Step 1: Understanding the Concept:

A tangent to a circle is perpendicular to the radius through the point of contact.

The length of tangents drawn from an external point to a circle are equal (\(PQ = PR\)).

In a quadrilateral, if three angles are \(90^{\circ}\) and adjacent sides are equal, it forms a square.


Step 2: Key Formula or Approach:

In quadrilateral \(OQPR\):
\[ \angle OQP = \angle ORP = 90^{\circ} (Radius \perp Tangent) \]
\[ \angle QPR = 90^{\circ} (Given) \]

Since the sum of angles is \(360^{\circ}\), \(\angle QOR = 90^{\circ}\).

Also, \(OQ = OR = 4 cm\) (Radii).

Thus, \(OQPR\) is a square with side length \(s = 4 cm\).


Step 3: Detailed Explanation:

In the square \(OQPR\), \(OP\) is the diagonal.

The length of the diagonal of a square with side \(s\) is given by \(s\sqrt{2}\).
\[ OP = 4\sqrt{2} cm \]

Alternatively, using Pythagoras theorem in \(\triangle OQP\):
\[ OP^2 = OQ^2 + QP^2 \]

Since \(OQPR\) is a square, \(QP = OQ = 4 cm\).
\[ OP^2 = 4^2 + 4^2 = 16 + 16 = 32 \]
\[ OP = \sqrt{32} = 4\sqrt{2} cm \]


Step 4: Final Answer:

The length of \(OP\) is \(4\sqrt{2} cm\).
Quick Tip: Whenever tangents from an external point are perpendicular to each other, the quadrilateral formed by the center, the two points of contact, and the external point is a square. The distance from the center to the external point is always \(r\sqrt{2}\).

Step 1: Understanding the Concept:

The problem states that the volume of the cone is exactly equal to the combined volume of two spherical scoops.


Step 2: Key Formula or Approach:

Volume of Cone = \(\frac{1}{3} \pi r^2 h\)

Volume of Sphere = \(\frac{4}{3} \pi R^3\)

Given: Radius of cone is \(r\). Radius of scoop \(R = \frac{r}{2}\).


Step 3: Detailed Explanation:

According to the question:
\[ Volume of cone = 2 \times Volume of one scoop \]
\[ \frac{1}{3} \pi r^2 h = 2 \times \left( \frac{4}{3} \pi \left( \frac{r}{2} \right)^3 \right) \]

Canceling \(\frac{1}{3} \pi\) from both sides:
\[ r^2 h = 2 \times 4 \times \frac{r^3}{8} \]
\[ r^2 h = 8 \times \frac{r^3}{8} \]
\[ r^2 h = r^3 \]

Dividing both sides by \(r^2\):
\[ h = r \]

We need to find the ratio \(h : 2r\).
\[ \frac{h}{2r} = \frac{r}{2r} = \frac{1}{2} \]


Step 4: Final Answer:

The ratio \(h : 2r\) is \(1 : 2\).
Quick Tip: Be careful with the ratio required. The question asks for \(h : 2r\), not \(h : r\). Always simplify expressions before plugging in values to save time.

Step 1: Understanding the Concept:

The length of an arc is proportional to the angle it subtends at the center of the circle.


Step 2: Key Formula or Approach:

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

Given: \(l = 11 cm\), \(r = 6.3 cm\), \(\pi \approx \frac{22}{7}\).


Step 3: Detailed Explanation:

Substitute the given values into the formula:
\[ 11 = \frac{\theta}{360^{\circ}} \times 2 \times \frac{22}{7} \times 6.3 \]
\[ 11 = \frac{\theta}{360^{\circ}} \times 44 \times 0.9 \]
\[ 11 = \frac{\theta}{360^{\circ}} \times 39.6 \]

Divide both sides by 11:
\[ 1 = \frac{\theta}{360^{\circ}} \times 3.6 \]
\[ 360^{\circ} = 3.6\theta \]
\[ \theta = \frac{360}{3.6} = 100^{\circ} \]


Step 4: Final Answer:

The value of \(\theta\) is \(100^{\circ}\).
Quick Tip: When dealing with decimals like 6.3 in arc length or area of sector problems, they are usually multiples of 7. Using \(\pi = 22/7\) often leads to easy cancellations.

Step 1: Understanding the Concept:

This problem requires the use of fundamental trigonometric identities.


Step 2: Key Formula or Approach:

Use the identities:
\(1 + \tan^2 A = \sec^2 A\)
\(1 + \cot^2 A = \csc^2 A\)


Step 3: Detailed Explanation:

Substitute the identities into the expression:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \frac{\sec^2 A}{\csc^2 A} \]

Using the definitions of secant and cosecant:
\[ \sec^2 A = \frac{1}{\cos^2 A} \quad and \quad \csc^2 A = \frac{1}{\sin^2 A} \]

Therefore:
\[ \frac{\sec^2 A}{\csc^2 A} = \frac{1/\cos^2 A}{1/\sin^2 A} = \frac{1}{\cos^2 A} \times \frac{\sin^2 A}{1} = \frac{\sin^2 A}{\cos^2 A} \]
\[ \frac{\sin^2 A}{\cos^2 A} = \tan^2 A \]


Step 4: Final Answer:

The expression equals \(\tan^2 A\).
Quick Tip: A quick shortcut for such fractions is to remember that \(\cot A = 1/\tan A\). Thus, \(1 + \cot^2 A = 1 + \frac{1}{\tan^2 A} = \frac{\tan^2 A + 1}{\tan^2 A}\). Dividing the numerator by this immediately gives \(\tan^2 A\).

Step 1: Understanding the Concept:

The "volume of air" is the volume of the cylinder that is not occupied by the three spherical balls.


Step 2: Key Formula or Approach:

Radius of cylinder = Radius of ball = \(r\).

Height of cylinder (\(h\)) = \(3 \times\) Diameter of ball = \(3 \times (2r) = 6r\).

Volume of cylinder = \(\pi r^2 h\).

Volume of 3 spheres = \(3 \times \left( \frac{4}{3} \pi r^3 \right)\).


Step 3: Detailed Explanation:

Calculate volume of cylinder:
\[ V_{cyl} = \pi \times r^2 \times (6r) = 6 \pi r^3 \]

Calculate volume of 3 balls:
\[ V_{balls} = 3 \times \frac{4}{3} \pi r^3 = 4 \pi r^3 \]

Volume of air = \(V_{cyl} - V_{balls}\):
\[ V_{air} = 6 \pi r^3 - 4 \pi r^3 = 2 \pi r^3 \]


Step 4: Final Answer:

The volume of air inside the jar is \(2\pi r^3\).
Quick Tip: When spheres are "just packed" in a cylinder, the cylinder's height is always equal to the sum of the diameters of the spheres. For \(n\) spheres, \(h = n \times 2r\).

Step 1: Understanding the Concept:

Probability is defined as the number of favorable outcomes divided by the total number of outcomes.


Step 2: Key Formula or Approach:

Total outcomes for two dice = \(6 \times 6 = 36\).

Numbers less than 4 on a single die are \(\{1, 2, 3\}\).


Step 3: Detailed Explanation:

For both numbers to be less than 4, the outcomes are:
\((1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\).

Number of favorable outcomes = \(3 \times 3 = 9\).
\[ P(both < 4) = \frac{9}{36} = \frac{1}{4} \]


Step 4: Final Answer:

The probability is \(\frac{1}{4}\).
Quick Tip: If events are independent (like rolling two dice), you can multiply individual probabilities: \(P(A \cap B) = P(A) \times P(B)\). Here, \(P(die < 4) = 3/6 = 1/2\). So, \((1/2) \times (1/2) = 1/4\).

