{8}\).
Quick Tip: Always double-check the total count of letters and ensure you identify all vowels (A, E, I, O, U) correctly.

Step 1: Understanding the Concept:

For a number to end with the digit 0, its prime factorization must include both 2 and 5 as factors.


Step 2: Detailed Explanation:

Evaluate each option:

(a) \(4n\): If \(n = 5\), \(4 \times 5 = 20\). This ends in 0.

(b) \(4^n\): The prime factorization of 4 is \(2 \times 2\). Therefore, \(4^n = (2^2)^n = 2^{2n}\). It contains only the prime factor 2. Since 5 is missing, it can never end in 0.

(c) \(3^n + 1\): If \(n = 2\), \(3^2 + 1 = 9 + 1 = 10\). This ends in 0.

(d) \(10^{n+1}\): Since 10 is the base, it will always end in 0 for any natural \(n \ge 1\).

Thus, \(4^n\) is the only number that will never end in 0.


Step 3: Final Answer:

The number \(4^n\) will not end with 0 for any natural number \(n\).
Quick Tip: A number \(a^n\) ends in zero only if the base \(a\) is divisible by 10 or has both 2 and 5 in its prime factorization.

Step 1: Understanding the Concept:

A pair of linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\) represents two lines. A unique solution exists if the lines intersect at exactly one point.


Step 2: Key Formula or Approach:

The condition for a unique solution is:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]


Step 3: Detailed Explanation:

In the given system:
\(a_1 = p\), \(b_1 = q\)
\(a_2 = p_1\), \(b_2 = q_1\)

The condition becomes:
\[ \frac{p}{p_1} \neq \frac{q}{q_1} \]

Cross-multiplying the denominators, we get:
\[ pq_1 \neq qp_1 \]


Step 4: Final Answer:

The system has a unique solution if \(pq_1 \neq qp_1\).
Quick Tip: Think of the condition as the cross-product of the coefficients of \(x\) and \(y\) not being equal.

Step 1: Understanding the Concept:

A quadratic equation is a polynomial equation of degree 2, typically in the form \(ax^2 + bx + c = 0\) where \(a \neq 0\).


Step 2: Detailed Explanation:

Let's analyze each equation individually:

1. \(q_1 : x^2 + x = (x+1)^2 \Rightarrow x^2 + x = x^2 + 2x + 1 \).

After simplification: \(x = 2x + 1 \Rightarrow x + 1 = 0 \). This is a linear equation (degree 1).

2. \(q_2 : x - 1 = x^2 - 1 \Rightarrow x^2 - x = 0 \).

This is a polynomial of degree 2. Hence, it is a quadratic equation.

3. \(q_3 : x^4 = x^2 \). This is a polynomial of degree 4. It is not quadratic.

4. \(q_4 : \sqrt{x} = x^2\sqrt{x} + 1 \). This is not a polynomial equation because the exponent of \(x\) is not a non-negative integer (\(\sqrt{x} = x^{1/2}\)).

Therefore, only \(q_2\) is a quadratic equation.


Step 3: Final Answer:

Only \(q_2\) is a quadratic equation.
Quick Tip: Always simplify the equation completely before determining its degree. Terms with the highest power might cancel out.

Step 1: Understanding the Concept:

The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) determines the nature of the roots and is a part of the quadratic formula.


Step 2: Key Formula or Approach:

The discriminant (\(D\)) is given by:
\[ D = B^2 - 4AC \]


Step 3: Detailed Explanation:

For the given equation \(ax^2 + x + a = 0\):
\(A = a\)
\(B = 1\)
\(C = a\)

Substituting these into the discriminant formula:
\[ D = (1)^2 - 4(a)(a) \]
\[ D = 1 - 4a^2 \]


Step 4: Final Answer:

The discriminant is \(1 - 4a^2\).
Quick Tip: The discriminant is just the expression inside the square root of the quadratic formula. It does not include the square root sign itself.

Step 1: Understanding the Concept:

Distance between two points in a Cartesian plane can be found using the distance formula derived from the Pythagorean Theorem.


Step 2: Key Formula or Approach:

The distance \(d\) between points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]


Step 3: Detailed Explanation:

Given points are \((x_1, y_1) = (3, 0)\) and \((x_2, y_2) = (0, -3)\).

Substituting values:
\[ d = \sqrt{(0 - 3)^2 + (-3 - 0)^2} \]
\[ d = \sqrt{(-3)^2 + (-3)^2} \]
\[ d = \sqrt{9 + 9} \]
\[ d = \sqrt{18} units \]


Step 4: Final Answer:

The distance is \(\sqrt{18}\) units.
Quick Tip: If you see \(\sqrt{18}\) in the options, select it. Note that \(\sqrt{18}\) is also equal to \(3\sqrt{2}\).

Step 1: Understanding the Concept:

Similar triangles have proportional corresponding sides. The order of vertices in the similarity statement determines the correspondence.


Step 2: Detailed Explanation:

Given \(\triangle ABC \sim \triangle ADE\).

The correspondence is:

- Vertex A corresponds to Vertex A.

- Vertex B corresponds to Vertex D.

- Vertex C corresponds to Vertex E.

The ratio of corresponding sides is:
\[ \frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE} \]

Looking at the options, \(\frac{AB}{AD} = \frac{AC}{AE}\) matches this fundamental property.


Step 3: Final Answer:

The correct statement is \(\frac{AB}{AD} = \frac{AC}{AE}\).
Quick Tip: Match the first two letters of the first triangle with the first two of the second, and so on: \(AB/AD\), \(BC/DE\), \(AC/AE\).

Step 1: Understanding the Concept:

The Basic Proportionality Theorem (BPT) states that if a line is parallel to one side of a triangle, it divides the other two sides in the same ratio.


Step 2: Key Formula or Approach:

In \(\triangle PSR\), since \(EA \parallel SR\):
\[ \frac{PE}{ES} = \frac{PA}{AR} \]


Step 3: Detailed Explanation:

From the figure, we have:
\(PE = x\), \(ES = 1.8\), \(PA = 2\), \(AR = 1.5\).

