Step 1: Understanding the Concept:

To carve out a cone of maximum volume from a hemisphere, the base of the cone must coincide with the base of the hemisphere, and the vertex of the cone must be at the center of the hemisphere's base or the top of the dome. In this case, the radius of the cone (\(r\)) will be equal to the radius of the hemisphere (\(R\)), and the height of the cone (\(h\)) will also be equal to the radius of the hemisphere (\(R\)).


Step 2: Key Formula or Approach:

The curved surface area (CSA) of a cone is given by:
\[ CSA = \pi r l \]

where \(l\) is the slant height, calculated as \(l = \sqrt{r^2 + h^2}\).


Step 3: Detailed Explanation:

Given: Radius of hemisphere \(R = 10\) cm.

For maximum volume, for the cone:

Radius \(r = R = 10\) cm

Height \(h = R = 10\) cm

First, we find the slant height (\(l\)):
\[ l = \sqrt{10^2 + 10^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} cm \]

Now, calculate the Curved Surface Area:
\[ CSA = \pi \times 10 \times 10\sqrt{2} \]
\[ CSA = 3.14 \times 100\sqrt{2} \]
\[ CSA = 314\sqrt{2} cm^2 \]


Step 4: Final Answer:

The curved surface area of the cavity is \(314 \sqrt{2}\) \(cm^{2}\).
Quick Tip: For any cone inscribed in a hemisphere to have maximum volume, its height must be equal to its base radius, which are both equal to the radius of the hemisphere. The slant height will always be \(r\sqrt{2}\).

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. This difference is called the common difference (\(d\)). Any term \(a_n\) can be written as \(a + (n-1)d\).


Step 2: Key Formula or Approach:

The difference between the \(p^{th}\) and \(q^{th}\) terms of an A.P. is:
\[ a_p - a_q = (p - q)d \]


Step 3: Detailed Explanation:

The given A.P. is \(-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots\)

First term (\(a\)) = \(-\frac{15}{4}\)

Common difference (\(d\)) = \((-\frac{10}{4}) - (-\frac{15}{4}) = \frac{5}{4}\)

We need to find \(a_{16} - a_{12}\).

Using the formula:
\[ a_{16} - a_{12} = (16 - 12)d \]
\[ a_{16} - a_{12} = 4d \]

Substitute the value of \(d\):
\[ a_{16} - a_{12} = 4 \times \left(\frac{5}{4}\right) = 5 \]


Step 4: Final Answer:

The value of \(a_{16} - a_{12}\) is \(5\).
Quick Tip: Avoid calculating individual terms like \(a_{16}\) and \(a_{12}\) separately. Directly use the property \(a_n - a_m = (n-m)d\) to save time in MCQs.

Step 1: Understanding the Concept:

Probability of an event occurring is the ratio of the number of favorable outcomes to the total number of possible outcomes.


Step 2: Key Formula or Approach:
\[ P(Winning) = \frac{Number of tickets bought}{Total number of tickets sold} \]


Step 3: Detailed Explanation:

Given:

Probability of winning \(P(E) = 0.08\)

Total tickets sold = \(800\)

Let the number of tickets bought by Meena be \(x\).
\[ 0.08 = \frac{x}{800} \]

Multiply both sides by \(800\):
\[ x = 0.08 \times 800 \]
\[ x = \frac{8}{100} \times 800 \]
\[ x = 8 \times 8 = 64 \]


Step 4: Final Answer:

The number of tickets bought by her is \(64\).
Quick Tip: Think of probability as a percentage. \(0.08\) is simply \(8%\). Finding \(8%\) of \(800\) gives \(64\) instantly.

Step 1: Understanding the Concept:

The outer surface area of a hemispherical tent is its Curved Surface Area (CSA). However, since there is a door opening, that specific area must be subtracted from the total CSA to find the actual outer surface area of the material used.


Step 2: Key Formula or Approach:
\[ Net Outer Surface Area = (CSA of Hemisphere) - (Area of Door) \]
\[ CSA of Hemisphere = 2 \pi r^2 \]


Step 3: Detailed Explanation:

Given:

Radius \(r = 1.4\) m

Area of door = \(0.50\) \(m^2\)

Using \(\pi = \frac{22}{7}\):
\[ CSA = 2 \times \frac{22}{7} \times (1.4) \times (1.4) \]
\[ CSA = 44 \times 0.2 \times 1.4 \]
\[ CSA = 8.8 \times 1.4 = 12.32 m^2 \]

Now, Net Surface Area:
\[ Area = 12.32 - 0.50 = 11.82 m^2 \]


Step 4: Final Answer:

The outer surface area of the tent is \(11.82\) \(m^2\).
Quick Tip: Remember to use \(\pi = 22/7\) when radius is a multiple of \(7\) (like \(1.4\)) as it simplifies the calculations significantly.

Step 1: Understanding the Concept:

A tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the triangle formed by the center, the point of contact, and an external point is a right-angled triangle.


Step 2: Key Formula or Approach:

In \(\triangle OQP\), since \(PQ\) is tangent at \(Q\), \(\angle OQP = 90^{\circ}\).

By Pythagoras Theorem:
\[ OQ^2 + PQ^2 = OP^2 \]


Step 3: Detailed Explanation:

Given:
\(OQ = a\)
\(PQ = 2b\)
\(OP = a + 2\)

Applying Pythagoras Theorem:
\[ a^2 + (2b)^2 = (a + 2)^2 \]
\[ a^2 + 4b^2 = a^2 + 4a + 4 \]

Subtracting \(a^2\) from both sides:
\[ 4b^2 = 4a + 4 \]

Dividing the entire equation by \(4\):
\[ b^2 = a + 1 \]


Step 4: Final Answer:

The relation between \(a\) and \(b\) is \(b^2 = a + 1\).
Quick Tip: Always draw the radius to the point of contact of a tangent to identify the right-angled triangle and apply the Pythagorean property.

Step 1: Understanding the Concept:

This question involves using fundamental trigonometric identities to simplify an expression.


Step 2: Key Formula or Approach:

Recall the identity: \(1 + \tan^2 A = \sec^2 A \implies \sec^2 A - 1 = \tan^2 A\).


Step 3: Detailed Explanation:

The given expression is:
\[ \frac{\sec A}{\sqrt{\sec^2 A - 1}} \]

Substitute \(\sec^2 A - 1 = \tan^2 A\):
\[ = \frac{\sec A}{\sqrt{\tan^2 A}} \]
\[ = \frac{\sec A}{\tan A} \]

Expressing in terms of \(\sin A\) and \(\cos A\):
\[ = \frac{\frac{1}{\cos A}}{\frac{\sin A}{\cos A}} \]
\[ = \frac{1}{\cos A} \times \frac{\cos A}{\sin A} \]
\[ = \frac{1}{\sin A} = \csc A \]


Step 4: Final Answer:

The simplest form is \(\csc A\).
Quick Tip: Whenever you see \(\sqrt{\sec^2 A - 1}\) or \(\sqrt{1 - \sin^2 A}\), immediately replace them using basic identities (\(\tan A\) and \(\cos A\) respectively) to simplify.

Step 1: Understanding the Concept:

When a line segment is divided by the y-axis, the x-coordinate of the point of intersection is always \(0\). We use the section formula to find the ratio.


Step 2: Key Formula or Approach:

Let the ratio be \(k:1\). The x-coordinate of the point dividing the line joining \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ x = \frac{k x_2 + x_1}{k + 1} \]


Step 3: Detailed Explanation:

Points are \(P(-4, -2)\) and \(Q(10, 4)\).

Since the division is by the y-axis, the x-coordinate of the point is \(0\).

Let ratio be \(k:1\).
\[ 0 = \frac{k(10) + 1(-4)}{k + 1} \]
\[ 0 = 10k - 4 \]
\[ 10k = 4 \]
\[ k = \frac{4}{10} = \frac{2}{5} \]

The ratio \(k:1\) becomes \(\frac{2}{5}:1\), which is \(2:5\).


Step 4: Final Answer:

The ratio is \(2:5\).
Quick Tip: Shortcut: If a line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is divided by the y-axis, the ratio is simply \(-x_1 : x_2\). Here, \(-(-4) : 10 = 4 : 10 = 2 : 5\).

Step 1: Understanding the Concept:

The situation forms a right-angled triangle \(\triangle ABC\), where \(BC\) is the pole, \(AB\) is the distance on the ground, and \(AC\) is the wire (hypotenuse).


Step 2: Key Formula or Approach:

Using the cosine ratio in \(\triangle ABC\):
\[ \cos \theta = \frac{Adjacent}{Hypotenuse} = \frac{AB}{AC} \]


Step 3: Detailed Explanation:

Given:

Distance \(AB = 5\sqrt{3}\) m

Angle \(\angle A = 60^{\circ}\)

Let the length of the wire be \(AC\).
\[ \cos 60^{\circ} = \frac{AB}{AC} \]
\[ \frac{1}{2} = \frac{5\sqrt{3}}{AC} \]
\[ AC = 2 \times 5\sqrt{3} = 10\sqrt{3} m \]


Step 4: Final Answer:

The length of the wire is \(10\sqrt{3}\) m.
Quick Tip: In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the hypotenuse is always twice the length of the side adjacent to the \(60^{\circ}\) angle.