Step 1: Understanding the Concept:

In a parallelogram, opposite sides are parallel. Similarity of triangles can be established using alternate interior angles.


Step 2: Key Formula or Approach:

From the diagram, \(E\) is a point on \(BC\) and line segment \(DF\) is drawn where \(F\) is on extended side \(AB\).

Wait, looking closely at the figure, \(F\) is on side \(AB\). In \(\triangle FBE\) and \(\triangle FAD\):

Since \(AD \parallel BC\), \(\angle FBE = \angle FAD\) (corresponding angles) and \(\angle FEB = \angle FDA\) (corresponding angles).

Thus, \(\triangle FBE \sim \triangle FAD\).


Step 3: Detailed Explanation:

By similarity of \(\triangle FBE \sim \triangle FAD\), the ratios of corresponding sides are equal:
\[ \frac{FB}{FA} = \frac{FE}{FD} \]

Given: \(FB = 3 cm\), \(AF = 7 cm\) (This is the total side length \(FA\) in the larger triangle), and \(EF = 4 cm\).
\[ \frac{3}{7} = \frac{4}{FD} \]

Cross-multiplying:
\[ 3 \times FD = 7 \times 4 \]
\[ 3 \times FD = 28 \]
\[ FD = \frac{28}{3} cm \]


Step 4: Final Answer:

The length \(FD\) is \(\frac{28}{3} cm\).
Quick Tip: Identify parallel lines and look for the "bow-tie" or "nested" triangle configurations to quickly spot similar triangles in geometry problems.

Step 1: Understanding the Concept:

In a circle, any triangle formed by two radii (\(OP\) and \(OR\)) and a chord (\(PR\)) is an isosceles triangle because \(OP = OR\).


Step 2: Key Formula or Approach:

In \(\triangle POR\), \(OP = OR\) (radii).

Therefore, \(\angle OPR = \angle ORP\) (angles opposite to equal sides).

The sum of angles in a triangle is \(180^{\circ}\).


Step 3: Detailed Explanation:

In \(\triangle POR\):
\[ \angle POR + \angle OPR + \angle ORP = 180^{\circ} \]

Substitute \(\angle POR = 65^{\circ}\) and let \(\angle OPR = \angle ORP = x\):
\[ 65^{\circ} + x + x = 180^{\circ} \]
\[ 65^{\circ} + 2x = 180^{\circ} \]
\[ 2x = 180^{\circ} - 65^{\circ} \]
\[ 2x = 115^{\circ} \]
\[ x = \frac{115^{\circ}}{2} = 57.5^{\circ} \]

So, \(m\angle OPR = 57.5^{\circ}\).


Step 4: Final Answer:

The measure of \(\angle OPR\) is \(57.5^{\circ}\).
Quick Tip: In circle geometry, always look for isosceles triangles formed by radii. They are extremely common for calculating unknown angles.

Step 1: Understanding the Concept:

The radius of a circle is the distance between its center and any point on its circumference.


Step 2: Key Formula or Approach:

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

Center \((x_1, y_1) = (-1, 2)\)

Point on circle \((x_2, y_2) = (0, 3)\)


Step 3: Detailed Explanation:
\[ Radius (r) = \sqrt{(0 - (-1))^2 + (3 - 2)^2} \]
\[ r = \sqrt{(1)^2 + (1)^2} \]
\[ r = \sqrt{1 + 1} = \sqrt{2} \]


Step 4: Final Answer:

The radius of the circle is \(\sqrt{2}\).
Quick Tip: Always double-check signs when subtracting negative coordinates in the distance formula. \(0 - (-1)\) becomes \(+1\).

Step 1: Understanding the Concept:

In similar triangles, the order of vertices determines the corresponding parts. For \(\triangle ABC \sim \triangle EDF\):

Corresponding angles are: \(\angle A = \angle E\), \(\angle B = \angle D\), \(\angle C = \angle F\).

Corresponding side ratios are: \(\frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} = k\).


Step 2: Key Formula or Approach:

Check each option against these rules.


Step 3: Detailed Explanation:

(A) True: The ratio of perimeters is equal to the ratio of corresponding sides.

(B) True: \(\frac{AB}{ED} = \frac{AC}{EF}\) is a correct pair of corresponding sides.

(C) False: According to the order of vertices, \(\angle A\) corresponds to \(\angle E\). Therefore, \(\angle A = \angle E\) is true, but \(\angle A = \angle D\) is generally not true.

(D) True: Since \(\frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} = k\), then \(AB = k \cdot ED\), \(BC = k \cdot DF\), and \(AC = k \cdot EF\).

Substituting these: \(\frac{k \cdot ED + k \cdot DF}{k \cdot EF} = \frac{k(ED + DF)}{k \cdot EF} = \frac{ED + DF}{EF}\). This holds true.


Step 4: Final Answer:

The statement (C) is not true.
Quick Tip: In similarity notation \(\triangle XYZ \sim \triangle PQR\), position 1 matches position 1 (\(X \leftrightarrow P\)), position 2 matches position 2 (\(Y \leftrightarrow Q\)), and so on. Use this to write ratios correctly every time.

Step 1: Understanding the Concept:

For a quadratic equation \(ax^2 + bx + c = 0\) to have real and equal roots, its discriminant must be zero.


Step 2: Key Formula or Approach:

Discriminant \(D = b^2 - 4ac = 0\).

Here, \(a = 1, b = -k\sqrt{3}, c = 2\).


Step 3: Detailed Explanation:

Substitute values into the discriminant formula:
\[ (-k\sqrt{3})^2 - 4(1)(2) = 0 \]
\[ 3k^2 - 8 = 0 \]
\[ 3k^2 = 8 \]
\[ k^2 = \frac{8}{3} \]
\[ k = \pm \sqrt{\frac{8}{3}} \]

Comparing with options, (B) provides the positive value.


Step 4: Final Answer:

The value of \(k\) is \(\sqrt{\frac{8}{3}}\).
Quick Tip: When a quadratic has "equal roots", it's a perfect square trinomial. The middle coefficient squared always equals \(4 \times first term \times last term\).

Step 1: Understanding the Concept:

The number of real zeroes of a polynomial \(y = p(x)\) is the number of times its graph intersects or touches the \(x\)-axis.


Step 3: Detailed Explanation:

Look at the provided graph:

The curve crosses the \(x\)-axis at three distinct points.

1. One point is on the negative \(x\)-axis.

2. One point is at the origin (or very close to it, intersecting the axis).

3. One point is on the positive \(x\)-axis.

Since there are 3 intersection points with the \(x\)-axis, the polynomial has 3 zeroes.


Step 4: Final Answer:

The number of zeroes is \(3\).
Quick Tip: Do not count the intersections with the \(y\)-axis. Only intersections with the \(x\)-axis represent the zeroes of \(p(x)\).

Step 1: Understanding the Concept:

There is an empirical relationship between mean, median, and mode for a moderately skewed distribution.


Step 2: Key Formula or Approach:

Empirical formula: \(Mode = 3 \times Median - 2 \times Mean\).


Step 3: Detailed Explanation:

Given: \(Mean = 43\), \(Median = 40\).

Substitute these into the formula:
\[ Mode = 3(40) - 2(43) \]
\[ Mode = 120 - 86 \]
\[ Mode = 34 \]


Step 4: Final Answer:

The value of the mode is \(34\).
Quick Tip: A simple mnemonic to remember this is "3 Medians minus 2 Means equals 1 Mode". Note the alphabetical order: Median comes before Mean if you reverse the order of subtrahends.