Apply the BPT:
\[ \frac{x}{1.8} = \frac{2}{1.5} \]

Cross-multiply to solve for \(x\):
\[ 1.5x = 2 \times 1.8 \]
\[ 1.5x = 3.6 \]
\[ x = \frac{3.6}{1.5} = \frac{36}{15} = 2.4 cm \]

The question asks for the value of \(5x\):
\[ 5x = 5 \times 2.4 = 12 cm \]


Step 4: Final Answer:

The value of \(5x\) is 12 cm.
Quick Tip: Read the final requirement carefully. Many students calculate \(x\) and immediately pick option (a). Here, you need \(5x\).

Step 1: Understanding the Concept:

Zeroes of a polynomial are the x-coordinates of points where the graph intersects the x-axis. For zeroes to be positive, these intersections must occur to the right of the y-axis.


Step 2: Detailed Explanation:

Looking at the graphs:

- Graph (a) has one zero on the negative x-axis and one on the positive.

- Graph (b) also has one negative and one positive zero.

- Graph (c) is a parabola that intersects the x-axis at two distinct points, both of which are located to the right of the origin (positive x-values).

- Graph (d) does not touch the x-axis, indicating no real zeroes.

Therefore, Graph (c) represents both zeroes being positive.


Step 3: Final Answer:

Graph (c) represents both zeroes being positive.
Quick Tip: Check the intersection points relative to the origin (0,0). "Positive zeroes" mean the parabola must cross the x-axis in the region where \(x > 0\).

Step 1: Understanding the Concept:

A solution to a system of equations exists where the geometric representations (lines) of those equations intersect.


Step 2: Detailed Explanation:

- The equation \(x = 2\) represents a vertical line passing through 2 on the x-axis.

- The equation \(x = 3\) represents a vertical line passing through 3 on the x-axis.

- Since both are vertical lines, they are parallel to each other.

- Parallel lines do not intersect at any point.

- Because there is no intersection, there is no value of \(x\) that can be both 2 and 3 simultaneously.

Thus, the system has no solution.


Step 3: Final Answer:

The system has no solution.
Quick Tip: Linear equations like \(x=a\) and \(x=b\) where \(a \neq b\) are always parallel. Parallel lines mean "no solution".

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence where the difference between any two consecutive terms is constant. This constant is the common difference.


Step 2: Key Formula or Approach:

Common difference \(d\) is calculated as:
\[ d = a_2 - a_1 = a_3 - a_2 \]


Step 3: Detailed Explanation:

Let \(a_1 = x\), \(a_2 = x + 4\), and \(a_3 = x + 8\).

Calculate the difference between the second and first term:
\[ d = (x + 4) - x = 4 \]

Calculate the difference between the third and second term:
\[ d = (x + 8) - (x + 4) = x + 8 - x - 4 = 4 \]

Since the difference is constant and equal to 4, the common difference is 4.


Step 4: Final Answer:

The common difference is 4.
Quick Tip: The common difference is independent of the variable \(x\) in this case because the variable part is the same in every term.

Step 1: Understanding the Concept:

This question tests the knowledge of trigonometric values at standard angles and basic identities.


Step 2: Detailed Explanation:

Evaluate each statement:

(a) \(\sin 0^\circ = 0\) and \(\cos 0^\circ = 1\). Since \(0 \neq 1\), this statement is not true.

(b) \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) and \(\cot 60^\circ = \frac{1}{\sqrt{3}}\). This is true.

(c) \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 60^\circ = \frac{1}{2}\). This is true.

(d) \(\frac{1}{\sec 45^\circ} = \cos 45^\circ = \frac{1}{\sqrt{2}}\) and \(\sin 45^\circ = \frac{1}{\sqrt{2}}\). This is true.


Step 3: Final Answer:

The statement \(\sin 0^\circ = \cos 0^\circ\) is not true.
Quick Tip: Remember complementary angle properties: \(\sin \theta = \cos(90^\circ - \theta)\). This helps verify (b) and (c) without calculating values.

Step 1: Understanding the Concept:

The question involves solving a basic trigonometric equation relating the sine and cosine of an angle \(A\).

To find the value of the angle, we can transform the equation to involve a single trigonometric ratio, such as tangent (\(\tan A\)), which is defined as the ratio of sine to cosine.


Step 2: Key Formula or Approach:

We use the fundamental trigonometric relationship:
\[\tan A = \frac{\sin A}{\cos A}\]

Once the equation is in terms of \(\tan A\), we compare the result with standard trigonometric values to find the angle \(A\).


Step 3: Detailed Explanation:

Given the equation:
\[\sqrt{3} \sin A = \cos A\]

Divide both sides of the equation by \(\cos A\) (assuming \(\cos A \neq 0\)):
\[\frac{\sqrt{3} \sin A}{\cos A} = \frac{\cos A}{\cos A}\]

Substituting \(\frac{\sin A}{\cos A}\) with \(\tan A\), we get:
\[\sqrt{3} \tan A = 1\]

Now, divide both sides by \(\sqrt{3}\) to isolate \(\tan A\):
\[\tan A = \frac{1}{\sqrt{3}}\]

We know from the standard trigonometric table that for an acute angle:
\[\tan 30^{\circ} = \frac{1}{\sqrt{3}}\]

By comparing \(\tan A = \tan 30^{\circ}\), we find the value of angle \(A\):
\[A = 30^{\circ}\]


Step 4: Final Answer:

The measure of angle \(A\) is \(30^{\circ}\).
Quick Tip: When you see an equation with both \(\sin\) and \(\cos\), always try to convert it to \(\tan\) by dividing.
A useful mnemonic for the \(\tan\) values is: \(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\) (the "smaller" value for the smaller angle) and \(\tan 60^{\circ} = \sqrt{3}\) (the "larger" value for the larger angle).

Step 1: Understanding the Concept:

The angle of elevation of a point is the angle formed between the horizontal line of sight and the line of sight from the observer's eye to the point above the horizontal.

In the given right-angled triangle \(\triangle ABC\), \(\angle A = 90^{\circ}\).

The angle of elevation of point \(C\) from point \(B\) is the angle \(\angle ABC\), which is given as \(x\).