Step 1: Understanding the Concept:

Since \(AB \parallel EF\), the triangles \(\triangle DAB\) and \(\triangle DEF\) are similar by AA similarity criterion (\(\angle D\) is common and corresponding angles are equal).


Step 2: Key Formula or Approach:

For similar triangles, the ratio of corresponding sides is equal:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]


Step 3: Detailed Explanation:

Given: \(AB = 24\), \(EF = 36\), \(DA = 7\).

Let \(AE = x\). Then \(DE = DA + AE = 7 + x\).
\[ \frac{7}{7 + x} = \frac{24}{36} \]

Simplify the fraction:
\[ \frac{7}{7 + x} = \frac{2}{3} \]

Cross-multiply:
\[ 21 = 2(7 + x) \]
\[ 21 = 14 + 2x \]
\[ 2x = 21 - 14 = 7 \]
\[ x = \frac{7}{2} = 3.5 cm \]


Step 4: Final Answer:

The length of \(AE\) is \(3.5\) cm.
Quick Tip: Similar triangles often appear in "ladder" or "parallel line" diagrams. Identify the common vertex (D) to set up the correct side ratio.

Step 1: Understanding the Concept:

SAS (Side-Angle-Side) similarity criterion states that two triangles are similar if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal.


Step 2: Detailed Explanation:

In \(\triangle ABC\), the angle is \(\angle C\). The sides forming this angle are \(AC\) and \(BC\).

In \(\triangle PQR\), the corresponding angle is \(\angle R\). The sides forming this angle are \(PR\) and \(QR\).

For SAS similarity to hold with \(\angle C = \angle R\):
\[ \frac{AC}{PR} = \frac{BC}{QR} \]

Rearranging the terms (alternando):
\[ \frac{AC}{BC} = \frac{PR}{QR} \]

This matches option (D).


Step 3: Final Answer:

The required condition is \(\frac{AC}{BC} = \frac{PR}{QR}\).
Quick Tip: Remember that for SAS, the sides must be the ones that actually form the angle. Just looking at the letters: \(\angle C\) involves sides with \(C\) (\(AC, BC\)); \(\angle R\) involves sides with \(R\) (\(PR, QR\)).

Step 1: Understanding the Concept:

The step deviation method simplifies the calculation of the mean for grouped data. The mean \(\bar{x}\) is related to the mean of deviations \(\bar{u}\).


Step 2: Key Formula or Approach:

The formula for the mean in the step-deviation method is:
\[ \bar{x} = a + h \bar{u} \]


Step 3: Detailed Explanation:

Given:
\(\bar{x} = 64\)
\(a = 62.5\)
\(h = 5\)

Substitute these into the formula:
\[ 64 = 62.5 + 5\bar{u} \]
\[ 64 - 62.5 = 5\bar{u} \]
\[ 1.5 = 5\bar{u} \]
\[ \bar{u} = \frac{1.5}{5} = 0.3 \]


Step 4: Final Answer:

The value of \(\bar{u}\) is \(0.3\).
Quick Tip: Always ensure that the units of \((x-a)\) match \(h \bar{u}\). Here, the difference \(1.5\) is less than \(h=5\), so \(\bar{u}\) must be a decimal less than \(1\).

Step 1: Understanding the Concept:
\(\csc \theta\) is the reciprocal of \(\sin \theta\). If \(\sin \theta = \frac{1}{x}\), then \(\csc \theta = x\).


Step 2: Detailed Explanation:

Given: \(\sin \theta = \frac{1}{9}\)

Therefore, \(\csc \theta = \frac{1}{\sin \theta} = 9\).

Now, evaluate the given expression:
\[ \frac{9 \csc \theta + 1}{9 \csc \theta - 1} \]

Substitute \(\csc \theta = 9\):
\[ = \frac{9(9) + 1}{9(9) - 1} \]
\[ = \frac{81 + 1}{81 - 1} \]
\[ = \frac{82}{80} \]


Step 3: Final Answer:

The value is \(\frac{82}{80}\).
Quick Tip: Direct substitution is fastest here. Don't waste time trying to find \(\cos \theta\) or other ratios.

Step 1: Understanding the Concept:

The probability \(P(E)\) of any event \(E\) must satisfy the condition: \(0 \le P(E) \le 1\). It can never be negative and never be greater than \(1\).


Step 2: Detailed Explanation:

(A) \(\frac{39}{100} = 0.39\) (In range \([0, 1]\))

(B) \(\frac{0.001}{20} = 0.00005\) (In range \([0, 1]\))

(C) \(\frac{10}{0.2} = \frac{100}{2} = 50\) (Greater than \(1\))

(D) \(10% = 0.10\) (In range \([0, 1]\))

Since \(50 > 1\), it cannot be a probability.


Step 3: Final Answer:

Option (C) \(\frac{10}{0.2}\) cannot be a probability.
Quick Tip: In probability questions, if you see a numerator larger than the denominator (after simplifying), that value is always invalid.

Step 1: Understanding the Concept:

A quadratic equation \(ax^2 + bx + c = 0\) has real and equal roots if and only if its discriminant (\(D\)) is zero.


Step 2: Key Formula or Approach:

The discriminant is given by:
\[ D = b^2 - 4ac = 0 \]


Step 3: Detailed Explanation:

For the equation \(3x^2 - 7x + m = 0\):
\(a = 3, b = -7, c = m\)

Set \(D = 0\):
\[ (-7)^2 - 4(3)(m) = 0 \]
\[ 49 - 12m = 0 \]
\[ 12m = 49 \]
\[ m = \frac{49}{12} \]


Step 4: Final Answer:

The value of \(m\) is \(\frac{49}{12}\).
Quick Tip: "Real and equal roots" \(\rightarrow D=0\).
"Real roots" \(\rightarrow D \ge 0\).
Make sure you distinguish between these two phrasing in exams.

Step 1: Understanding the Concept:

A polynomial with zeroes \(\alpha\) and \(\beta\) can be written in factored form as \(p(x) = k(x - \alpha)(x - \beta)\), where \(k\) is a non-zero constant.


Step 2: Detailed Explanation:

Given zeroes: \(\alpha = -3\) and \(\beta = 8\).

The factored form is:
\[ p(x) = k(x - (-3))(x - 8) \]
\[ p(x) = k(x + 3)(x - 8) \]

Let's check the options:

(A) \(x^2 + 5x - 4\): Sum of roots is \(-5\). Incorrect.

(B) \((x + 3)(-x + 8)\): Here, if we set the expression to \(0\), we get \(x + 3 = 0 \implies x = -3\) and \(-x + 8 = 0 \implies x = 8\). This matches the given zeroes perfectly. Note that this is just the form with \(k = -1\).

(C) \(a(x^2 + 5x - 24)\): For this, sum of roots is \(-5\). Incorrect.

(D) \(x^2 - 24\): Roots are \(\pm \sqrt{24}\). Incorrect.


Step 3: Final Answer:

The polynomial is \((x + 3)(-x + 8)\).
Quick Tip: Check roots by substitution! Plug \(x = -3\) and \(x = 8\) into the options. The one that results in zero for both values is the correct polynomial.

Step 1: Understanding the Concept:

For a quadratic equation \(ax^2 + bx + c = 0\) with rational coefficients, the roots are rational if and only if the discriminant \(D = b^2 - 4ac\) is a perfect square of a rational number.


Step 2: Key Formula or Approach:

Discriminant formula: \[ D = b^2 - 4ac \]

Given equation: \(x^2 - px + 6 = 0\)

Here, \(a = 1, b = -p, c = 6\).


Step 3: Detailed Explanation:

Substitute the values into the discriminant formula:
\[ D = (-p)^2 - 4(1)(6) \]
\[ D = p^2 - 24 \]

We test the given options for \(p\) to see which makes \(D\) a perfect square:

(A) If \(p = 1\): \(D = 1^2 - 24 = -23\) (Not a perfect square; roots are non-real).

(B) If \(p = -5\): \(D = (-5)^2 - 24 = 25 - 24 = 1\). Since \(1\) is a perfect square (\(1^2\)), the roots are rational.

(C) If \(p = 25\): \(D = 25^2 - 24 = 625 - 24 = 601\) (Not a perfect square).

(D) If \(p = \sqrt{5}\): \(D = (\sqrt{5})^2 - 24 = 5 - 24 = -19\) (Not a perfect square; roots are non-real).


Step 4: Final Answer:

The value of \(p\) is \(-5\).
Quick Tip: Roots are rational only if \(D \ge 0\) and \(D\) is a perfect square. If \(D\) is not a perfect square (like \(2\), \(3\), etc.), roots are irrational. If \(D < 0\), roots are non-real.

Step 1: Understanding the Concept:

The length of an arc of a circle is proportional to the angle it subtends at the center. The total circumference subtends \(360^{\circ}\).