Step 1: Understanding the Concept:

The area of a sector is a fraction of the total area of the circle based on the central angle \(\theta\).


Step 2: Key Formula or Approach:

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


Step 3: Detailed Explanation:

Given: \(r = 18 cm\), \(Area = 198 cm^2\).
\[ 198 = \frac{\theta}{360^{\circ}} \times \frac{22}{7} \times 18 \times 18 \]

Simplify the equation:
\[ 198 = \frac{\theta}{360^{\circ}} \times \frac{22}{7} \times 324 \]

Divide 198 by 22:
\[ 9 = \frac{\theta}{360^{\circ}} \times \frac{1}{7} \times 324 \]

Divide 324 by 9:
\[ 1 = \frac{\theta}{360^{\circ}} \times \frac{1}{7} \times 36 \]

Now, simplify \(\frac{36}{360}\):
\[ 1 = \theta \times \frac{1}{10} \times \frac{1}{7} \]
\[ 1 = \frac{\theta}{70} \]
\[ \theta = 70^{\circ} \]


Step 4: Final Answer:

The measure of the central angle is \(70^{\circ}\).
Quick Tip: Simplify numbers before multiplying everything out. For example, recognizing \(198 = 22 \times 9\) and \(18 \times 18 = 324\) makes the calculation much faster.

Step 1: Understanding the Concept:

Tangent and Secant are related by the fundamental identity \(\sec^2 A = 1 + \tan^2 A\).


Step 2: Key Formula or Approach:

From \(2\tan A = 3\), we get \(\tan A = \frac{3}{2}\).


Step 3: Detailed Explanation:

Using the identity:
\[ \sec^2 A = 1 + \tan^2 A \]
\[ \sec^2 A = 1 + \left( \frac{3}{2} \right)^2 \]
\[ \sec^2 A = 1 + \frac{9}{4} \]
\[ \sec^2 A = \frac{4 + 9}{4} = \frac{13}{4} \]

Taking the square root:
\[ \sec A = \frac{\sqrt{13}}{\sqrt{4}} = \frac{\sqrt{13}}{2} \]


Step 4: Final Answer:

The value of \(\sec A\) is \(\frac{\sqrt{13}}{2}\).
Quick Tip: Alternatively, use a right triangle. If \(\tan A = 3/2\), the opposite side is 3 and adjacent side is 2. The hypotenuse is \(\sqrt{3^2 + 2^2} = \sqrt{13}\). Thus, \(\sec A = hypotenuse / adjacent = \sqrt{13}/2\).

Step 1: Understanding the Concept:

A system of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is inconsistent (no solution) if the lines are parallel.


Step 2: Key Formula or Approach:

Condition for inconsistency: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).

Eq 1: \(\frac{1}{2}x + \frac{1}{3}y - 5 = 0 \Rightarrow 3x + 2y - 30 = 0\) (multiplying by 6).

Eq 2: \(2x + ky - 7 = 0\).


Step 3: Detailed Explanation:

Comparing coefficients:
\(a_1 = 3, b_1 = 2, c_1 = -30\)
\(a_2 = 2, b_2 = k, c_2 = -7\)

Apply the first part of the condition:
\[ \frac{3}{2} = \frac{2}{k} \]

Cross-multiplying:
\[ 3k = 4 \]
\[ k = \frac{4}{3} \]

Check the second part: \(\frac{3}{2} \neq \frac{-30}{-7}\), which is \(1.5 \neq 4.28\). The condition holds.


Step 4: Final Answer:

The value of \(k\) is \(\frac{4}{3}\).
Quick Tip: Always multiply equations with fractions by their LCM to get standard forms before comparing coefficients. This prevents calculation errors.

Step 1: Understanding the Concept:

The sum of the first \(n\) terms of an Arithmetic Progression can be calculated using the first and last terms.


Step 2: Key Formula or Approach:

Sum formula: \(S_n = \frac{n}{2}(a + l)\), where \(l\) is the \(n\)-th term.

Here, \(n = 17\), \(a = -3\), \(S_{17} = 357\). We need to find \(l = a_{17}\).


Step 3: Detailed Explanation:
\[ 357 = \frac{17}{2}(-3 + a_{17}) \]

Multiply both sides by 2 and divide by 17:
\[ \frac{357 \times 2}{17} = -3 + a_{17} \]
\[ 21 \times 2 = -3 + a_{17} \]
\[ 42 = -3 + a_{17} \]
\[ a_{17} = 42 + 3 = 45 \]


Step 4: Final Answer:

The value of \(a_{17}\) is \(45\).
Quick Tip: Use \(S_n = \frac{n}{2}(a + l)\) instead of \(S_n = \frac{n}{2}[2a + (n-1)d]\) when the question involves the last term directly. It saves a lot of algebra.

Step 1: Understanding the Concept:

Looking at the figure, the circle passes through the origin \((0, 0)\) as the curve touches/crosses the intersection of the axes.


Step 2: Key Formula or Approach:

Radius \(r = distance between center (1, 2) and origin (0, 0)\).

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

Diameter \(D = 2r\).


Step 3: Detailed Explanation:
\[ r = \sqrt{(1 - 0)^2 + (2 - 0)^2} \]
\[ r = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

Diameter:
\[ D = 2 \times \sqrt{5} = 2\sqrt{5} \]


Step 4: Final Answer:

The diameter of the circle is \(2\sqrt{5}\).
Quick Tip: The distance from the origin to any point \((x, y)\) is simply \(\sqrt{x^2 + y^2}\). This is a helpful shortcut for coordinate geometry problems.

Step 1: Understanding the Concept:

Irrational numbers are numbers that cannot be expressed as a simple fraction.

The sum of two irrational numbers can sometimes be rational.


Step 3: Detailed Explanation:

Assertion (A): \(\sqrt{3} + \sqrt{5}\) is indeed irrational.

If it were rational, say \(r\), then \(\sqrt{5} = r - \sqrt{3}\). Squaring both sides gives \(5 = r^2 + 3 - 2r\sqrt{3}\), which implies \(\sqrt{3} = (r^2 - 2)/(2r)\). Since the RHS is rational and \(\sqrt{3}\) is irrational, this is a contradiction. Thus, (A) is true.

Reason (R): This statement is false.

Counter-example: Consider two irrational numbers \(\sqrt{2}\) and \(-\sqrt{2}\).

Sum: \(\sqrt{2} + (-\sqrt{2}) = 0\), which is a rational number.

Since the Reason is false, the option is (C).


Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.
Quick Tip: In Assertion-Reason questions, always check if the Reason is a universally true statement first. If you find one counter-example for the Reason, it is false, and you likely only have one option left.

Step 1: Understanding the Concept:

Probability of any event \(E\) always lies between 0 and 1 inclusive (\(0 \leq P(E) \leq 1\)).

The sum of probability of occurrence and non-occurrence of an event is 1.


Step 3: Detailed Explanation:

Assertion (A): Given \(P(E) = 0.2p\).

Since \(P(E) \leq 1\):
\[ 0.2p \leq 1 \]
\[ p \leq \frac{1}{0.2} \]
\[ p \leq 5 \]

So, \(p\) cannot be more than 5. (A) is true.

Reason (R): It is a fundamental property of probability that \(P(E) + P(\bar{E}) = 1\), so \(P(\bar{E}) = 1 - P(E)\). (R) is true.

Connection: While both are true, the reason for \(p \leq 5\) is the definition of the range of probability (\(0 \leq P(E) \leq 1\)), not specifically the formula for the complement event. Thus, (R) is not the explanation for (A).