Step 2: Key Formula or Approach:

We use the Angle Sum Property of a triangle, which states that the sum of all interior angles of a triangle is \(180^{\circ}\).
\[\angle A + \angle B + \angle C = 180^{\circ}\]


Step 3: Detailed Explanation:

From the figure, we have the following angle measures:
\(\angle A = 90^{\circ}\) (as indicated by the square symbol at vertex \(A\))
\(\angle B = x\)
\(\angle C = 3x\)

Applying the angle sum property:
\[90^{\circ} + x + 3x = 180^{\circ}\]
\[90^{\circ} + 4x = 180^{\circ}\]

Subtracting \(90^{\circ}\) from both sides:
\[4x = 180^{\circ} - 90^{\circ}\]
\[4x = 90^{\circ}\]

Dividing by 4:
\[x = \frac{90^{\circ}}{4}\]
\[x = 22.5^{\circ}\]

The angle of elevation of point \(C\) from point \(B\) is \(\angle ABC = x = 22.5^{\circ}\).


Step 4: Final Answer:

The angle of elevation is \(22.5^{\circ}\).
Quick Tip: In a right-angled triangle, the sum of the two acute angles is always \(90^{\circ}\).
Here, \(x + 3x = 90^{\circ} \Rightarrow 4x = 90^{\circ} \Rightarrow x = 22.5^{\circ}\).
This saves a step by not involving the \(180^{\circ}\) total.

Step 1: Understanding the Concept:

The figure represents a composite solid consisting of a cone mounted on a hemisphere.

To find the slant height of the conical part, we need to identify the vertical height (\(h\)) and the base radius (\(r\)) of the cone.


Step 2: Key Formula or Approach:

The slant height (\(l\)) of a cone is related to its vertical height (\(h\)) and radius (\(r\)) by the Pythagorean theorem, as they form a right-angled triangle.
\[l = \sqrt{r^{2} + h^{2}}\]


Step 3: Detailed Explanation:

From the given figure:

Vertical height of the cone (\(h\)) = \(4 cm\)

Radius of the hemisphere (which is also the radius of the cone's base, \(r\)) = \(3 cm\)

Now, substituting these values into the slant height formula:
\[l = \sqrt{3^{2} + 4^{2}}\]
\[l = \sqrt{9 + 16}\]
\[l = \sqrt{25}\]
\[l = 5 cm\]


Step 4: Final Answer:

The slant height of the conical part is \(5 cm\).
Quick Tip: Remember the standard Pythagorean triplet (3, 4, 5).
If the legs of a right triangle are 3 and 4, the hypotenuse (slant height in this case) must be 5.

Step 1: Understanding the Concept:

The median class is the class interval whose cumulative frequency is greater than or equal to \(\frac{N}{2}\), where \(N\) is the total frequency.


Step 2: Key Formula or Approach:

1. Calculate the total frequency (\(N\)).

2. Calculate \(\frac{N}{2}\).

3. Find the cumulative frequencies (\(cf\)) for each class.

4. Identify the class where \(cf \ge \frac{N}{2}\).


Step 3: Detailed Explanation:

First, let's construct a cumulative frequency table:


\begin{tabular{|c|c|c|
\hline
Class Interval & Frequency (\(f\)) & Cumulative Frequency (\(cf\))

\hline
0-10 & 3 & 3

10-20 & 5 & \(3 + 5 = 8\)

20-30 & 7 & \(8 + 7 = 15\)

30-40 & 9 & \(15 + 9 = 24\)

40-50 & 11 & \(24 + 11 = 35\)

\hline
\end{tabular

Total frequency (\(N\)) = 35.
\[\frac{N}{2} = \frac{35}{2} = 17.5\]

We look for the class whose cumulative frequency is just greater than or equal to 17.5.

- For class 20-30, \(cf = 15\) (which is \(< 17.5\)).

- For class 30-40, \(cf = 24\) (which is \(\ge 17.5\)).

Therefore, the median class is 30-40.

The question asks for the upper limit of the median class.

In the interval 30-40, the lower limit is 30 and the upper limit is 40.


Step 4: Final Answer:

The upper limit of the median class is 40.
Quick Tip: Always double-check your cumulative frequency additions.
The last cumulative frequency must equal the sum of all individual frequencies.
In this case, \(3 + 5 + 7 + 9 + 11 = 35\), which matches.

Step 1: Understanding the Concept:

The three measures of central tendency (Mean, Median, and Mode) for a frequency distribution are related by an empirical formula.


Step 2: Key Formula or Approach:

The empirical relationship is:
\[ Mode = 3 \times Median - 2 \times Mean \]


Step 3: Detailed Explanation:

Given:

Median = 5

Mode = 4

Let Mean be \( \bar{x} \).

Substituting the given values into the formula:
\[ 4 = 3(5) - 2(\bar{x}) \]
\[ 4 = 15 - 2\bar{x} \]

Transpose \( 2\bar{x} \) to the left side and 4 to the right side:
\[ 2\bar{x} = 15 - 4 \]
\[ 2\bar{x} = 11 \]
\[ \bar{x} = \frac{11}{2} \]


Step 4: Final Answer:

The mean of the data is \(\frac{11}{2}\).
Quick Tip: Remember the formula using the "3-2-1" rule: 3 Median - 2 Mean = 1 Mode. Also, note that "Median" has more letters than "Mean", and it is multiplied by the larger number (3).

Step 1: Understanding the Concept:

The probability of a non-occurrence of an event is the complement of the probability of its occurrence.


Step 2: Detailed Explanation:

For Assertion (A):

Total number of balls = \( 5 (red) + 2 (white) + 3 (green) = 10 \).

Number of white balls = 2.

Probability of drawing a white ball, \( P(W) = \frac{2}{10} = \frac{1}{5} \).

Probability of drawing a non-white ball = \( 1 - P(W) = 1 - \frac{1}{5} = \frac{4}{5} \).

Alternatively, non-white balls = Red + Green = \( 5 + 3 = 8 \).
\( P(non-white) = \frac{8}{10} = \frac{4}{5} \).

Thus, Assertion (A) is true.

For Reason (R):

The sum of probability of an event and its complement is always 1. This is a fundamental property.

Thus, Reason (R) is true and correctly explains the logic used in Assertion (A).


Step 3: Final Answer:

Both (A) and (R) are true and (R) is the correct explanation of (A).
Quick Tip: In probability, the phrase "non-white" is exactly the complement of "white". Always check if it is faster to calculate the event directly or via its complement.

Step 1: Understanding the Concept:

A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. A prime number has exactly two factors.


Step 2: Detailed Explanation:

For Assertion (A):

Evaluate the expression: \( 7 \times 2 + 3 = 14 + 3 = 17 \).

17 is a prime number because its only factors are 1 and 17.