Step 2: Key Formula or Approach:

Length of arc (\(l\)) is given by:
\[ l = \frac{\theta}{360^{\circ}} \times 2\pi r \]


Step 3: Detailed Explanation:

Given:

Arc length \(l = 2.2\) cm

Radius \(r = 2.8\) cm

Using \(\pi = \frac{22}{7}\):
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 2 \times \frac{22}{7} \times 2.8 \]
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 44 \times 0.4 \]
\[ 2.2 = \frac{\theta}{360^{\circ}} \times 17.6 \]

Rearranging to solve for \(\theta\):
\[ \theta = \frac{2.2 \times 360^{\circ}}{17.6} \]
\[ \theta = \frac{360^{\circ}}{8} = 45^{\circ} \]


Step 4: Final Answer:

The value of \(\theta\) is \(45^{\circ}\).
Quick Tip: Notice that \(17.6\) is exactly \(8\) times \(2.2\). Identifying such ratios quickly makes simplifying fractions much easier in competitive exams.

Step 1: Understanding the Concept:

When two dice are rolled, there are \(6 \times 6 = 36\) total possible outcomes. For any pair \((x, y)\), there are three possibilities: \(x = y\), \(x > y\), or \(x < y\).


Step 2: Detailed Explanation:

1. Case \(x = y\): The outcomes are \((1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\). Total = \(6\).

2. Total remaining outcomes where \(x \neq y\) is \(36 - 6 = 30\).

3. By symmetry, the number of outcomes where \(x > y\) must be equal to the number of outcomes where \(y > x\).

Number of outcomes where \(x > y = \frac{30}{2} = 15\).

4. Probability Calculation:
\[ P(x > y) = \frac{Number of favorable outcomes}{Total outcomes} \]
\[ P(x > y) = \frac{15}{36} \]

Dividing by \(3\):
\[ P(x > y) = \frac{5}{12} \]


Step 3: Final Answer:

The probability is \(\frac{5}{12}\).
Quick Tip: Instead of listing all outcomes, remember: for two dice, \(P(x > y) = P(x < y) = \frac{36 - 6}{2 \times 36} = \frac{15}{36}\).

Step 1: Understanding the Concept:

Highest Common Factor (H.C.F.) is the largest positive integer that divides each of the integers. For monomials, H.C.F. is the product of the H.C.F. of coefficients and the lowest power of each common variable.


Step 2: Detailed Explanation:

Evaluating Assertion (A):

The expressions are \(36m^2\) and \(18m\).

Since \(m\) is a prime number, \(m \ge 2\). Thus \(36m^2\) is a multiple of \(18m\).
\[ 36m^2 = 18m \times 2m \]

If one number is a factor of another, the smaller number is the H.C.F.

So, \(H.C.F.(36m^2, 18m) = 18m\).

Assertion (A) is True.


Evaluating Reason (R):

By definition, H.C.F. of two positive integers \(a\) and \(b\) is always \(\le a\) and \(\le b\). Therefore, it is always less than or equal to the smaller of the two numbers.

Reason (R) is True.


Relationship Analysis:

While both statements are true, Reason (R) is a general property of H.C.F. and does not specifically explain why the H.C.F. of \(36m^2\) and \(18m\) is exactly \(18m\). The correct explanation for (A) would be that \(18m\) is a factor of \(36m^2\).


Step 3: Final Answer:

Both (A) and (R) are true but (R) is not the correct explanation.
Quick Tip: For any two numbers \(x\) and \(y\), if \(x\) divides \(y\), then \(HCF(x, y) = x\) and \(LCM(x, y) = y\). This is the specific logic often tested in such questions.

Step 1: Understanding the Concept:

A system of linear equations is inconsistent if it has no solution. This occurs when the lines are parallel. For equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the condition for no solution is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]


Step 2: Detailed Explanation:

Evaluating Assertion (A):

Eq 1: \(3x - 5y + 7 = 0 \implies a_1=3, b_1=-5, c_1=7\)

Eq 2: \(-6x + 10y + 14 = 0 \implies a_2=-6, b_2=10, c_2=14\)

Check the ratios:
\[ \frac{a_1}{a_2} = \frac{3}{-6} = -\frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2} \]
\[ \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \]

Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) (\(-\frac{1}{2} = -\frac{1}{2} \neq \frac{1}{2}\)), the system has no solution.

Thus, the system is inconsistent. Assertion (A) is True.


Evaluating Reason (R):

A system "doesn't have a unique solution" when it has either no solution (parallel lines) or infinitely many solutions (coincident lines). Therefore, it is incorrect to say it \textit{always represents parallel lines.

Reason (R) is False.


Step 3: Final Answer:

Assertion (A) is true and Reason (R) is false.
Quick Tip: Be careful with absolute words like "always" in Reason statements. If there is even one exception (like coincident lines having infinite solutions), the "always" makes the statement false.

Step 1: Understanding the Concept:

To find the length of the segment \(AD\), we first need to find the coordinates of point \(D\) using the section formula.

Once the coordinates of \(D\) are known, we use the distance formula between points \(A\) and \(D\).


Step 2: Key Formula or Approach:

Section Formula for a point dividing a line in ratio \(m:n\):
\[ D(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Distance Formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]


Step 3: Detailed Explanation:

Given coordinates: \(B(-2, 1)\), \(C(4, 2)\), and ratio \(m:n = 1:2\).

Let \(D\) have coordinates \((x, y)\).
\[ x = \frac{1(4) + 2(-2)}{1+2} = \frac{4 - 4}{3} = 0 \]
\[ y = \frac{1(2) + 2(1)}{1+2} = \frac{2 + 2}{3} = \frac{4}{3} \]

So, coordinates of \(D\) are \((0, \frac{4}{3})\).

Now, we find the length \(AD\) where \(A\) is \((1, 5)\):
\[ AD = \sqrt{(0 - 1)^2 + \left( \frac{4}{3} - 5 \right)^2} \]
\[ AD = \sqrt{(-1)^2 + \left( \frac{4 - 15}{3} \right)^2} \]
\[ AD = \sqrt{1 + \left( \frac{-11}{3} \right)^2} \]
\[ AD = \sqrt{1 + \frac{121}{9}} \]
\[ AD = \sqrt{\frac{9 + 121}{9}} = \sqrt{\frac{130}{9}} \]
\[ AD = \frac{\sqrt{130}}{3} units \]


Step 4: Final Answer:

The length of \(AD\) is \(\frac{\sqrt{130}}{3}\) units.
Quick Tip: Always double-check the order of \(m\) and \(n\) in the section formula relative to points \(B\) and \(C\). Since the ratio is \(1:2\) from \(B\) to \(C\), \(m=1\) multiplies the coordinates of \(C\).

Step 1: Understanding the Concept:

This problem requires substituting the standard trigonometric values for \(60^{\circ}\), \(30^{\circ}\), and \(45^{\circ}\) and simplifying the resulting fraction.


Step 2: Key Formula or Approach:

Standard values:
\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)
\(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\)
\(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\)


Step 3: Detailed Explanation:

Numerator:
\[ \sin^3 60^{\circ} - \tan 30^{\circ} = \left(\frac{\sqrt{3}}{2}\right)^3 - \frac{1}{\sqrt{3}} \]
\[ = \frac{3\sqrt{3}}{8} - \frac{1}{\sqrt{3}} \]

Take the L.C.M. of the denominators (\(8\) and \(\sqrt{3}\)):
\[ = \frac{3\sqrt{3}(\sqrt{3}) - 1(8)}{8\sqrt{3}} = \frac{3(3) - 8}{8\sqrt{3}} = \frac{9 - 8}{8\sqrt{3}} = \frac{1}{8\sqrt{3}} \]

Denominator:
\[ \cos^2 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

Now, evaluate the full expression:
\[ Value = \frac{\frac{1}{8\sqrt{3}}}{\frac{1}{2}} = \frac{1}{8\sqrt{3}} \times 2 = \frac{1}{4\sqrt{3}} \]

Rationalizing the denominator:
\[ = \frac{1}{4\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{4 \times 3} = \frac{\sqrt{3}}{12} \]


Step 4: Final Answer:

The evaluated value is \(\frac{\sqrt{3}}{12}\).
Quick Tip: Remember that \(\sin^3 \theta\) means \((\sin \theta)^3\). When simplifying fractions with surds, it is often helpful to rationalize the final answer.

Step 1: Understanding the Concept:

We use the inverse values of trigonometric functions for standard angles to form a system of two linear equations in terms of \(A\) and \(B\).


Step 2: Detailed Explanation:

From \(\tan (A + 2B) = \sqrt{3}\):

Since \(\tan 60^{\circ} = \sqrt{3}\), we have:
\[ A + 2B = 60^{\circ} \quad \dots(1) \]

From \(\sin (2A + B) = \frac{1}{\sqrt{2}}\):

Since \(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\), we have:
\[ 2A + B = 45^{\circ} \quad \dots(2) \]

Solving the equations:

Multiply equation (2) by 2:
\[ 4A + 2B = 90^{\circ} \quad \dots(3) \]

Subtract equation (1) from (3):
\[ (4A + 2B) - (A + 2B) = 90^{\circ} - 60^{\circ} \]
\[ 3A = 30^{\circ} \implies A = 10^{\circ} \]

Substitute \(A = 10^{\circ}\) in equation (2):
\[ 2(10^{\circ}) + B = 45^{\circ} \]
\[ 20^{\circ} + B = 45^{\circ} \implies B = 25^{\circ} \]


Step 3: Final Answer:

The measures of the angles are \(A = 10^{\circ}\) and \(B = 25^{\circ}\).
Quick Tip: Always check if the calculated angles satisfy the "acute" condition mentioned in the question. Here, \(A+2B = 10+50 = 60^{\circ}\) and \(2A+B = 20+25 = 45^{\circ}\), both are acute.