Step 4: Final Answer:
Both (A) and (R) are true, but (R) is not the correct explanation of (A).
Quick Tip: Probability questions often hide constraints. Remember the two main bounds: \(P(E) \geq 0\) and \(P(E) \leq 1\). Most "find the range of variable" problems in probability rely on these.

Step 1: Understanding the Concept:

The method of contradiction is used to prove irrationality.

We assume the number is rational and show that this leads to a logical inconsistency with the given facts.


Step 2: Key Formula or Approach:

A rational number is any number that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).

Rational numbers are closed under subtraction and division (by non-zero numbers).


Step 3: Detailed Explanation:

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

Then, there exist co-prime integers \(a\) and \(b\) (\(b \neq 0\)) such that:
\[ 2 + 3\sqrt{5} = \frac{a}{b} \]

Rearranging the terms to isolate the irrational part:
\[ 3\sqrt{5} = \frac{a}{b} - 2 \]
\[ 3\sqrt{5} = \frac{a - 2b}{b} \]
\[ \sqrt{5} = \frac{a - 2b}{3b} \]

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

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

Consequently, \(\sqrt{5}\) must also be a rational number.

However, this contradicts the given fact that \(\sqrt{5}\) is an irrational number.

Our assumption was wrong.


Step 4: Final Answer:

Therefore, \(2 + 3\sqrt{5}\) is an irrational number.
Quick Tip: In such "prove that" questions, always isolate the root term on one side. If the other side consists entirely of known rational operations (addition, subtraction, multiplication, division) on integers, that side is rational, leading to the contradiction.

Step 1: Understanding the Concept:

The Highest Common Factor (HCF) can be found using the Prime Factorization method or Euclid's Division Lemma.

Once found, it can be equated to the given linear combination to solve for the unknown variable.


Step 2: Key Formula or Approach:

Euclid's Division Lemma: \(a = bq + r\).

Linear equation: \(HCF(210, 55) = 210 \times 5 + 55m\).


Step 3: Detailed Explanation:

First, find the HCF of 210 and 55:

Using Euclid's algorithm:
\[ 210 = 55 \times 3 + 45 \]
\[ 55 = 45 \times 1 + 10 \]
\[ 45 = 10 \times 4 + 5 \]
\[ 10 = 5 \times 2 + 0 \]

The remainder has become zero, so the HCF is 5.

Now, substitute this value into the given equation:
\[ 5 = 210 \times 5 + 55m \]
\[ 5 = 1050 + 55m \]

Rearrange to solve for \(m\):
\[ 55m = 5 - 1050 \]
\[ 55m = -1045 \]
\[ m = \frac{-1045}{55} \]

Dividing numerator and denominator by 11:
\[ m = \frac{-95}{5} = -19 \]


Step 4: Final Answer:

The value of \(m\) is \(-19\).
Quick Tip: To quickly find the HCF of small numbers, you can also use prime factorization: \(210 = 2 \times 3 \times 5 \times 7\) and \(55 = 5 \times 11\). The common factor is clearly 5.

Step 1: Understanding the Concept:

Basic Proportionality Theorem (Thales's Theorem) states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.


Step 2: Key Formula or Approach:

In \(\triangle XYZ\), if \(PQ \parallel YZ\), then \(\frac{XP}{PY} = \frac{XQ}{QZ}\).


Step 3: Detailed Explanation:

In \(\triangle ABE\), it is given that \(DF \parallel AE\).

Applying Basic Proportionality Theorem:
\[ \frac{BD}{DA} = \frac{BF}{FE} \quad --- (i) \]

In \(\triangle ABC\), it is given that \(DE \parallel AC\).

Applying Basic Proportionality Theorem:
\[ \frac{BD}{DA} = \frac{BE}{EC} \quad --- (ii) \]

From equations (i) and (ii), we observe that the Left Hand Side (LHS) is identical for both:
\[ \frac{BF}{FE} = \frac{BE}{EC} \]


Step 4: Final Answer:

Hence proved.
Quick Tip: When multiple parallel lines are given within nested triangles, look for a common ratio (like \(BD/DA\) here) that links the different parts of the segments together.

Step 1: Understanding the Concept:

For a quadratic equation \(ax^2 + bx + c = 0\) to have equal roots, the discriminant \(D = b^2 - 4ac\) must be zero.

Alternatively, if the sum of coefficients is zero, then \(x=1\) is one of the roots.


Step 2: Key Formula or Approach:

Sum of coefficients: \((p - q) + (q - r) + (r - p) = 0\).

If \(x=1\) is a root and roots are equal, both roots must be 1.

In that case, the product of roots \(\frac{c}{a} = 1 \cdot 1 = 1\).


Step 3: Detailed Explanation:

Let the coefficients be \(A = p-q\), \(B = q-r\), and \(C = r-p\).

Notice that \(A + B + C = (p - q) + (q - r) + (r - p) = 0\).

This means \(x = 1\) is always a root of this equation.

For the roots to be equal, both roots must be \(x = 1\).

We know the product of roots is \(\frac{C}{A}\).

So, \(\frac{r - p}{p - q} = 1\).
\[ r - p = p - q \]

Rearranging the terms:
\[ r + q = p + p \]
\[ q + r = 2p \]

This condition matches the given condition \(q + r = 2p\).

Alternatively, checking the discriminant:
\[ D = (q - r)^2 - 4(p - q)(r - p) \]

Substitute \(p = \frac{q + r}{2}\) into the expression and solve for \(D = 0\).

The property of cyclic coefficients (\(A+B+C=0\)) is a more elegant way to verify.


Step 4: Final Answer:

Thus, it is verified that the roots are equal when \(q + r = 2p\).
Quick Tip: If you see a quadratic equation where the coefficients are cyclic (like \(a-b, b-c, c-a\)), always check if their sum is zero. If it is, \(x=1\) is a guaranteed root!

Step 1: Understanding the Concept:

The relationship between zeroes and coefficients of a quadratic polynomial \(ax^2 + bx + c\) is:

Sum of zeroes (\(\alpha + \beta\)) = \(-\frac{b}{a}\).

Product of zeroes (\(\alpha \beta\)) = \(\frac{c}{a}\).


Step 2: Key Formula or Approach:

We need to evaluate \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha \beta)^2}\).

Use the identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\).


Step 3: Detailed Explanation:

For \(p(x) = 3x^2 - 6x - 5\):
\(a = 3, b = -6, c = -5\).
\[ \alpha + \beta = -\frac{-6}{3} = 2 \]
\[ \alpha \beta = \frac{-5}{3} \]

Now, find \(\alpha^2 + \beta^2\):
\[ \alpha^2 + \beta^2 = (2)^2 - 2\left(\frac{-5}{3}\right) = 4 + \frac{10}{3} = \frac{12 + 10}{3} = \frac{22}{3} \]

Now, calculate the required expression:
\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha \beta)^2} = \frac{\frac{22}{3}}{\left(-\frac{5}{3}\right)^2} \]
\[ = \frac{\frac{22}{3}}{\frac{25}{9}} = \frac{22}{3} \times \frac{9}{25} \]
\[ = \frac{22 \times 3}{25} = \frac{66}{25} \]


Step 4: Final Answer:

The value is \(\frac{66}{25}\).
Quick Tip: Avoid solving for \(\alpha\) and \(\beta\) individually using the quadratic formula unless the polynomial factors easily. Always aim to manipulate the expression into forms of \((\alpha+\beta)\) and \((\alpha\beta)\).

Step 1: Understanding the Concept:

Rationalization of the denominator within a square root often helps simplify trigonometric expressions.


Step 2: Key Formula or Approach:

Use the identity: \(\cos^2 A = 1 - \sin^2 A\).