Therefore, it is NOT a composite number. Assertion (A) is false.

For Reason (R):

By definition, a composite number is an integer greater than 1 that has more than two factors.

Therefore, Reason (R) is true.


Step 3: Final Answer:

Assertion (A) is false, but Reason (R) is true.
Quick Tip: Don't confuse composite numbers with odd numbers. While 17 is odd, it is prime. While 9 is odd, it is composite. Always evaluate the numerical expression before judging.

Step 1: Understanding the Concept:

The coordinates of a point dividing a line segment in a given ratio can be determined using the internal section formula.


Step 2: Key Formula or Approach:

Section Formula: If a point \( P(x, y) \) divides the segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in ratio \( m : n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n} \]


Step 3: Detailed Explanation:

Given: \( A(-6, 10) \implies x_1 = -6, y_1 = 10 \).
\( B(3, -8) \implies x_2 = 3, y_2 = -8 \).

Ratio \( m : n = 2 : 7 \).

Applying the formula for \( x \):
\[ x = \frac{2(3) + 7(-6)}{2 + 7} = \frac{6 - 42}{9} = \frac{-36}{9} = -4 \]

Applying the formula for \( y \):
\[ y = \frac{2(-8) + 7(10)}{2 + 7} = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \]

The coordinates of the point are \( (-4, 6) \).


Step 4: Final Answer:

The required point is \( (-4, 6) \).
Quick Tip: Be very careful with negative signs during calculation. In the numerator, multiply the ratio 'm' with the 'far' coordinate \(x_2\), and 'n' with the 'near' coordinate \(x_1\).

Step 1: Understanding the Concept:

A quadratic polynomial with zeroes \( \alpha \) and \( \beta \) is given by \( p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta] \).


Step 2: Detailed Explanation:

Let the zeroes be \( \alpha \) and \( 2\alpha \).

Given: Sum of zeroes = \( -6 \).
\[ \alpha + 2\alpha = -6 \]
\[ 3\alpha = -6 \implies \alpha = -2 \]

So, the zeroes are \( -2 \) and \( 2(-2) = -4 \).

Product of zeroes \( = (-2) \times (-4) = 8 \).

The quadratic polynomial is:
\[ p(x) = k[x^2 - (sum of zeroes)x + (product of zeroes)] \]
\[ p(x) = k[x^2 - (-6)x + 8] \]
\[ p(x) = k(x^2 + 6x + 8) \]


Step 3: Final Answer:

The polynomial is \( x^2 + 6x + 8 \) (taking \( k = 1 \)).
Quick Tip: If you know the sum and product of zeroes, the polynomial is always \( x^2 - Sx + P \). Don't forget the negative sign before the sum term.

Step 1: Understanding the Concept:

If \( \alpha \) is a zero of \( p(x) \), then \( p(\alpha) = 0 \). The relationship between zeroes and coefficients is \( \alpha + \beta = -b/a \).


Step 2: Detailed Explanation:

Given \( p(x) = x^2 - 5x - c \).

Since \( -1 \) is a zero:
\[ (-1)^2 - 5(-1) - c = 0 \]
\[ 1 + 5 - c = 0 \implies 6 - c = 0 \implies c = 6 \]

Now, the polynomial is \( x^2 - 5x - 6 = 0 \).

Let the zeroes be \( \alpha = -1 \) and \( \beta \).

Sum of zeroes \( = -b/a = -(-5)/1 = 5 \).
\[ -1 + \beta = 5 \implies \beta = 6 \]


Step 3: Final Answer:

The value of \( c \) is 6 and the other zero is 6.
Quick Tip: You can also find the other zero using the product of zeroes: \( \alpha\beta = c/a = -6/1 = -6 \). Since \( \alpha = -1 \), \( \beta = -6/(-1) = 6 \).

Step 1: Understanding the Concept:

The Basic Proportionality Theorem (BPT) states that if a line is parallel to one side of a triangle, it divides the other two sides in the same ratio.


Step 2: Detailed Explanation:

In \( \triangle ABC \), \( PQ \parallel BC \).

By BPT:
\[ \frac{AP}{PB} = \frac{AQ}{QC} \]

Given \( AP = \frac{1}{2}AB \).

This means \( P \) is the midpoint of \( AB \), so \( AP = PB \).
\[ \frac{AP}{AP} = \frac{AQ}{QC} \implies 1 = \frac{AQ}{QC} \implies AQ = QC \]

Since \( QC = 3 \) cm, then \( AQ = 3 \) cm.

The total length of \( AC = AQ + QC = 3 + 3 = 6 \) cm.


Step 3: Final Answer:

The length of \( AC \) is 6 cm.
Quick Tip: If a line is parallel to the base and starts at the midpoint of one side, it will hit the midpoint of the other side. This is the Converse of the Midpoint Theorem.

Step 1: Understanding the Concept:

Trigonometric values for specific standard angles (\( 30^\circ, 45^\circ, 60^\circ \)) are used to evaluate algebraic expressions involving them.


Step 2: Detailed Explanation:

The standard values are:
\( \sin 30^\circ = \frac{1}{2} \)
\( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
\( \cot 60^\circ = \frac{1}{\sqrt{3}} \)

Substituting these values into the expression:
\[ \left( \frac{1}{2} \right)^2 - \left( \frac{1}{\sqrt{2}} \right)^2 + \left( \frac{1}{\sqrt{3}} \right)^2 \]
\[ = \frac{1}{4} - \frac{1}{2} + \frac{1}{3} \]

Taking the LCM of 4, 2, and 3, which is 12:
\[ = \frac{3 - 6 + 4}{12} = \frac{1}{12} \]


Step 3: Final Answer:

The value of the expression is \( \frac{1}{12} \).
Quick Tip: Always memorize the trigonometric table for \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \). It saves time and ensures accuracy in basic calculation questions.

Step 1: Understanding the Concept:

Trigonometric ratios can be equated to find angles if the value corresponds to a standard angle value.


Step 2: Detailed Explanation:

Given: \( \sin(A + 2B) = 2 \cos 60^\circ \).

We know \( \cos 60^\circ = \frac{1}{2} \).
\[ \sin(A + 2B) = 2 \times \frac{1}{2} = 1 \]

Since \( \sin 90^\circ = 1 \), we can equate the angles:
\[ A + 2B = 90^\circ \quad \dots (1) \]

Also given: \( A = 3B \).