Step 1: Understanding the Concept:

Probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes.


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


Step 3: Detailed Explanation:

Total number of balls in the bag = 25.

Let the number of green balls be \(G\).

Given, \(P(Green ball) = \frac{3}{5}\).
\[ \frac{G}{25} = \frac{3}{5} \]

Cross-multiplying to find \(G\):
\[ G = \frac{3}{5} \times 25 = 3 \times 5 = 15 \]

The number of green balls is 15.

Since the rest are yellow balls, the number of yellow balls is:
\[ Number of yellow balls = Total balls - Green balls \]
\[ Number of yellow balls = 25 - 15 = 10 \]


Step 4: Final Answer:

The number of yellow balls is 10.
Quick Tip: You can also find the probability of getting a yellow ball first: \(P(Yellow) = 1 - P(Green) = 1 - 3/5 = 2/5\). Then, number of yellow balls = \(2/5 \times 25 = 10\).

Step 1: Understanding the Concept:

Parallel lines create equal corresponding angles when intersected by a transversal. This allows us to prove triangle similarity via the AA (Angle-Angle) criterion.


Step 2: Detailed Explanation:

Part 1: Showing Similarity

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

1. Since \(AB \parallel DE\) and \(EF\) is a transversal, \(\angle ABC = \angle DEF\) (Corresponding angles).

2. Since \(AC \parallel DF\) and \(EF\) is a transversal, \(\angle ACB = \angle DFE\) (Corresponding angles).

Therefore, \(\triangle ABC \sim \triangle DEF\) by the AA Similarity Criterion.


Part 2: Finding Length DE

Since the triangles are similar, their corresponding sides are proportional:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]

Given: \(BC = 10\) cm, \(EB = 5\) cm, \(CF = 5\) cm.

From the figure, \(EF = EB + BC + CF\).
\[ EF = 5 + 10 + 5 = 20 cm \]

Now, substitute the values into the proportionality equation:
\[ \frac{7}{DE} = \frac{10}{20} \]
\[ \frac{7}{DE} = \frac{1}{2} \]
\[ DE = 7 \times 2 = 14 cm \]


Step 3: Final Answer:

The length of \(DE\) is 14 cm.
Quick Tip: When proving similarity with parallel lines, look for the 'F' shape for corresponding angles. Always ensure you add up the segments correctly to find the full length of the side of the larger triangle.

Step 1: Understanding the Concept:

We use the method of contradiction. We assume the number is rational and show that this leads to a logical inconsistency with the given fact that \(\sqrt{3}\) is irrational.


Step 2: Detailed Explanation:

Let us assume that \(14 - 2\sqrt{3}\) is a rational number.

If it is rational, it can be represented as \(x\), where \(x\) is rational.
\[ 14 - 2\sqrt{3} = x \]

Rearrange the equation to isolate the irrational term:
\[ 14 - x = 2\sqrt{3} \]
\[ \frac{14 - x}{2} = \sqrt{3} \]

Now, evaluate both sides:

On the Left Hand Side (L.H.S.):

Since \(14\) and \(2\) are rational integers and we assumed \(x\) is rational, the difference and division of rational numbers is also rational.

Thus, \(\frac{14 - x}{2}\) is a rational number.

On the Right Hand Side (R.H.S.):

We are given that \(\sqrt{3}\) is an irrational number.

This leads to a contradiction: \(Rational = Irrational\).

Our initial assumption that \(14 - 2\sqrt{3}\) is rational is incorrect.

Therefore, \(14 - 2\sqrt{3}\) is an irrational number.


Step 3: Final Answer:

Hence, \(14 - 2\sqrt{3}\) is proved to be irrational.
Quick Tip: In these proofs, always isolate the surd (like \(\sqrt{3}\)) on one side. Remember the properties: Rational \(\pm\) Rational = Rational; Rational / Rational = Rational (if divisor \(\neq 0\)).

Step 1: Understanding the Concept:

Since the circle passes through points \(A\) and \(B\), the distance from the center \(O(2, 1)\) to point \(A\) and point \(B\) must be equal to the radius of the circle.

Thus, \(OA = OB\).


Step 2: Key Formula or Approach:

We use the distance formula between two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\):
\[ d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \]


Step 3: Detailed Explanation:

First, calculate the radius squared (\(OA^2\)):
\[ OA^{2} = (5 - 2)^{2} + (6 - 1)^{2} \]
\[ OA^{2} = (3)^{2} + (5)^{2} = 9 + 25 = 34 \]

Now, calculate \(OB^2\) and set it equal to 34:
\[ OB^{2} = (-3 - 2)^{2} + (K - 1)^{2} = 34 \]
\[ (-5)^{2} + (K - 1)^{2} = 34 \]
\[ 25 + (K - 1)^{2} = 34 \]
\[ (K - 1)^{2} = 9 \]

Taking the square root on both sides:
\[ K - 1 = \pm 3 \]

Case 1: \(K - 1 = 3 \implies K = 4\)

Case 2: \(K - 1 = -3 \implies K = -2\)

Calculating length of chord AB:

If \(K = 4\), points are \(A(5, 6)\) and \(B(-3, 4)\):
\[ AB = \sqrt{(-3 - 5)^{2} + (4 - 6)^{2}} = \sqrt{(-8)^{2} + (-2)^{2}} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} units \]

If \(K = -2\), points are \(A(5, 6)\) and \(B(-3, -2)\):
\[ AB = \sqrt{(-3 - 5)^{2} + (-2 - 6)^{2}} = \sqrt{(-8)^{2} + (-8)^{2}} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} units \]


Step 4: Final Answer:

The values of \(K\) are 4 and -2.

The corresponding lengths of chord \(AB\) are \(2\sqrt{17}\) units and \(8\sqrt{2}\) units.
Quick Tip: For questions involving distances from the center, always work with \(d^2\) instead of \(d\) to avoid carrying square roots through your algebraic steps.

Step 1: Understanding the Concept:

We use the section formula to find the coordinates of point \(P\). Then, we substitute these coordinates into the given line equation to verify if it satisfies the equation. Finally, we use the distance formula for \(PA\) and \(PB\).


Step 2: Key Formula or Approach:

Section Formula: \(P(x, y) = \left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}\right)\)

Distance Formula: \(d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\)


Step 3: Detailed Explanation:

Find coordinates of \(P\) with \(m=3, n=2, A(-1, 7), B(4, -3)\):
\[ x = \frac{3(4) + 2(-1)}{3 + 2} = \frac{12 - 2}{5} = \frac{10}{5} = 2 \]
\[ y = \frac{3(-3) + 2(7)}{3 + 2} = \frac{-9 + 14}{5} = \frac{5}{5} = 1 \]

So, \(P\) is \((2, 1)\).

Verification: Substitute \(P(2, 1)\) into \(x - 3y = -1\):

L.H.S. \(= 2 - 3(1) = 2 - 3 = -1\).

Since L.H.S. \(=\) R.H.S., point \(P\) lies on the line.

Calculating lengths:
\[ PA = \sqrt{(2 - (-1))^{2} + (1 - 7)^{2}} = \sqrt{3^{2} + (-6)^{2}} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} units \]
\[ PB = \sqrt{(4 - 2)^{2} + (-3 - 1)^{2}} = \sqrt{2^{2} + (-4)^{2}} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} units \]


Step 4: Final Answer:

Point \(P(2, 1)\) satisfies the equation \(x - 3y = -1\), proving it lies on the line.

The lengths are \(PA = 3\sqrt{5}\) units and \(PB = 2\sqrt{5}\) units.
Quick Tip: If you find the ratio of lengths \(PA/PB\), it must equal the given section ratio (3:2). In this case, \(3\sqrt{5} / 2\sqrt{5} = 3/2\). This is a great way to verify your answer!

Step 1: Understanding the Concept:

To solve a system of equations graphically, we plot both lines on the Cartesian plane. The point where the two lines intersect is the solution to the system.


Step 2: Detailed Explanation:

Plotting \(x = -3\):

This is a vertical line passing through the point \((-3, 0)\) on the x-axis.

Plotting \(5x - 2y = -5\):

Find at least two points for this line:

1. Let \(x = -1\): \(5(-1) - 2y = -5 \implies -5 - 2y = -5 \implies -2y = 0 \implies y = 0\). Point is \((-1, 0)\).

2. Let \(x = 1\): \(5(1) - 2y = -5 \implies 5 - 2y = -5 \implies -2y = -10 \implies y = 5\). Point is \((1, 5)\).

3. To find intersection with \(x = -3\), substitute \(x = -3\) into the second equation:
\(5(-3) - 2y = -5\)
\(-15 - 2y = -5\)
\(-2y = -5 + 15\)
\(-2y = 10 \implies y = -5\)

The line passes through \((-3, -5)\).