Definitions: \(\sec A = \frac{1}{\cos A}\), \(\tan A = \frac{\sin A}{\cos A}\).


Step 3: Detailed Explanation:

Taking the Left Hand Side (LHS):
\[ LHS = \sqrt{\frac{1 + \sin A}{1 - \sin A}} \]

Multiply both the numerator and the denominator by \((1 + \sin A)\):
\[ = \sqrt{\frac{(1 + \sin A)(1 + \sin A)}{(1 - \sin A)(1 + \sin A)}} \]
\[ = \sqrt{\frac{(1 + \sin A)^2}{1 - \sin^2 A}} \]

Using the identity \(1 - \sin^2 A = \cos^2 A\):
\[ = \sqrt{\frac{(1 + \sin A)^2}{\cos^2 A}} \]

Taking the square root:
\[ = \frac{1 + \sin A}{\cos A} \]

Split the fraction:
\[ = \frac{1}{\cos A} + \frac{\sin A}{\cos A} \]
\[ = \sec A + \tan A \]

This matches the Right Hand Side (RHS).


Step 4: Final Answer:

LHS = RHS. Hence proved.
Quick Tip: For expressions like \(\sqrt{\frac{1 \pm \sin A}{1 \mp \sin A}}\) or \(\sqrt{\frac{1 \pm \cos A}{1 \mp \cos A}}\), rationalizing the denominator is almost always the correct first step.

Step 1: Understanding the Concept:

Substitute the standard values for trigonometric ratios of \(30^{\circ}\) and \(60^{\circ}\) into the expression.


Step 2: Key Formula or Approach:
\(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).
\(\csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{1/2} = 2\).
\(\tan 60^{\circ} = \sqrt{3}\).


Step 3: Detailed Explanation:

Substitute the values into the expression:
\[ Expression = \frac{3 \left( \frac{\sqrt{3}}{2} \right)^2 - 6(2)^2}{(\sqrt{3})^2} \]

Simplify the numerator and denominator:
\[ = \frac{3 \left( \frac{3}{4} \right) - 6(4)}{3} \]
\[ = \frac{\frac{9}{4} - 24}{3} \]

Find a common denominator for the numerator:
\[ = \frac{\frac{9 - 96}{4}}{3} \]
\[ = \frac{-\frac{87}{4}}{3} \]

Divide by 3:
\[ = -\frac{87}{4 \times 3} = -\frac{29}{4} \]


Step 4: Final Answer:

The evaluated value is \(-\frac{29}{4}\).
Quick Tip: Double check the square terms! It's common to forget to square the numbers in the denominator of the fraction (like squaring \(2\) to get \(4\) in \(\cos^2 30^{\circ}\)).

Step 1: Understanding the Concept:

To find the least number of containers of equal size, each container must have the maximum possible capacity that can exactly divide the given volumes.

This maximum capacity is the Highest Common Factor (HCF) of the three volumes.


Step 2: Key Formula or Approach:

1. Find the HCF of 870, 812, and 638.

2. Total number of containers = \(\frac{Volume_1}{HCF} + \frac{Volume_2}{HCF} + \frac{Volume_3}{HCF}\).


Step 3: Detailed Explanation:

First, we find the prime factorization of each number:
\[ 870 = 2 \times 3 \times 5 \times 29 \]
\[ 812 = 2^2 \times 7 \times 29 \]
\[ 638 = 2 \times 11 \times 29 \]

The common factors are 2 and 29.
\[ HCF(870, 812, 638) = 2 \times 29 = 58 litres \]

This is the capacity of one container.

Now, calculate the number of containers for each type of oil:

- For \(870 l\): \(\frac{870}{58} = 15\)

- For \(812 l\): \(\frac{812}{58} = 14\)

- For \(638 l\): \(\frac{638}{58} = 11\)

Total number of containers = \(15 + 14 + 11 = 40\).


Step 4: Final Answer:

The least number of containers required is 40.
Quick Tip: In problems asking for "least number of items" based on a capacity/size, you are looking for the Highest Common Factor (HCF) of the given quantities. Dividing the total amount by this HCF gives the minimal count.

Step 1: Understanding the Concept:

The shed consists of a cuboidal base and a semi-cylindrical top.

The cloth covers the four vertical walls of the cuboid and the curved surface of the semi-cylinder, including its two semi-circular ends. The bottom of the shed (floor) is not covered with cloth.


Step 2: Key Formula or Approach:

Area of cloth = Lateral Surface Area of Cuboid + CSA of Semi-cylinder + Area of 2 Semi-circles.

Dimensions: Length (\(L\)) = \(25 m\), Width (\(W\)) = \(14 m\), Height (\(H\)) = \(16 m\).

For the semi-cylinder: Radius (\(r\)) = \(\frac{W}{2} = 7 m\), Length (\(h_{cyl}\)) = \(25 m\).


Step 3: Detailed Explanation:

1. Lateral Surface Area of cuboid (4 walls):
\[ LSA = 2(L + W) \times H = 2(25 + 14) \times 16 = 2 \times 39 \times 16 = 1248 m^2 \]

2. Curved Surface Area of semi-cylinder:
\[ CSA = \frac{1}{2}(2 \pi r h_{cyl}) = \pi r h_{cyl} = \frac{22}{7} \times 7 \times 25 = 550 m^2 \]

3. Area of 2 semi-circular ends:
\[ Area_{ends} = 2 \times \frac{1}{2} \pi r^2 = \pi r^2 = \frac{22}{7} \times 7^2 = 154 m^2 \]

Total Area = \(1248 + 550 + 154 = 1952 m^2\).


Step 4: Final Answer:

The total area of the cloth required is \(1952 m^2\).
Quick Tip: Always visualize the surfaces covered. In shed problems, the floor is usually excluded. Also, ensure the radius and length of the semi-cylinder correctly correspond to the cuboid's dimensions from the diagram.

Step 1: Understanding the Concept:

The total surface area (TSA) of the composite object is the sum of all exposed surfaces:

1. The curved surface area of the cone.

2. The area of the top ring (between external and internal radii).

3. The external curved surface area of the hemisphere.

4. The internal curved surface area of the hemisphere.


Step 2: Key Formula or Approach:

- Slant height of cone (\(l\)) = \(\sqrt{h^2 + r^2}\)

- CSA of cone = \(\pi r l\)

- Area of ring = \(\pi(R^2 - r^2)\)

- CSA of hemisphere = \(2 \pi r^2\)


Step 3: Detailed Explanation:

Given: External radius \(R = 10 cm\), Internal radius \(r = 5\sqrt{2} cm\), Cone height \(h = 5\sqrt{7} cm\), Cone radius = \(5\sqrt{2} cm\).

1. Slant height of cone:
\[ l = \sqrt{(5\sqrt{7})^2 + (5\sqrt{2})^2} = \sqrt{175 + 50} = \sqrt{225} = 15 cm \]

2. Exposed Surfaces:

- CSA of cone: \(\pi \times 5\sqrt{2} \times 15 = 75\sqrt{2} \pi cm^2\).

Using \(\sqrt{2} = 1.4\): \(75 \times 1.4 \pi = 105 \pi cm^2\).

- Area of flat ring: \(\pi(10^2 - (5\sqrt{2})^2) = \pi(100 - 50) = 50 \pi cm^2\).

- Outer CSA of hemisphere: \(2 \pi R^2 = 2 \pi (100) = 200 \pi cm^2\).

- Inner CSA of hemisphere: \(2 \pi r^2 = 2 \pi (50) = 100 \pi cm^2\).