Substitute \( A = 3B \) into equation (1):
\[ 3B + 2B = 90^\circ \]
\[ 5B = 90^\circ \implies B = 18^\circ \]

Now find \( A \):
\[ A = 3(18^\circ) = 54^\circ \]


Step 3: Final Answer:

The angles are \( A = 54^\circ \) and \( B = 18^\circ \).
Quick Tip: When \( \sin \theta = 1 \), \( \theta = 90^\circ \). When \( \cos \theta = 1 \), \( \theta = 0^\circ \). Always solve the numerical part on the RHS first.

Step 1: Understanding the Concept:

Probability = (Favorable outcomes) / (Total outcomes).


Step 2: Detailed Explanation:

Total wall clocks = 60.

Good clocks = 40.

Clocks with minor defects = 15.

Broken clocks = \( 60 - (40 + 15) = 60 - 55 = 5 \).

(i) The box will be rejected:

The box is rejected if the selected clock is broken.

Favorable outcomes (Broken) = 5.
\[ P(Rejected) = \frac{5}{60} = \frac{1}{12} \]

(ii) The clock has minor defect:

Favorable outcomes (Minor defect) = 15.
\[ P(Minor defect) = \frac{15}{60} = \frac{1}{4} \]


Step 3: Final Answer:

(i) \( 1/12 \), (ii) \( 1/4 \).
Quick Tip: Always simplify your fractions to the lowest terms. Probability values must always be between 0 and 1 inclusive.

Step 1: Understanding the Concept:

The sum and product of rational and irrational numbers follow specific properties. We usually prove irrationality using the method of contradiction.


Step 2: Detailed Explanation:

Let us assume that \( 3 + 2\sqrt{5} \) is a rational number.

Then, it can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
\[ 3 + 2\sqrt{5} = \frac{a}{b} \]

Subtract 3 from both sides:
\[ 2\sqrt{5} = \frac{a}{b} - 3 = \frac{a - 3b}{b} \]

Divide by 2:
\[ \sqrt{5} = \frac{a - 3b}{2b} \]

Since \( a \) and \( b \) are integers, \( a - 3b \) and \( 2b \) are also integers.

This means \( \frac{a - 3b}{2b} \) is a rational number.

This implies \( \sqrt{5} \) is a rational number.

But it is given that \( \sqrt{5} \) is irrational.

This is a contradiction to our assumption.

Therefore, \( 3 + 2\sqrt{5} \) must be an irrational number.


Step 3: Final Answer:
\( 3 + 2\sqrt{5} \) is irrational.
Quick Tip: In such proofs, isolate the irrational term (\( \sqrt{5} \)) on one side and show that the other side is a rational expression (combination of integers).

Step 1: Understanding the Concept:

Solving graphically involves plotting lines and finding their intersection. The area of a triangle with vertices on axes is calculated using \( \frac{1}{2} \times Base \times Height \).


Step 2: Detailed Explanation:

For \( x + 3y = 6 \):

- If \( x = 0, y = 2 \). Point: \( (0, 2) \).

- If \( y = 0, x = 6 \). Point: \( (6, 0) \).

For \( 2x - 3y = 12 \):

- If \( x = 0, y = -4 \). Point: \( (0, -4) \).

- If \( y = 0, x = 6 \). Point: \( (6, 0) \).

Plotting these points, the two lines intersect at \( (6, 0) \).

So, the solution is \( x = 6, y = 0 \).

The second part asks for the area of the triangle formed by \( x + 3y = 6 \), \( x = 0 \) (y-axis) and \( y = 0 \) (x-axis).

The vertices of this triangle are \( (0, 0), (6, 0) \), and \( (0, 2) \).

Base = 6 units (on x-axis), Height = 2 units (on y-axis).
\[ Area = \frac{1}{2} \times 6 \times 2 = 6 sq. units \]


Step 3: Final Answer:

Intersection point is \( (6, 0) \) and the area is 6 sq. units.
Quick Tip: For area with the axes, simply find the x-intercept and y-intercept of the line. The product of these intercepts divided by 2 gives the area.

Step 1: Understanding the Concept:

Supplementary angles are two angles whose sum is \( 180^\circ \).


Step 2: Detailed Explanation:

Let the two angles be \( x \) and \( y \).

From the definition of supplementary angles:
\[ x + y = 180^\circ \quad \dots (1) \]

Given: One angle exceeds the other by \( 120^\circ \).
\[ x - y = 120^\circ \quad \dots (2) \]

Adding equations (1) and (2):
\[ (x + y) + (x - y) = 180^\circ + 120^\circ \]
\[ 2x = 300^\circ \implies x = 150^\circ \]

Substituting \( x = 150^\circ \) in equation (1):
\[ 150^\circ + y = 180^\circ \implies y = 30^\circ \]


Step 3: Final Answer:

The angles are \( 150^\circ \) and \( 30^\circ \).
Quick Tip: Remember: Supplementary = \( 180^\circ \), Complementary = \( 90^\circ \). A quick check: \( 150 + 30 = 180 \) and \( 150 - 30 = 120 \).

Step 1: Understanding the Concept:

If \( P \) is equidistant from \( A \) and \( B \), then the distance \( PA = PB \). We use the distance formula.


Step 2: Detailed Explanation:

Let \( A = (3, 6) \) and \( B = (-3, 4) \).
\[ PA = PB \implies PA^2 = PB^2 \]

Using distance formula \( d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \):
\[ (x - 3)^2 + (y - 6)^2 = (x + 3)^2 + (y - 4)^2 \]
\[ x^2 - 6x + 9 + y^2 - 12y + 36 = x^2 + 6x + 9 + y^2 - 8y + 16 \]

Cancel \( x^2, y^2, 9 \) from both sides:
\[ -6x - 12y + 36 = 6x - 8y + 16 \]
\[ -12x - 4y = 16 - 36 \]
\[ -12x - 4y = -20 \]

Divide by \(-4 \):
\[ 3x + y = 5 \quad \dots (The relation) \]

If \( P \) lies on the x-axis, then \( y = 0 \).
\[ 3x + 0 = 5 \implies x = \frac{5}{3} \]


Step 3: Final Answer:

The relation is \( 3x + y = 5 \) and the point is \( (5/3, 0) \).
Quick Tip: Points on the x-axis always have \( y = 0 \). This simplification usually leads directly to the answer once the relation is found.