Graphing:

Draw the axes and plot the vertical line \(x = -3\).

Plot the points \((-1, 0), (1, 5)\), and \((-3, -5)\) and draw the line for \(5x - 2y = -5\).

The intersection point is clearly seen at \((-3, -5)\).


Step 3: Final Answer:

The solution to the system is \(x = -3\) and \(y = -5\).
Quick Tip: When one equation is of the form \(x = c\), you already know the x-coordinate of the solution. Simply find the corresponding y-value in the second equation to verify your graph.

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is defined by its first term (\(a\)) and common difference (\(d\)). We use the general term and sum formulas to establish equations.


Step 2: Key Formula or Approach:
\(n^{th}\) term: \(a_{n} = a + (n - 1)d\)

Sum of \(n\) terms: \(S_{n} = \frac{n}{2}[2a + (n - 1)d]\)


Step 3: Detailed Explanation:

Given: \(a_{15} - a_{8} = 21\)
\[ (a + 14d) - (a + 7d) = 21 \]
\[ 7d = 21 \implies d = 3 \]

Given: \(S_{10} = 55\)
\[ \frac{10}{2}[2a + (10 - 1)d] = 55 \]
\[ 5[2a + 9(3)] = 55 \]
\[ 2a + 27 = 11 \]
\[ 2a = 11 - 27 = -16 \]
\[ a = -8 \]

The A.P. is:

Term 1: \(a = -8\)

Term 2: \(a + d = -8 + 3 = -5\)

Term 3: \(a + 2d = -8 + 6 = -2\)

Term 4: \(a + 3d = -8 + 9 = 1\)


Step 4: Final Answer:

The A.P. is \(-8, -5, -2, 1, \dots\)
Quick Tip: The difference between any two terms \(a_{p}\) and \(a_{q}\) is simply \((p - q)d\). Here, \(a_{15} - a_{8} = 7d\), which allows you to find \(d\) instantly.

Step 1: Understanding the Concept:

The \(n^{th}\) term of a sequence can be found from the sum of terms using the relation \(a_{n} = S_{n} - S_{n-1}\).


Step 2: Detailed Explanation:

Given: \(S_{n} = 2n^{2} + 13n\)

Find \(S_{n-1}\):
\[ S_{n-1} = 2(n - 1)^{2} + 13(n - 1) \]
\[ S_{n-1} = 2(n^{2} - 2n + 1) + 13n - 13 \]
\[ S_{n-1} = 2n^{2} - 4n + 2 + 13n - 13 = 2n^{2} + 9n - 11 \]

Calculate \(a_{n}\):
\[ a_{n} = S_{n} - S_{n-1} \]
\[ a_{n} = (2n^{2} + 13n) - (2n^{2} + 9n - 11) \]
\[ a_{n} = 2n^{2} + 13n - 2n^{2} - 9n + 11 = 4n + 11 \]

Find \(10^{th}\) term (\(a_{10}\)):
\[ a_{10} = 4(10) + 11 = 40 + 11 = 51 \]


Step 3: Final Answer:

The \(n^{th}\) term is \(4n + 11\) and the \(10^{th}\) term is 51.
Quick Tip: Shortcut for \(S_{n} = An^2 + Bn\):
Common difference \(d = 2A\) and first term \(a = A + B\).
Here \(d = 2(2) = 4\) and \(a = 2 + 13 = 15\).
\(a_{n} = a + (n-1)d = 15 + (n-1)4 = 4n + 11\).

Step 1: Understanding the Concept:

To create complete squares of "maximum size" that fit exactly into the rectangular dimensions, the side of the square must be the Highest Common Factor (H.C.F.) of the length and width of the window.


Step 2: Detailed Explanation:

We need to find H.C.F.(156, 216).

Prime factorization of 156:
\[ 156 = 2 \times 78 = 2^{2} \times 39 = 2^{2} \times 3 \times 13 \]

Prime factorization of 216:
\[ 216 = 2 \times 108 = 2^{2} \times 54 = 2^{3} \times 27 = 2^{3} \times 3^{3} \]

H.C.F. is the product of the lowest powers of common prime factors:
\[ H.C.F. = 2^{2} \times 3 = 4 \times 3 = 12 \]

Thus, the side length of the maximum square is 12 cm.

Finding number of squares:
\[ Number of squares = \frac{Area of window}{Area of one square} \]
\[ Number of squares = \frac{156 \times 216}{12 \times 12} \]
\[ Number of squares = \left(\frac{156}{12}\right) \times \left(\frac{216}{12}\right) = 13 \times 18 = 234 \]


Step 3: Final Answer:

The maximum side length of the square is 12 cm and the total number of squares formed is 234.
Quick Tip: Whenever a problem asks for "maximum size" or "largest possible" pieces to be cut or fitted without remainder, it is a hint to calculate the H.C.F.

Step 1: Understanding the Concept:

We convert all terms into \(\tan \theta\) to simplify the expression on the L.H.S. and then use algebraic identities to reach the R.H.S.


Step 2: Detailed Explanation:

L.H.S. \(= \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta}\)

Substitute \(\cot \theta = \frac{1}{\tan \theta}\):
\[ = \frac{\tan \theta}{1 - \frac{1}{\tan \theta}} + \frac{\frac{1}{\tan \theta}}{1 - \tan \theta} \]
\[ = \frac{\tan \theta}{\frac{\tan \theta - 1}{\tan \theta}} + \frac{1}{\tan \theta (1 - \tan \theta)} \]
\[ = \frac{\tan^{2} \theta}{\tan \theta - 1} - \frac{1}{\tan \theta (\tan \theta - 1)} \]

Take L.C.M. \(\tan \theta (\tan \theta - 1)\):
\[ = \frac{\tan^{3} \theta - 1}{\tan \theta (\tan \theta - 1)} \]

Using identity \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\):
\[ = \frac{(\tan \theta - 1)(\tan^{2} \theta + \tan \theta + 1)}{\tan \theta (\tan \theta - 1)} \]
\[ = \frac{\tan^{2} \theta + \tan \theta + 1}{\tan \theta} \]

Divide each term in the numerator by \(\tan \theta\):
\[ = \frac{\tan^{2} \theta}{\tan \theta} + \frac{\tan \theta}{\tan \theta} + \frac{1}{\tan \theta} \]
\[ = \tan \theta + 1 + \cot \theta \]

L.H.S. \(=\) R.H.S. Hence Proved.


Step 3: Final Answer:

The trigonometric identity is proved.
Quick Tip: When dealing with mixed \(\tan\) and \(\cot\) identities, converting everything to \(\tan \theta\) or \(\sin \theta, \cos \theta\) is usually the most effective strategy.

Step 1: Understanding the Concept:

A sector is a part of a circle bounded by two radii and an arc. A segment is bounded by a chord and an arc.


Step 2: Key Formula or Approach:

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

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

Perimeter of segment \(=\) Length of arc \(+\) Length of chord.


Step 3: Detailed Explanation:

Given: \(r = 14\) cm, \(\theta = 60^{\circ}\).

Area of Sector:
\[ Area = \frac{60}{360} \times \frac{22}{7} \times 14 \times 14 \]
\[ Area = \frac{1}{6} \times 22 \times 2 \times 14 = \frac{308}{3} \approx 102.67 cm^{2} \]

Perimeter of Segment:

1. Length of arc \(AB = \frac{60}{360} \times 2 \times \frac{22}{7} \times 14 = \frac{1}{6} \times 44 \times 2 = \frac{44}{3} \approx 14.67\) cm.

2. Length of chord \(AB\): Since \(\theta = 60^{\circ}\) and \(OA = OB = 14\), \(\triangle OAB\) is equilateral.

Thus, chord \(AB = 14\) cm.

Perimeter \(= Arc length + Chord length\)
\[ Perimeter = 14.67 + 14 = 28.67 cm \]


Step 4: Final Answer:

The area of the smaller sector is \(102.67 cm^{2}\) and the perimeter of the smaller segment is 28.67 cm.
Quick Tip: If the central angle is \(60^{\circ}\), the triangle formed by the radii and the chord is always equilateral. If the angle is \(90^{\circ}\), use Pythagoras theorem to find the chord length.

Step 1: Understanding the Concept:

A quadrilateral is a parallelogram if its diagonals bisect each other.

We will use the properties of midpoints and given congruences to show the diagonals \(BC\) and \(OG\) bisect each other.


Step 2: Detailed Explanation:

In the quadrilateral \(OBGC\):

1. It is given that \(D\) is the mid-point of side \(BC\).

This implies that \(BD = DC\).

2. It is also given that \(AD\) is produced to \(G\) such that \(OD = DG\).

This implies that \(D\) is the mid-point of segment \(OG\).

3. In quadrilateral \(OBGC\), the diagonals are \(BC\) and \(OG\).

Since both diagonals intersect at \(D\) and are bisected at \(D\) (\(BD = DC\) and \(OD = DG\)), the diagonals bisect each other.