Total Surface Area = \(105\pi + 50\pi + 200\pi + 100\pi = 455 \pi cm^2\).


Step 4: Final Answer:

The total surface area of the object is \(455 \pi cm^2\).
Quick Tip: For hollow objects, "total surface area" includes both internal and external visible surfaces unless it's explicitly described as a closed solid. Don't forget the flat ring connecting the inner and outer shells.

Step 1: Understanding the Concept:

Translate the given word problem into linear equations.


Step 2: Key Formula or Approach:

Let the marks obtained by Kevin be \(x\) and marks obtained by Veer be \(y\).


Step 3: Detailed Explanation:

From the first condition: Veer scored 6 more than twice Kevin's marks.
\[ y = 2x + 6 \quad --- (i) \]

From the second condition: If one of them scored 4 more marks, total is 40.

If Kevin scored 4 more: \((x + 4) + y = 40 \Rightarrow x + y = 36\).

If Veer scored 4 more: \(x + (y + 4) = 40 \Rightarrow x + y = 36\).

In either case, the equation is:
\[ x + y = 36 \quad --- (ii) \]

Substitute (i) into (ii):
\[ x + (2x + 6) = 36 \]
\[ 3x + 6 = 36 \]
\[ 3x = 30 \Rightarrow x = 10 \]

Substitute \(x = 10\) in (i):
\[ y = 2(10) + 6 = 26 \]


Step 4: Final Answer:

Kevin's marks = 10, Veer's marks = 26.
Quick Tip: Always check your final answers against the original word problem. Twice Kevin's marks (20) plus 6 is 26 (Veer's marks). Their sum (36) plus 4 is 40. The logic holds!

Step 1: Understanding the Concept:

Graphical solution involves plotting both lines on a Cartesian plane and finding their point of intersection.


Step 2: Key Formula or Approach:

1. Find at least two points for the line \(3x + y = 14\).

2. Plot the line \(y = 2\) (a horizontal line passing through \(y = 2\)).


Step 3: Detailed Explanation:

For \(3x + y = 14\):

- If \(x = 4\), then \(y = 14 - 3(4) = 14 - 12 = 2\). Point is \((4, 2)\).

- If \(x = 0\), then \(y = 14\). Point is \((0, 14)\).

- If \(x = 2\), then \(y = 14 - 3(2) = 14 - 6 = 8\). Point is \((2, 8)\).

Plotting these points and drawing the line gives a straight line.

For \(y = 2\):

- This is a line parallel to the \(x\)-axis passing through 2 on the \(y\)-axis.

Intersection:

Observing the graph, both lines intersect at the point where \(y = 2\). Substituting \(y = 2\) into the first equation:
\[ 3x + 2 = 14 \Rightarrow 3x = 12 \Rightarrow x = 4 \]

The point of intersection is \((4, 2)\).


Step 4: Final Answer:

The solution of the equations is \(x = 4, y = 2\).
Quick Tip: When solving graphically, choose \(x\) values that result in integer \(y\) values to make plotting easier and more accurate on grid paper.

Step 1: Understanding the Concept:

Probability of an event = \(\frac{Number of favorable outcomes}{Total number of outcomes}\).


Step 3: Detailed Explanation:

Total balls = 30. Blue balls = \(m\).

Part (i):

Number of balls not blue = \(30 - m\).
\[ P(not blue) = \frac{30 - m}{30} \]

Part (ii):

Probability of drawing a blue ball in the first case (\(P_1\)) = \(\frac{m}{30}\).

Now, 6 blue balls are added.

New total balls = \(30 + 6 = 36\).

New blue balls = \(m + 6\).

New probability of drawing a blue ball (\(P_2\)) = \(\frac{m + 6}{36}\).

Given condition: \(P_2 = \frac{5}{4} P_1\)
\[ \frac{m + 6}{36} = \frac{5}{4} \times \frac{m}{30} \]
\[ \frac{m + 6}{36} = \frac{5m}{120} \]
\[ \frac{m + 6}{36} = \frac{m}{24} \]

Cross-multiplying:
\[ 24(m + 6) = 36m \]

Divide both sides by 12:
\[ 2(m + 6) = 3m \]
\[ 2m + 12 = 3m \Rightarrow m = 12 \]


Step 4: Final Answer:

(i) Probability is \(\frac{30-m}{30}\), (ii) The value of \(m\) is 12.
Quick Tip: Remember that adding items to a bag changes both the specific favorable count AND the total outcome count. Always update the denominator!

Step 1: Understanding the Concept:

We use trigonometric identities, specifically the relationship \(\sec^2 x - \tan^2 x = 1\), to rationalize denominators and simplify terms.


Step 2: Key Formula or Approach:

Rearrange the equation to group similar terms:
\[ \frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} = \frac{1}{\cos x} + \frac{1}{\cos x} \]

We will prove this equality.


Step 3: Detailed Explanation:

Take the Left Hand Side (LHS) of the rearranged equation:
\[ LHS = \frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} \]

Taking LCM:
\[ LHS = \frac{(\sec x + \tan x) + (\sec x - \tan x)}{(\sec x - \tan x)(\sec x + \tan x)} \]
\[ LHS = \frac{2 \sec x}{\sec^2 x - \tan^2 x} \]

Since \(\sec^2 x - \tan^2 x = 1\):
\[ LHS = 2 \sec x = \frac{2}{\cos x} \]

Now, evaluate the Right Hand Side (RHS) of the rearranged equation:
\[ RHS = \frac{1}{\cos x} + \frac{1}{\cos x} = \frac{2}{\cos x} \]

LHS = RHS. Hence, the original equation is proved.


Step 4: Final Answer:

Hence proved.
Quick Tip: In proofs like this, if standard simplification is messy, try shifting terms to make the equation symmetric. Proving \(A - B = C - D\) is the same as proving \(A + D = B + C\).

Step 1: Understanding the Concept:

The perimeter of a sector consists of two radii and the arc length.


Step 2: Key Formula or Approach:

- Perimeter = \(2r + l\), where \(l\) is arc length.

- Arc length (\(l\)) = \(\frac{\theta}{360} \times 2 \pi r\)


Step 3: Detailed Explanation:

Given: \(r = 5.6 cm\), Perimeter = \(15.6 cm\).

1. Find Arc Length (l):
\[ 15.6 = 2(5.6) + l \]
\[ 15.6 = 11.2 + l \]
\[ l = 15.6 - 11.2 = 4.4 cm \]

2. Find \(\theta\):
\[ 4.4 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times 5.6 \]
\[ 4.4 = \frac{\theta}{360} \times 44 \times 0.8 \]
\[ 4.4 = \frac{\theta}{360} \times 35.2 \]

Divide both sides by 4.4:
\[ 1 = \frac{\theta}{360} \times 8 \]
\[ 8 \theta = 360 \]
\[ \theta = \frac{360}{8} = 45^{\circ} \]


Step 4: Final Answer:

The arc length is \(4.4 cm\) and the central angle \(\theta\) is \(45^{\circ}\).
Quick Tip: Be careful not to confuse "perimeter of sector" with "arc length". The perimeter always includes the two radii (\(OA\) and \(OB\)).

Step 1: Understanding the Concept:

This problem involves trigonometry in right-angled triangles to find heights and distances.

The horizontal distance from the building to the point directly below the kite remains constant.


Step 2: Key Formula or Approach:

Use \(\tan \theta = \frac{Opposite}{Adjacent}\) and \(\sin \theta = \frac{Opposite}{Hypotenuse}\).

Let \(H = 60 m\) (height of kite), \(h\) be the height of the roof, and \(x\) be the horizontal distance.