Step 1: Understanding the Concept:

To prove trigonometric identities, convert terms into their basic components (\( \sin \) and \( \cos \)) or use reciprocal identities.


Step 2: Detailed Explanation:

LHS \( = \frac{\sin A - \tan A}{\sin A + \tan A} \)

Substitute \( \tan A = \frac{\sin A}{\cos A} \):
\[ = \frac{\sin A - \frac{\sin A}{\cos A}}{\sin A + \frac{\sin A}{\cos A}} \]

Factor out \( \sin A \) from numerator and denominator:
\[ = \frac{\sin A \left( 1 - \frac{1}{\cos A} \right)}{\sin A \left( 1 + \frac{1}{\cos A} \right)} \]

Cancel \( \sin A \):
\[ = \frac{1 - \frac{1}{\cos A}}{1 + \frac{1}{\cos A}} \]

We know \( \frac{1}{\cos A} = \sec A \):
\[ = \frac{1 - \sec A}{1 + \sec A} = RHS \]

Hence Proved.


Step 3: Final Answer:

LHS = RHS.
Quick Tip: Often, "taking common" is the hidden step in simplifying trigonometric fractions. Look for terms like \( \sin A \) or \( \cos A \) that appear in both parts of a numerator/denominator.

Step 1: Understanding the Concept:

Use the fundamental identity \( \sin^2 x + \cos^2 x = 1 \) to find related trigonometric ratios.


Step 2: Detailed Explanation:

Given: \( \sin x = p \).

Then \( \cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - p^2} \).

(i) For \( \cot x \):
\[ \cot x = \frac{\cos x}{\sin x} = \frac{\sqrt{1 - p^2}}{p} \]

Hence Proved.

(ii) For the expression:

LHS \( = \frac{1 + \tan^2 x}{1 + \cot^2 x} \)

Using identities \( 1 + \tan^2 x = \sec^2 x \) and \( 1 + \cot^2 x = \csc^2 x \):
\[ = \frac{\sec^2 x}{\csc^2 x} = \frac{1/\cos^2 x}{1/\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x \]

Substitute values of \( \sin x \) and \( \cos x \):
\[ = \frac{(p)^2}{(\sqrt{1 - p^2})^2} = \frac{p^2}{1 - p^2} = RHS \]

Hence Proved.


Step 3: Final Answer:

LHS = RHS for both parts.
Quick Tip: The identity \( \frac{1+\tan^2 A}{1+\cot^2 A} = \tan^2 A \) is very useful and appears frequently in competitive exams.

Step 1: Understanding the Concept:

This theorem is a fundamental property of circles stating that two tangents drawn from a single point outside the circle to the points of tangency have identical lengths.


Step 2: Key Formula or Approach:

The proof relies on establishing the congruence of two right-angled triangles formed by the radii, the tangents, and the line connecting the external point to the center.


Step 3: Detailed Explanation:

Given: A circle with centre \( O \), an external point \( P \), and two tangents \( PA \) and \( PB \) meeting the circle at \( A \) and \( B \) respectively.

To Prove: \( PA = PB \).

Construction: Join \( OA, OB, \) and \( OP \).

Proof:

In \( \triangle OAP \) and \( \triangle OBP \):

1. \( \angle OAP = \angle OBP = 90^\circ \)

(A radius is always perpendicular to the tangent at the point of contact).

2. \( OA = OB \)

(Radii of the same circle are equal).

3. \( OP = OP \)

(Common hypotenuse for both triangles).

Therefore, \( \triangle OAP \cong \triangle OBP \) by the RHS (Right angle-Hypotenuse-Side) congruence criterion.

By CPCT (Corresponding Parts of Congruent Triangles):
\[ PA = PB \]


Step 4: Final Answer:

Hence, the lengths of tangents drawn from an external point to a circle are proved to be equal.
Quick Tip: Remember that \( \triangle OAP \cong \triangle OBP \) also implies \( \angle APO = \angle BPO \), meaning the line joining the external point to the center bisects the angle between the tangents.

Step 1: Understanding the Concept:

Tangents at the endpoints of a diameter are perpendicular to that diameter. For the second part, since the tangents are parallel, the quadrilateral formed behaves as a trapezium.


Step 2: Detailed Explanation:

Part 1: Prove \( p \parallel q \)

- Tangent \( p \) is drawn at \( A \), so \( OA \perp p \), implying \( \angle PAB = 90^\circ \).

- Tangent \( q \) is drawn at \( B \), so \( OB \perp q \), implying \( \angle QBA = 90^\circ \).

- Since \( \angle PAB + \angle QBA = 90^\circ + 90^\circ = 180^\circ \), and these are consecutive interior angles, lines \( p \) and \( q \) must be parallel.


Part 2: Find \( \angle ADC \)

- Given that tangents \( p \) and \( q \) are parallel, line segment \( AD \) lies on tangent \( p \) and \( BC \) lies on tangent \( q \).

- Thus, \( AD \parallel BC \).

- In a pair of parallel lines, the sum of consecutive interior angles is \( 180^\circ \).

- Therefore, \( \angle BCD + \angle ADC = 180^\circ \).

- Given \( \angle BCD = 110^\circ \).
\[ 110^\circ + \angle ADC = 180^\circ \]
\[ \angle ADC = 180^\circ - 110^\circ = 70^\circ \]


Step 3: Final Answer:

The measure of \( \angle ADC \) is \( 70^\circ \).
Quick Tip: Whenever you see tangents at the end of a diameter, automatically think of parallel lines and properties like interior angles or alternate interior angles.

Step 1: Understanding the Concept:

To convert a rational equation into standard quadratic form \( ax^2 + bx + c = 0 \), we eliminate fractions by multiplying through by the least common denominator.


Step 2: Key Formula or Approach:

Standard form: \( ax^2 + bx + c = 0 \).

Discriminant \( D = b^2 - 4ac \).

Quadratic Formula: \( x = \frac{-b \pm \sqrt{D}}{2a} \).