4. By the property of quadrilaterals, if the diagonals bisect each other, the quadrilateral is a parallelogram.

Therefore, \(OBGC\) is a parallelogram.


Step 3: Final Answer:

Since the diagonals \(BC\) and \(OG\) bisect each other at point \(D\), quadrilateral \(OBGC\) is a parallelogram.
Quick Tip: To prove a quadrilateral is a parallelogram, diagonal bisection is often the fastest method when midpoints are given.

Step 1: Understanding the Concept:

To prove \(EF \parallel BC\), we can use the converse of Basic Proportionality Theorem (BPT) in \(\triangle ABC\).

We need to show that \(\frac{AE}{EB} = \frac{AF}{FC}\).


Step 2: Detailed Explanation:

1. From part (i), we know \(OBGC\) is a parallelogram.

Therefore, \(BG \parallel OC\) and \(GC \parallel OB\).

2. Since \(BG \parallel OC\) and \(O, E, C\) are collinear, we have \(BG \parallel OE\).

In \(\triangle ABG\), \(OE\) is a line segment parallel to the base \(BG\).

By Basic Proportionality Theorem (BPT):

\[ \frac{AE}{EB} = \frac{AO}{OG} \quad \dots(1) \]

3. Similarly, since \(GC \parallel OB\) and \(O, F, B\) are collinear, we have \(GC \parallel OF\).

In \(\triangle ACG\), \(OF\) is parallel to the base \(GC\).

By BPT:

\[ \frac{AF}{FC} = \frac{AO}{OG} \quad \dots(2) \]

4. From equations (1) and (2), we equate the ratios:

\[ \frac{AE}{EB} = \frac{AF}{FC} \]

5. In \(\triangle ABC\), since the line segment \(EF\) divides sides \(AB\) and \(AC\) in the same ratio, by the converse of BPT:

\[ EF \parallel BC \]


Step 3: Final Answer:

Using BPT in \(\triangle ABG\) and \(\triangle ACG\), we established \(\frac{AE}{EB} = \frac{AF}{FC}\), which proves \(EF \parallel BC\) by the converse of BPT.
Quick Tip: Whenever you need to prove lines are parallel in a triangle, look for a common ratio using BPT. Here, \(AO/OG\) acts as the bridge between the two sides.

Step 1: Understanding the Concept:

Two triangles are similar if their corresponding angles are equal (AA similarity criterion).


Step 2: Detailed Explanation:

1. In \(\triangle AEF\) and \(\triangle ABC\):

- \(\angle EAF = \angle BAC\) (Common angle for both triangles).

2. From part (ii), we proved that \(EF \parallel BC\).

3. When parallel lines are intersected by a transversal, corresponding angles are equal.

- \(\angle AEF = \angle ABC\) (Corresponding angles).

- \(\angle AFE = \angle ACB\) (Corresponding angles).

4. Since two corresponding angles are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion.

\[ \triangle AEF \sim \triangle ABC \]


Step 3: Final Answer:

The similarity is proven by the AA criterion because \(\angle A\) is common and \(\angle AEF = \angle ABC\) due to \(EF \parallel BC\).
Quick Tip: Parallel lines within a triangle always create a smaller triangle similar to the original one.

Step 1: Understanding the Concept:

To prove \(AQ = QR\), we show that \(\triangle ADQ\) and \(\triangle RCQ\) are congruent.


Step 3: Detailed Explanation:

In \(\triangle ADQ\) and \(\triangle RCQ\):

1. \(DQ = QC\) (Given that \(Q\) is the mid-point of \(CD\)).

2. \(\angle ADQ = \angle RCQ\) (Alternate interior angles, as \(AD \parallel BR\)).

3. \(\angle AQD = \angle RQC\) (Vertically opposite angles).

4. Therefore, \(\triangle ADQ \cong \triangle RCQ\) by ASA (Angle-Side-Angle) congruence rule.

5. By CPCT (Corresponding Parts of Congruent Triangles):

- \(AQ = QR\)

- \(AD = CR\)


Step 4: Final Answer:

By ASA congruence between \(\triangle ADQ\) and \(\triangle RCQ\), we find \(AQ = QR\).
Quick Tip: In parallelograms, extending a side and using a midpoint often creates congruent triangles.

Step 1: Understanding the Concept:

We will use the similarity of triangles \(\triangle APD\) and \(\triangle RPB\) and the midpoint property.


Step 3: Detailed Explanation:

1. In \(\triangle APD\) and \(\triangle RPB\):

- \(\angle PAD = \angle PRB\) (Alternate interior angles, \(AD \parallel BR\)).

- \(\angle ADP = \angle RBP\) (Alternate interior angles, \(AD \parallel BR\)).

- Thus, \(\triangle APD \sim \triangle RPB\) by AA similarity.

2. Therefore, the ratio of corresponding sides is equal:

\[ \frac{AP}{PR} = \frac{AD}{BR} \]

3. We know \(BR = BC + CR\).

In parallelogram \(ABCD\), \(AD = BC\).

From part (i), \(\triangle ADQ \cong \triangle RCQ \implies AD = CR\).

So, \(BR = AD + AD = 2AD\).

4. Substitute \(BR = 2AD\) into the ratio:

\[ \frac{AP}{PR} = \frac{AD}{2AD} = \frac{1}{2} \implies PR = 2AP \]

5. From part (i), \(Q\) is the midpoint of \(AR\), so \(AR = 2AQ\).

Also, \(AR = AP + PR = AP + 2AP = 3AP\).

So, \(3AP = 2AQ \implies AP = \frac{2}{3}AQ\).

6. Now, find \(PQ\):

\[ PQ = AQ - AP = AQ - \frac{2}{3}AQ = \frac{1}{3}AQ \]

7. Compare \(AP\) and \(PQ\):

\[ AP = \frac{2}{3}AQ = 2 \times \left( \frac{1}{3}AQ \right) = 2PQ \]


Step 4: Final Answer:

Since \(AP = \frac{2}{3}AQ\) and \(PQ = \frac{1}{3}AQ\), it follows that \(AP = 2PQ\).
Quick Tip: Express all segments in terms of one major segment (like \(AQ\)) to compare them easily.

Step 1: Understanding the Concept:

This follows directly from the similarity of triangles established in part (ii).


Step 3: Detailed Explanation:

1. As shown in part (ii), \(\triangle APD \sim \triangle RPB\).

2. The ratio of their sides is:

\[ \frac{AP}{PR} = \frac{AD}{BR} \]

3. Since \(ABCD\) is a parallelogram, \(AD = BC\).

4. From the congruence \(\triangle ADQ \cong \triangle RCQ\), we have \(AD = CR\).

5. Point \(R\) is on the extension of \(BC\), so \(BR = BC + CR = AD + AD = 2AD\).

6. Substitute this back into the similarity ratio:

\[ \frac{AP}{PR} = \frac{AD}{2AD} = \frac{1}{2} \]

7. Cross-multiplying gives:

\[ PR = 2AP \]


Step 4: Final Answer:

The similarity ratio \(\frac{AP}{PR} = \frac{1}{2}\) leads directly to \(PR = 2AP\).
Quick Tip: The ratio of corresponding sides of similar triangles is constant. Always relate the base segments (\(AD\) and \(BR\)) first.

Step 1: Understanding the Concept:

Mean is the weighted average of class marks. Mode is the value with the highest frequency.


Step 2: Key Formula or Approach:

Mean \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)

Mode \(= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)


Step 3: Detailed Explanation:

Calculating Mean and Finding p:

\begin{tabular{|c|c|c|c|
\hline
Class Interval & Frequency (\(f_i\)) & Class Mark (\(x_i\)) & \(f_i x_i\)
\hline
0-20 & 12 & 10 & 120

20-40 & 15 & 30 & 450

40-60 & p & 50 & 50p

60-80 & 28 & 70 & 1960

80-100 & 13 & 90 & 1170
\hline
Total & \(68+p\) & & \(3700+50p\)
\hline
\end{tabular

Given Mean \(= 53\).
\[ 53 = \frac{3700 + 50p}{68 + p} \]
\[ 53(68 + p) = 3700 + 50p \implies 3604 + 53p = 3700 + 50p \]
\[ 3p = 3700 - 3604 = 96 \implies p = 32 \]

Calculating Mode:

Since \(p = 32\), the maximum frequency is 32.

Modal Class: 40-60.
\(l = 40, f_1 = 32, f_0 = 15, f_2 = 28, h = 20\).
\[ Mode = 40 + \left( \frac{32 - 15}{2(32) - 15 - 28} \right) \times 20 \]
\[ Mode = 40 + \left( \frac{17}{64 - 43} \right) \times 20 = 40 + \frac{17}{21} \times 20 \]
\[ Mode = 40 + \frac{340}{21} \approx 40 + 16.19 = 56.19 \]


Step 4: Final Answer:

The missing frequency \(p\) is 32 and the mode of the distribution is approximately 56.19.
Quick Tip: Double-check the modal class after finding missing frequencies, as it might change depending on the calculated value.