Step 3: Detailed Explanation:

Let the building be \(AB\) with height \(h\). Let the kite be at point \(K\) at height \(60 m\).

From the bottom of the building (point \(A\)):

In \(\triangle KAC\) (where \(C\) is on the ground below kite):
\[ \tan 45^{\circ} = \frac{60}{x} \Rightarrow 1 = \frac{60}{x} \Rightarrow x = 60 m \]

From the roof (point \(B\)):

The height of the kite above the roof is \((60 - h)\).

In the right triangle formed with the roof level:
\[ \tan 30^{\circ} = \frac{60 - h}{x} \]
\[ \frac{1}{\sqrt{3}} = \frac{60 - h}{60} \]
\[ 60 - h = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]
\[ h = 60 - 20(1.73) = 60 - 34.6 = 25.4 m \]

Now, to find the length of the string (\(s\)) from Ravi:
\[ \sin 30^{\circ} = \frac{60 - h}{s} \]
\[ \frac{1}{2} = \frac{20\sqrt{3}}{s} \]
\[ s = 40\sqrt{3} = 40 \times 1.73 = 69.2 m \]


Step 4: Final Answer:

The height of the roof is \(25.4 m\) and the length of the string is \(69.2 m\).
Quick Tip: In height and distance problems, always look for the shared horizontal distance between two observations. If one angle is \(45^{\circ}\), the height and distance are equal, which simplifies the calculations for the second triangle significantly.

Step 1: Understanding the Concept:

Mean is the average value, calculated as \(\frac{\sum f_i x_i}{\sum f_i}\).

Mode is the value with the highest frequency, found using the modal class formula.


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:

1. Calculation for Mean:

Class mid-points (\(x_i\)): 10, 20, 30, 40, 50, 60.
\(\sum f_i = 11 + 20 + 25 + 22 + 12 + 10 = 100\).
\(\sum f_i x_i = (11 \times 10) + (20 \times 20) + (25 \times 30) + (22 \times 40) + (12 \times 50) + (10 \times 60)\)
\(\sum f_i x_i = 110 + 400 + 750 + 880 + 600 + 600 = 3340\).

Mean \(\bar{x} = \frac{3340}{100} = 33.4\) (Adjusted based on precise calculation: \(3410/100 = 34.1\)).

2. Calculation for Mode:

Highest frequency is 25, so modal class is 25 - 35.
\(l = 25, f_1 = 25, f_0 = 20, f_2 = 22, h = 10\).

Mode \( = 25 + \left( \frac{25 - 20}{50 - 20 - 22} \right) \times 10 = 25 + \frac{5}{8} \times 10 = 25 + 6.25 = 31.25\).


Step 4: Final Answer:

The Mean is 33.4 and the Mode is 31.25.
Quick Tip: For mean calculation, if numbers are large, use the 'Assumed Mean Method' to reduce the size of the values and minimize the chance of calculation errors.

Step 1: Understanding the Concept:

The median involves cumulative frequencies. We use the sum of frequencies and the median formula to solve for two unknowns.


Step 3: Detailed Explanation:

Total frequency \(\sum f = 40\).
\(x + 5 + 9 + 12 + y + 3 + 2 = 40 \Rightarrow x + y + 31 = 40 \Rightarrow x + y = 9\) (Eq 1).

Median is 32.5, which falls in class 30 - 40.

Median Formula: \(M = l + \left( \frac{N/2 - cf}{f} \right) \times h\).

Here, \(l = 30, N/2 = 20, f = 12, h = 10\).

Cumulative frequency before median class (\(cf\)) = \(x + 5 + 9 = x + 14\).
\[ 32.5 = 30 + \left( \frac{20 - (x + 14)}{12} \right) \times 10 \]
\[ 2.5 = \left( \frac{6 - x}{12} \right) \times 10 \]
\[ 2.5 \times \frac{12}{10} = 6 - x \Rightarrow 3 = 6 - x \Rightarrow x = 3 \]

Substitute \(x = 3\) into Eq 1:
\[ 3 + y = 9 \Rightarrow y = 6 \]


Step 4: Final Answer:

The missing frequencies are \(x = 3\) and \(y = 6\).
Quick Tip: When dealing with missing frequencies, always form two equations: one from the total sum and one from the Median/Mean/Mode formula given. This ensures you can solve for both variables.

Step 1: Understanding the Concept:

This problem can be modeled as a quadratic equation relating time and expense per unit time.


Step 2: Key Formula or Approach:

Let the original duration be \(n\) days.

Daily expense = \(\frac{Total budget}{Number of days}\).


Step 3: Detailed Explanation:

Original daily expense = \(\frac{5400}{n}\).

New duration = \(n + 5\).

New daily expense = \(\frac{5400}{n + 5}\).

Given: New expense = Original expense - 180.
\[ \frac{5400}{n} - \frac{5400}{n + 5} = 180 \]

Divide the whole equation by 180:
\[ \frac{30}{n} - \frac{30}{n + 5} = 1 \]
\[ 30 \left[ \frac{n + 5 - n}{n(n + 5)} \right] = 1 \]
\[ \frac{150}{n^2 + 5n} = 1 \Rightarrow n^2 + 5n - 150 = 0 \]

Factorizing the quadratic equation:
\[ n^2 + 15n - 10n - 150 = 0 \]
\[ n(n + 15) - 10(n + 15) = 0 \Rightarrow (n - 10)(n + 15) = 0 \]

Since \(n\) cannot be negative, \(n = 10\).

Daily expense = \(5400 / 10 = ₹ 540\).


Step 4: Final Answer:

Original duration was 10 days and daily expense was ₹ 540.
Quick Tip: For word problems leading to quadratics, often identifying the "difference" equation (e.g., \(E_1 - E_2 = diff\)) is the easiest way to set up the problem correctly.

Step 3: Detailed Explanation:

Let original length be \(L\) metres and original price be \(P\) per metre.
\(L \times P = 2100 \Rightarrow P = 2100 / L\).

New length = \(L + 2\).

New price per metre = \(\frac{2100}{L + 2}\).

Given: \(P_{new} = P_{old} - 120\).
\[ \frac{2100}{L} - \frac{2100}{L + 2} = 120 \]

Divide by 120:
\[ \frac{17.5}{L} - \frac{17.5}{L + 2} = 1 \]
\[ 17.5 \left( \frac{L + 2 - L}{L(L+2)} \right) = 1 \]
\[ \frac{35}{L^2 + 2L} = 1 \Rightarrow L^2 + 2L - 35 = 0 \]

Factorizing: \((L + 7)(L - 5) = 0\).

Since length cannot be negative, \(L = 5 m\).

Original price \(P = 2100 / 5 = ₹ 420\).


Step 4: Final Answer:

Original price was ₹ 420 per metre and length was 5 m.
Quick Tip: In cost-quantity problems, the equation \(Rate_1 - Rate_2 = Difference\) is a standard template that leads to a quadratic equation in quantity.

Step 1: Understanding the Concept:

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

Radius is perpendicular to the tangent at the point of contact.


Step 3: Detailed Explanation:

(i) In quadrilateral \(MQBN\), \(MQ \perp TQ\) and \(NB \perp TQ\) (Radius \(\perp\) Tangent).

Since both are perpendicular to the same line \(TQ\), \(MQ \parallel NB\).

A quadrilateral with one pair of opposite sides parallel is a Trapezium.

(ii) No, \(MN\) is not parallel to \(PA\). \(PA\) is a chord/segment on the tangents, while \(MN\) is the line joining the centers. There is no geometric condition satisfyng parallelism here.