Step 3: Detailed Explanation:

Given equation:
\[ \frac{24}{18-x} - \frac{24}{18+x} = 1 \]

Take 24 as a common factor:
\[ 24 \left( \frac{(18+x) - (18-x)}{(18-x)(18+x)} \right) = 1 \]
\[ 24 \left( \frac{18 + x - 18 + x}{324 - x^2} \right) = 1 \]
\[ 24 \left( \frac{2x}{324 - x^2} \right) = 1 \]
\[ 48x = 324 - x^2 \]
\[ x^2 + 48x - 324 = 0 \]

This is the standard form.

Find Discriminant (D):

Here \( a = 1, b = 48, c = -324 \).
\[ D = (48)^2 - 4(1)(-324) \]
\[ D = 2304 + 1296 = 3600 \]

Find Roots:
\[ x = \frac{-48 \pm \sqrt{3600}}{2(1)} \]
\[ x = \frac{-48 \pm 60}{2} \]
\( x = \frac{-48 + 60}{2} = \frac{12}{2} = 6 \)
\( x = \frac{-48 - 60}{2} = \frac{-108}{2} = -54 \)


Step 4: Final Answer:

Standard form: \( x^2 + 48x - 324 = 0 \).

Discriminant = 3600.

Roots = 6 and -54.
Quick Tip: When dealing with speed/distance/time problems resulting in this format, discard the negative root as speed or time cannot be negative.

Step 1: Understanding the Concept:

We use algebraic modeling to define two variables based on their relationship and then create a quadratic equation based on the condition given.


Step 2: Detailed Explanation:

Let the smaller positive number be \( x \).

Since the other number exceeds it by 2, the larger number is \( x + 2 \).

According to the question:
\[ x^2 + (x+2)^2 = 100 \]
\[ x^2 + x^2 + 4x + 4 = 100 \]
\[ 2x^2 + 4x - 96 = 0 \]

Divide by 2 to simplify:
\[ x^2 + 2x - 48 = 0 \]

Factorizing the equation:

Find two numbers whose product is -48 and sum is 2. These are 8 and -6.
\[ x^2 + 8x - 6x - 48 = 0 \]
\[ x(x + 8) - 6(x + 8) = 0 \]
\[ (x - 6)(x + 8) = 0 \]
\( x = 6 \) or \( x = -8 \).

Since the question specifies "positive numbers", we reject \( x = -8 \).

So, \( x = 6 \).

The other number is \( x + 2 = 6 + 2 = 8 \).


Step 3: Final Answer:

The two positive numbers are 6 and 8.
Quick Tip: Recognize Pythagorean triplets! \( 6^2 + 8^2 = 100 = 10^2 \). This can often help you guess the answer in multiple-choice questions.

Step 1: Understanding the Concept:

Congruent triangles have equal corresponding sides and angles (CPCT). These equalities are used to create the proportional ratios needed for similarity.


Step 2: Detailed Explanation:

Given \( \triangle ABE \cong \triangle ACD \).

By CPCT:

- \( AB = AC \)

- \( AE = AD \implies AD = AE \)


(i) To Prove \( \triangle ADE \sim \triangle ABC \):

In \( \triangle ADE \) and \( \triangle ABC \):

1. \( \angle A = \angle A \) (Common angle).

2. Since \( AB = AC \) and \( AD = AE \), we can write:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]

By the SAS (Side-Angle-Side) similarity criterion, \( \triangle ADE \sim \triangle ABC \).


(ii) To Prove \( \triangle BOD \sim \triangle COE \):

From \( \triangle ABE \cong \triangle ACD \), \( \angle ABE = \angle ACD \).

Also, in isosceles \( \triangle ABC \) (since \( AB = AC \)), \( \angle ABC = \angle ACB \).

Subtracting equals from equals:
\( \angle ABC - \angle ABE = \angle ACB - \angle ACD \)
\( \angle DBO = \angle ECO \).

In \( \triangle BOD \) and \( \triangle COE \):

1. \( \angle DBO = \angle ECO \) (Proven above).

2. \( \angle BOD = \angle COE \) (Vertically opposite angles).

By the AA similarity criterion, \( \triangle BOD \sim \triangle COE \).


Step 3: Final Answer:

Both similarity relations have been proved based on CPCT properties.
Quick Tip: Whenever triangles are congruent, start by listing all 6 CPCT equalities. Usually, 1 or 2 of them are the key to the next part of the proof.

Step 1: Understanding the Concept:

The shaded region is a major sector of the circle. We calculate the central angle of the sector first, then use the area formula to find the radius.


Step 2: Key Formula or Approach:

Area of Sector = \( \frac{\theta}{360} \times \pi r^2 \).

Length of Arc = \( \frac{\theta}{360} \times 2\pi r \).


Step 3: Detailed Explanation:

- Since \( \triangle OAB \) is equilateral, \( \angle AOB = 60^\circ \).

- The shaded region is a major sector. Its central angle \( \theta = 360^\circ - 60^\circ = 300^\circ \).

- Area of shaded region = \( 750\pi \).
\[ \frac{300}{360} \times \pi r^2 = 750 \pi \]
\[ \frac{5}{6} r^2 = 750 \implies r^2 = \frac{750 \times 6}{5} = 900 \]
\[ r = 30 cm \]

- The sides of the equilateral triangle are also equal to the radius, so \( OA = OB = AB = 30 cm \).

- Perimeter of shaded region = Length of major arc AB + Radius OA + Radius OB.

- Length of major arc = \( \frac{300}{360} \times 2 \times \pi \times 30 = \frac{5}{6} \times 60\pi = 50\pi cm \).

- Total Perimeter = \( 50\pi + 30 + 30 = 50\pi + 60 cm \).


Step 4: Final Answer:

The perimeter of the shaded region is \( (50\pi + 60) cm \).
Quick Tip: Don't forget to add the radii! The "perimeter of a sector" is not just the arc length; it's the entire boundary.

Step 1: Understanding the Concept:

The area of a sector is directly proportional to its central angle when the radii are equal.


Step 2: Detailed Explanation:

- Radius for both circles = \( r \).

- Central angle for fig (i), \( \theta_1 = 120^\circ \).

- Central angle for fig (ii), \( \theta_2 = 60^\circ \).

- Area of sector (i) = \( \frac{120}{360} \pi r^2 = \frac{1}{3} \pi r^2 \).

- Area of sector (ii) = \( \frac{60}{360} \pi r^2 = \frac{1}{6} \pi r^2 \).

- Ratio = \( \frac{Area (i)}{Area (ii)} = \frac{(1/3)\pi r^2}{(1/6)\pi r^2} = \frac{6}{3} = \frac{2}{1} \).