Step 1: Understanding the Concept:

To find the median from mid-values, first convert them into class intervals. The interval size \(h\) is the difference between consecutive mid-values.


Step 2: Key Formula or Approach:

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


Step 3: Detailed Explanation:

Difference between mid-values \(= 125 - 115 = 10\). So \(h = 10\).

Class boundaries for 115: \(115 \pm \frac{10}{2} \implies 110-120\).

\begin{tabular{|c|c|c|c|
\hline
Class Interval & Mid-value & Frequency (\(f\)) & Cumulative Freq (\(cf\))
\hline
110-120 & 115 & 12 & 12

120-130 & 125 & 15 & 27

130-140 & 135 & 20 & 47

140-150 & 145 & 16 & 63

150-160 & 155 & 10 & 73

160-170 & 165 & 16 & 89

170-180 & 175 & 11 & 100
\hline
Total (\(N\)) & & 100 &
\hline
\end{tabular
\(N = 100, \frac{N}{2} = 50\).

The cumulative frequency just greater than 50 is 63, so the Median Class is 140-150.
\(l = 140, f = 16, cf = 47, h = 10\).
\[ Median = 140 + \left( \frac{50 - 47}{16} \right) \times 10 \]
\[ Median = 140 + \frac{3 \times 10}{16} = 140 + \frac{30}{16} = 140 + 1.875 = 141.875 \]


Step 4: Final Answer:

The median of the given data is 141.875.
Quick Tip: Class interval \(= [Mid-value - h/2, Mid-value + h/2]\). Correctly forming the table is 90% of the work.

Step 1: Understanding the Concept:

1. Tangents from an external point to a circle are equal in length.

2. Tangent is perpendicular to the radius at the point of contact.


Step 3: Detailed Explanation:

1. Finding PQ: In rt \(\triangle OQP\), \(OQ = 5\) cm (radius), \(OP = 13\) cm.

\[ QP^2 = OP^2 - OQ^2 = 13^2 - 5^2 = 169 - 25 = 144 \implies QP = 12 cm \]

2. Since \(PQ\) and \(PR\) are tangents from \(P\), \(PQ = PR = 12\) cm.

3. Finding CP: \(OC\) is radius, so \(OC = 5\) cm.

\[ CP = OP - OC = 13 - 5 = 8 cm \]

4. Since \(AB\) is a tangent at \(C\) on \(OP\), \(OP \perp AB\), so \(\triangle ACP\) is a right-angled triangle.

5. Let \(AC = x\). Since \(AC\) and \(AQ\) are tangents from point \(A\) to the circle:

\[ AQ = AC = x \]

6. Now, \(PA = PQ - AQ = 12 - x\).

7. In rt \(\triangle ACP\) (right angled at \(C\)):

\[ AP^2 = AC^2 + CP^2 \]

\[ (12 - x)^2 = x^2 + 8^2 \]

\[ 144 + x^2 - 24x = x^2 + 64 \]

\[ 24x = 144 - 64 = 80 \implies x = \frac{80}{24} = \frac{10}{3} cm \]

8. Calculating lengths:

- \(PA = 12 - x = 12 - \frac{10}{3} = \frac{36 - 10}{3} = \frac{26}{3} cm \).

- By symmetry, \(\triangle ACP \cong \triangle BCP\), so \(AC = BC = \frac{10}{3}\) cm.

- \(AB = AC + BC = \frac{10}{3} + \frac{10}{3} = \frac{20}{3} cm \).


Step 4: Final Answer:

The length of \(AB\) is \(\frac{20}{3}\) cm and the length of \(PA\) is \(\frac{26}{3}\) cm.
Quick Tip: Remember: \(AQ=AC\) and \(BR=BC\). This is the standard trick for problems involving triangles circumscribing circles or tangents intersecting.

Step 1: Understanding the Concept:

The rate of filling is the reciprocal of the time taken. For two taps working together, the sum of their separate rates equals the combined rate.


Step 2: Key Formula or Approach:

If time taken is \(t\), work rate is \(1/t\).


Step 3: Detailed Explanation:

1. Let the time taken by the smaller tap to fill the tank be \(x\) hours.

2. Then, the time taken by the larger tap \(= (x - 4)\) hours.

3. Combined time \(= 8\frac{8}{9} = \frac{80}{9}\) hours.

4. Combined rate \(= \frac{9}{80}\) tank/hour.

5. According to the problem:

\[ \frac{1}{x} + \frac{1}{x - 4} = \frac{9}{80} \]

\[ \frac{(x - 4) + x}{x(x - 4)} = \frac{9}{80} \implies \frac{2x - 4}{x^2 - 4x} = \frac{9}{80} \]

\[ 80(2x - 4) = 9(x^2 - 4x) \implies 160x - 320 = 9x^2 - 36x \]

\[ 9x^2 - 36x - 160x + 320 = 0 \implies 9x^2 - 196x + 320 = 0 \]

6. Solving using quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

\[ x = \frac{196 \pm \sqrt{(-196)^2 - 4(9)(320)}}{18} = \frac{196 \pm \sqrt{38416 - 11520}}{18} \]

\[ x = \frac{196 \pm \sqrt{26896}}{18} = \frac{196 \pm 164}{18} \]

- \(x = \frac{360}{18} = 20\)

- \(x = \frac{32}{18} \approx 1.77\) (Rejected as \(x-4\) would be negative).

7. Time for smaller tap \(= 20\) hours.

Time for larger tap \(= 20 - 4 = 16\) hours.


Step 4: Final Answer:

The smaller tap fills the tank in 20 hours and the larger tap fills it in 16 hours.
Quick Tip: When solving quadratic equations for time and work, always reject the value that makes any of the individual times negative or realistically impossible.

Step 1: Understanding the Concept:

In the given diagram, the window is at point \(W\).

The distance \(AX\) represents the height of the window \(d\) from the ground (assuming \(A\) is the foot of the tank tower on the ground).

Triangle \(\triangle AWX\) is a right-angled triangle where \(AW = y\) is the hypotenuse and \(AX = d\) is the opposite side to the angle of depression of \(30^{\circ}\).


Step 2: Key Formula or Approach:

We use the trigonometric ratio of sine:
\[ \sin \theta = \frac{Opposite}{Hypotenuse} \]


Step 3: Detailed Explanation:

In right-angled triangle \(\triangle AWX\):

The angle of depression is \(30^{\circ}\), so \(\angle AWX = 30^{\circ}\).
\[ \sin 30^{\circ} = \frac{AX}{AW} \]

Substituting the given values:
\[ \frac{1}{2} = \frac{d}{y} \]

Rearranging to find the relation:
\[ y = 2d \]


Step 4: Final Answer:

The relation between \(d\) and \(y\) is \(y = 2d\).
Quick Tip: In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the side opposite to the \(30^{\circ}\) angle is always half the length of the hypotenuse.

Step 1: Understanding the Concept:

The value \(h\) represents the height of the upper part of the water tank tower (\(BX\)) relative to the horizontal line of sight from the window \(W\).


Step 2: Key Formula or Approach:

We use the trigonometric ratio of tangent in \(\triangle BWX\):
\[ \tan \theta = \frac{Opposite}{Adjacent} \]


Step 3: Detailed Explanation:

In right-angled triangle \(\triangle BWX\):

The angle of elevation is \(45^{\circ}\), and the horizontal distance \(XW\) is equal to the distance between the building and the tank, which is 54 m.
\[ \tan 45^{\circ} = \frac{BX}{XW} \]

Substituting the values:
\[ 1 = \frac{h}{54} \]
\[ h = 54 m \]


Step 4: Final Answer:

The value of \(h\) is 54 m.
Quick Tip: Whenever the angle of elevation or depression is \(45^{\circ}\), the height (opposite) is always equal to the horizontal distance (adjacent).

Step 1: Understanding the Concept:

The total height of the water tank is the sum of the portions above and below the horizontal level of the window, i.e., \(AB = BX + XA = h + d\).


Step 2: Key Formula or Approach:

We find \(d\) using the tangent ratio in \(\triangle AWX\):
\[ \tan \theta = \frac{Opposite}{Adjacent} \]


Step 3: Detailed Explanation:

From part (ii), we know \(h = 54\) m.

In right-angled triangle \(\triangle AWX\):
\[ \tan 30^{\circ} = \frac{AX}{XW} = \frac{d}{54} \]
\[ \frac{1}{\sqrt{3}} = \frac{d}{54} \]
\[ d = \frac{54}{\sqrt{3}} = \frac{54 \times \sqrt{3}}{3} = 18\sqrt{3} m \]

Approximate value of \(d \approx 18 \times 1.732 = 31.176\) m.

Total height \(AB = h + d = 54 + 18\sqrt{3}\) m.

Total height \(\approx 54 + 31.18 = 85.18 m\).


Step 4: Final Answer:

The height of the water tank is \(54 + 18\sqrt{3}\) m.
Quick Tip: Always rationalize the denominator when dealing with surds like \(\sqrt{3}\) to make further additions easier.

Step 1: Understanding the Concept:

The value \(x\) is the line-of-sight distance from the window to the top of the tank (\(BW\)).

The height of the window above ground level is the distance \(AX = d\).


Step 2: Detailed Explanation:

1. Finding \(x\):

In right-angled triangle \(\triangle BWX\):
\[ \cos 45^{\circ} = \frac{XW}{BW} = \frac{54}{x} \]
\[ \frac{1}{\sqrt{2}} = \frac{54}{x} \implies x = 54\sqrt{2} m \]

2. Finding height of the window:

As calculated in part (iii)(a):
\[ \tan 30^{\circ} = \frac{d}{54} \implies d = 18\sqrt{3} m \]

This \(d\) is the height of the window above ground level.


Step 3: Final Answer:

The value of \(x\) is \(54\sqrt{2}\) m and the height of the window is \(18\sqrt{3}\) m.
Quick Tip: For an isosceles right triangle (\(45^{\circ}\) triangle), the hypotenuse is always \(\sqrt{2}\) times the side length.

Step 1: Understanding the Concept:

Point \(A\) is the peak (vertex) of the parabolic arch. For a quadratic polynomial \(ax^{2} + bx + c\), the \(x\)-coordinate of the vertex is given by \(x = -b/(2a)\).


Step 2: Detailed Explanation:

The polynomial is \(p(x) = -0.0025x^{2} - 0.025x + 136\).

Comparing with \(ax^{2} + bx + c\):
\(a = -0.0025\), \(b = -0.025\), \(c = 136\).

Find the \(x\)-coordinate:
\[ x = \frac{-(-0.025)}{2 \times (-0.0025)} = \frac{0.025}{-0.005} = -5 \]

Now, find the \(y\)-coordinate by substituting \(x = -5\) into \(p(x)\):
\[ p(-5) = -0.0025(-5)^{2} - 0.025(-5) + 136 \]
\[ p(-5) = -0.0025(25) + 0.125 + 136 \]
\[ p(-5) = -0.0625 + 0.125 + 136 = 136.0625 \]


Step 3: Final Answer:

The co-ordinates of point \(A\) are \((-5, 136.0625)\).
Quick Tip: The vertex of a downward-opening parabola (\(a < 0\)) is its maximum point. The coordinates are always \(\left(\frac{-b}{2a}, p\left(\frac{-b}{2a}\right)\right)\).

Step 1: Understanding the Concept:

The span of the arch is the horizontal distance between the two points where the arch meets the ground (or axis), which are points \(Q\) and \(P\).


Step 2: Detailed Explanation:

From the given diagram:

Point \(Q\) is at \((-238.5, 0)\).

Point \(P\) is at \((228.5, 0)\).

The span is the distance between these two \(x\)-coordinates on the \(x\)-axis.
\[ Span = |228.5 - (-238.5)| \]
\[ Span = 228.5 + 238.5 = 467 units \]


Step 3: Final Answer:

The span of the arch is 467 units.
Quick Tip: Distance between two points \((x_{1}, 0)\) and \((x_{2}, 0)\) on the horizontal axis is simply \(|x_{2} - x_{1}|\).

Step 1: Understanding the Concept:

The zeroes of a polynomial are the \(x\)-values where the graph intersects the \(x\)-axis. For a quadratic polynomial, the sum of zeroes \(\alpha + \beta = -b/a\).


Step 2: Detailed Explanation:

1. Identifying zeroes from the diagram:

The graph intersects the \(x\)-axis at \(x = -238.5\) and \(x = 228.5\).

So, \(\alpha = -238.5\) and \(\beta = 228.5\).

2. Sum of zeroes from the diagram:
\[ \alpha + \beta = -238.5 + 228.5 = -10 \]

3. Verification using polynomial coefficients:

From \(p(x) = -0.0025x^{2} - 0.025x + 136\), we have \(a = -0.0025\) and \(b = -0.025\).
\[ Relationship: Sum of zeroes = \frac{-b}{a} \]
\[ Sum = \frac{-(-0.025)}{-0.0025} = \frac{0.025}{-0.0025} = -10 \]

Since both calculations yield \(-10\), the relationship is verified.


Step 3: Final Answer:

The zeroes are \(-238.5\) and \(228.5\), and the sum relationship is verified as \(-10\).
Quick Tip: Zeroes of a polynomial \(p(x)\) are synonymous with the \(x\)-intercepts of the graph \(y = p(x)\).

Step 1: Understanding the Concept:

We substitute the specific values of \(x\) into the polynomial and evaluate. A parabola is symmetric about \(x = 0\) only if the coefficient \(b = 0\).


Step 2: Detailed Explanation:

Polynomial: \(p(x) = -0.0025x^{2} - 0.025x + 136\)

1. For \(x = 100\):
\[ p(100) = -0.0025(100)^{2} - 0.025(100) + 136 \]
\[ p(100) = -0.0025(10000) - 2.5 + 136 \]
\[ p(100) = -25 - 2.5 + 136 = 108.5 \]

2. For \(x = -100\):
\[ p(-100) = -0.0025(-100)^{2} - 0.025(-100) + 136 \]
\[ p(-100) = -0.0025(10000) + 2.5 + 136 \]
\[ p(-100) = -25 + 2.5 + 136 = 113.5 \]

3. Comparison:
\(108.5 \neq 113.5\). The values are not the same.


Step 3: Final Answer:

The values are \(p(100) = 108.5\) and \(p(-100) = 113.5\). They are not the same because the parabola's axis of symmetry is \(x = -5\), not \(x = 0\).
Quick Tip: A function \(f(x) = ax^{2} + bx + c\) will have \(f(k) = f(-k)\) if and only if \(b = 0\).

Step 1: Understanding the Concept:

The bulb is spherical in shape. We need to find its total surface area using its diameter.


Step 2: Key Formula or Approach:

Surface area of a sphere:
\[ SA = 4\pi r^{2} or \pi d^{2} \]


Step 3: Detailed Explanation:

Given: Diameter \(d = 7\) cm.

Radius \(r = \frac{d}{2} = 3.5\) cm.

Using \(\pi = \frac{22}{7}\):
\[ SA = 4 \times \frac{22}{7} \times 3.5 \times 3.5 \]
\[ SA = 4 \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \]
\[ SA = \frac{4 \times 22 \times 49}{7 \times 4} = 22 \times 7 = 154 cm^{2} \]


Step 4: Final Answer:

The surface area of the bulb is 154 \(cm^{2}\).
Quick Tip: Using the diameter directly in the formula \(\pi d^{2}\) is faster than converting to radius for spheres when the diameter is a multiple of 7.

Step 1: Understanding the Concept:

To fit inside the cuboidal space with a specific gap on all sides, the diameter must be less than the dimensions of the cuboid minus the required gaps.


Step 2: Detailed Explanation:

The cuboid dimensions are: Length \(L = 24\) cm, Width \(W = 12\) cm, Height \(H = 17\) cm.

A gap of 1 cm must be left from each side. This means for each dimension, we lose \(1 + 1 = 2\) cm of available space.

1. Along the width: Max diameter \(\le 12 - 2 = 10\) cm.

2. Along the length: Max diameter \(\le 24 - 2 = 22\) cm.

3. Along the height: The bulb can be anywhere, but it's restricted by the narrower sides of the box.

The most restrictive dimension is the width (12 cm).

Thus, the maximum diameter possible is \(12 - 2 = 10\) cm.


Step 3: Final Answer:

The maximum diameter of the bulb is 10 cm.
Quick Tip: When fitting an object inside another, the smallest dimension of the container usually determines the maximum size of the object.

Step 1: Understanding the Concept:

The lamp is cuboidal but open at the top and bottom. The area of fabric used is the lateral surface area of the cuboid. The folds mean the actual height of fabric cut is more than the height of the lamp.


Step 2: Detailed Explanation:

Dimensions of cuboid: \(24\) cm \(\times 12\) cm \(\times 17\) cm.

Perimeter of the base \(= 2 \times (24 + 12) = 2 \times 36 = 72\) cm.

The height of the lamp is 17 cm.

Since there is a fold of 2 cm on the top edge and a fold of 2 cm on the bottom edge, the total height of the fabric required is:
\[ Fabric Height = 17 + 2 + 2 = 21 cm \]

Area of fabric \(= Perimeter \times Fabric Height \)
\[ Area = 72 \times 21 = 1512 cm^{2} \]


Step 3: Final Answer:

The area of the fabric used is 1512 \(cm^{2}\).
Quick Tip: For open-ended prisms or cuboids, the material area is always the product of the base perimeter and the total material height.

Step 1: Understanding the Concept:

The space available inside the lamp refers to the volume enclosed by the cuboidal fabric frame.


Step 2: Key Formula or Approach:

Volume of a cuboid:
\[ V = L \times W \times H \]


Step 3: Detailed Explanation:

The dimensions of the cuboidal frame are:

Length \(L = 24\) cm

Width \(W = 12\) cm

Height \(H = 17\) cm
\[ Volume = 24 \times 12 \times 17 \]
\[ Volume = 288 \times 17 = 4896 cm^{3} \]


Step 4: Final Answer:

The space available inside the lamp is 4896 \(cm^{3}\).
Quick Tip: Space available is synonymous with the internal volume of the shape.