Step 3: Final Answer:

The ratio of the areas is \( 2 : 1 \).
Quick Tip: If radii are the same, the ratio of areas is simply the ratio of angles: \( 120/60 = 2 \). You don't need to plug in the full formula.

Step 1: Understanding the Concept:

Mean is the weighted average of the class marks. Median is the value corresponding to the middle frequency \( N/2 \).


Step 2: Detailed Explanation:

Calculation for Mean:

Total frequency \( \sum f = 7+8+2+2+1 = 20 \).

Class marks (\( x_i \)): 2, 4, 6, 8, 10.
\( \sum f_i x_i = (7 \times 2) + (8 \times 4) + (2 \times 6) + (2 \times 8) + (1 \times 10) \)
\( \sum f_i x_i = 14 + 32 + 12 + 16 + 10 = 84 \).

Mean \( = \frac{\sum f_i x_i}{\sum f_i} = \frac{84}{20} = 4.2 \).


Calculation for Median:

Cumulative frequencies (CF): 7, 15, 17, 19, 20.
\( N/2 = 20/2 = 10 \).

CF just greater than 10 is 15. The corresponding class is 3-5.

Lower limit \( l = 3 \), CF of preceding class \( cf = 7 \), frequency \( f = 8 \), class width \( h = 2 \).

Median \( = l + \left[ \frac{(N/2 - cf)}{f} \right] \times h \)

Median \( = 3 + \left[ \frac{(10 - 7)}{8} \right] \times 2 = 3 + \frac{3 \times 2}{8} = 3 + 0.75 = 3.75 \).


Step 3: Final Answer:

Mean = 4.2 and Median = 3.75.
Quick Tip: You can use the empirical relation \( Mode \approx 3Median - 2Mean \) to roughly verify your answers.

Step 1: Understanding the Concept:

The arrangement of watermelons follows an Arithmetic Progression (AP) where the number of items changes by a constant amount in each step.


Step 2: Detailed Explanation:

(i) Let the bottom row be the 1st term, \( a = 101 \).

Total rows \( n = 21 \). Last term \( a_{21} = 1 \).
\( a_n = a + (n-1)d \implies 1 = 101 + (21-1)d \)
\( 1 - 101 = 20d \implies -100 = 20d \implies d = -5 \).


(ii) 15th row from bottom is \( a_{15} \).
\( a_{15} = 101 + (15-1)(-5) = 101 - 70 = 31 \).


(iii)(a) Total watermelons \( S_{21} \).
\( S_n = \frac{n}{2}(a + l) = \frac{21}{2}(101 + 1) = \frac{21}{2} \times 102 = 1071 \).


(iii)(b) Let row from top be \( b_n \) where \( b_1 = 1, d = 5 \).

Row from bottom is \( a_n \) where \( a_1 = 101, d = -5 \).
\( b_n = a_n \implies 1 + (n-1)5 = 101 + (n-1)(-5) \)
\( 1 + 5n - 5 = 101 - 5n + 5 \implies 10n = 110 \implies n = 11 \).


Step 3: Final Answer:

(i) \( d = -5 \), (ii) 31, (iii)(a) 1071, (iii)(b) \( n = 11 \).
Quick Tip: In symmetric AP problems like (iii)(b), the common term is usually the middle term of the series. \( (21+1)/2 = 11 \).

Step 1: Understanding the Concept:

This problem involves calculating surface area and volume of combined solid shapes (cylinder and hemisphere).


Step 2: Detailed Explanation:

(i) Diameter = 1.4 m, so Radius \( r = 0.7 m \).


(ii) Volume of hemisphere = \( \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (0.7)^3 = \frac{0.686}{3} \pi \approx 0.2287 \pi m^3 \).


(iii)(a) Height above ground \( h = 2 - 0.6 = 1.4 m \).

Surface area = CSA of cylinder + Area of top ring? (Assume thin rim) + CSA of hemisphere.

Actually, TSA above ground = CSA of cylinder + CSA of hemisphere.

TSA \( = 2\pi r h + 2\pi r^2 = 2\pi(0.7)(1.4) + 2\pi(0.7)^2 = 1.96\pi + 0.98\pi = 2.94\pi m^2 \).

(Wait, if we exclude the ground base, it's just side + internal bowl).


(iii)(b) Volume of wood = Volume of cylinder above ground - Volume of hemisphere.

Volume \( = \pi r^2 h - \frac{2}{3} \pi r^3 = \pi (0.7)^2 (1.4) - 0.2287\pi = 0.686\pi - 0.2287\pi = 0.4573\pi m^3 \).


Step 3: Final Answer:

Values found for radius, volume of water, surface area, and wood volume.
Quick Tip: When a shape is "scooped out", the surface area \textbf{increases} (new internal surface) while the volume \textbf{decreases}.

Step 1: Understanding the Concept:

This is a standard Trigonometry (Heights and Distances) problem involving two right-angled triangles sharing a common base \( x \).


Step 2: Detailed Explanation:

(i) In small triangle: \( \tan 30^\circ = \frac{10}{x} \implies \frac{1}{1.732} = \frac{10}{x} \implies x = 17.32 m \).


(ii) Total height \( H = 10 + 7.32 = 17.32 m \).

In large triangle: \( \tan \theta = \frac{17.32}{x} = \frac{17.32}{17.32} = 1 \).

Since \( \tan 45^\circ = 1 \), \( \theta = 45^\circ \).


(iii)(a) Find \( l_1 \): \( \cos 30^\circ = \frac{x}{l_1} \implies \frac{1.732}{2} = \frac{17.32}{l_1} \implies l_1 = 20 m \).

Find \( l_2 \): \( \cos 45^\circ = \frac{x}{l_2} \implies \frac{1}{1.4} \approx \frac{17.32}{l_2} \implies l_2 = 17.32 \times 1.4 = 24.25 m \).

Total length \( = 20 + 24.25 = 44.25 m \).


(iii)(b) \( l_2 - l_1 = 24.25 - 20 = 4.25 m \).


Step 3: Final Answer:

Distance \( x = 17.32 m \), Angle \( \theta = 45^\circ \), Lengths found as 20 m and 24.25 m.
Quick Tip: If you find that the Height = Base in a right-angled triangle, the angle of elevation is always \( 45^\circ \).