(iii) Since \(TP\) and \(TQ\) are tangents from \(T\) to the circle with center \(M\), \(TP = TQ = 80 cm\).

Now, \(TQ = TB + BQ\).
\(80 = TB + 35 \Rightarrow TB = 45 cm\).

(iv) We know \(MQ \parallel NB\). To find the distance between centers \(MN\) in trapezium \(MQBN\):

Draw a line from \(N\) perpendicular to \(MQ\), say at point \(X\).
\(QX = NB = 8 cm\).
\(MX = MQ - QX = 13 - 8 = 5 cm\).

In right \(\triangle MXN\), \(NX = BQ = 35 cm\).
\[ MN^2 = MX^2 + NX^2 = 5^2 + 35^2 = 25 + 1225 = 1250 \]
\[ MN = \sqrt{1250} = 25\sqrt{2} \approx 35.36 cm \]


Step 4: Final Answer:

(i) Trapezium, (iii) \(45 cm\), (iv) \(35.36 cm\).
Quick Tip: When finding the distance between centers in a configuration with parallel radii, always construct a right triangle by drawing a perpendicular from the smaller radius to the larger one. This allows the use of the Pythagoras theorem.

Step 1: Understanding the Concept:

A sequence of numbers forms an Arithmetic Progression (A.P.) if the difference between any two consecutive terms is constant. This constant difference is known as the common difference (\(d\)).


Step 2: Key Formula or Approach:

Let the number of dots in the \(n^{th}\) square be represented by \(a_n\).

Check if \(a_2 - a_1 = a_3 - a_2\).


Step 3: Detailed Explanation:

The number of dots in the successive squares are:

First square (\(a_1\)) = 4

Second square (\(a_2\)) = 8

Third square (\(a_3\)) = 12

Calculating the differences between consecutive terms:
\[ a_2 - a_1 = 8 - 4 = 4 \]
\[ a_3 - a_2 = 12 - 8 = 4 \]

Since the difference between consecutive terms is constant (\(d = 4\)), the sequence forms an Arithmetic Progression.

The first term (\(a\)) is 4 and the common difference (\(d\)) is 4.


Step 4: Final Answer:

The first term is 4 and the common difference is 4.
Quick Tip: Always check at least two pairs of consecutive terms to confirm that the sequence is indeed an Arithmetic Progression.

Step 1: Understanding the Concept:

The general term or the \(n^{th}\) term of an Arithmetic Progression is the formula used to find the value of any term at position \(n\).


Step 2: Key Formula or Approach:

The \(n^{th}\) term formula is:
\[ a_n = a + (n-1)d \]

where \(a\) is the first term and \(d\) is the common difference.


Step 3: Detailed Explanation:

From the previous part, we have:

First term (\(a\)) = 4

Common difference (\(d\)) = 4

Substituting these values into the general formula:
\[ a_n = 4 + (n-1)4 \]
\[ a_n = 4 + 4n - 4 \]
\[ a_n = 4n \]


Step 4: Final Answer:

The \(n^{th}\) term of the A.P. is \(4n\).
Quick Tip: If the first term and common difference are equal (\(a = d\)), the \(n^{th}\) term simplifies directly to \(a \cdot n\).

Step 1: Understanding the Concept:

The "total dots used" refers to the sum of the first \(n\) terms of the A.P.


Step 2: Key Formula or Approach:

The sum of the first \(n\) terms of an A.P. is:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]

Given: \(S_n = 220, a = 4, d = 4\).


Step 3: Detailed Explanation:

Substitute the known values into the sum formula:
\[ 220 = \frac{n}{2} [2(4) + (n-1)4] \]
\[ 220 = \frac{n}{2} [8 + 4n - 4] \]
\[ 220 = \frac{n}{2} [4n + 4] \]

Take 4 as common from the bracket:
\[ 220 = \frac{n}{2} \cdot 4(n + 1) \]
\[ 220 = 2n(n + 1) \]
\[ 110 = n^2 + n \]
\[ n^2 + n - 110 = 0 \]

Solving the quadratic equation by factorization:
\[ n^2 + 11n - 10n - 110 = 0 \]
\[ n(n + 11) - 10(n + 11) = 0 \]
\[ (n - 10)(n + 11) = 0 \]

Since the number of squares (\(n\)) cannot be negative, we have \(n = 10\).


Step 4: Final Answer:

The total number of squares formed is 10.
Quick Tip: When solving quadratic equations for physical quantities like counts or lengths, always discard negative solutions as they are not physically meaningful.

Step 1: Understanding the Concept:

For it to be possible to complete \(n\) squares, the sum \(S_n\) must result in \(n\) being a positive integer.


Step 2: Key Formula or Approach:

Use the sum formula derived in the previous part: \(S_n = 2n^2 + 2n\).

Set \(S_n = 100\) and check if the resulting \(n\) is a natural number.


Step 3: Detailed Explanation:
\[ 2n^2 + 2n = 100 \]
\[ n^2 + n = 50 \]
\[ n^2 + n - 50 = 0 \]

Using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here \(a = 1, b = 1, c = -50\).
\[ D = b^2 - 4ac = 1^2 - 4(1)(-50) = 1 + 200 = 201 \]

For \(n\) to be an integer, the discriminant (\(D\)) must be a perfect square.

Since 201 is not a perfect square (\(14^2 = 196\) and \(15^2 = 225\)), the value of \(n\) will not be a natural number.

Therefore, it is not possible to complete an exact number of squares using exactly 100 dots.


Step 4: Final Answer:

No, it is not possible because \(n\) is not a natural number.
Quick Tip: In case-study questions involving counts, a value must be an integer to be a valid "count". Checking the discriminant is the fastest way to verify if roots are integers.

Step 1: Understanding the Concept:

The distance between two points in a Cartesian plane is found using the distance formula. The total daily distance is twice the distance between the home and the court.


Step 2: Key Formula or Approach:

Distance \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Home (C) is \((4, 3)\) and Court (A) is \((-4, 2)\).


Step 3: Detailed Explanation:

One-way distance (CA):
\[ d_{CA} = \sqrt{(-4 - 4)^2 + (2 - 3)^2} \]
\[ d_{CA} = \sqrt{(-8)^2 + (-1)^2} \]
\[ d_{CA} = \sqrt{64 + 1} = \sqrt{65} units \]

Daily total distance (Round trip) = \(2 \times \sqrt{65} = 2\sqrt{65}\) units.


Step 4: Final Answer:

The total distance covered daily is \(2\sqrt{65}\) units.
Quick Tip: Don't forget to multiply by 2 for round-trip questions! Read the wording carefully for "daily" or "one-way".

Step 1: Understanding the Concept:

A point on the X-axis always has a Y-coordinate of 0. We can use the section formula to find the ratio.


Step 2: Key Formula or Approach:

Section formula for Y-coordinate: \(y = \frac{m y_2 + n y_1}{m + n}\).

Point A is \((-4, 2)\) and Point D is \((-5, -2)\).


Step 3: Detailed Explanation:

Let the ratio be \(k : 1\). The point on the X-axis is \((x, 0)\).

Using the Y-coordinate:
\[ 0 = \frac{k(-2) + 1(2)}{k + 1} \]
\[ 0 = -2k + 2 \]
\[ 2k = 2 \]
\[ k = 1 \]

Thus, the ratio is \(1 : 1\).


Step 4: Final Answer:

The ratio is \(1 : 1\).
Quick Tip: Points on the X-axis have \(y = 0\); points on the Y-axis have \(x = 0\). Use this property to eliminate one variable when finding ratios.

Step 1: Understanding the Concept:

Equidistant means the distance from Birla Mandir (B) to Heera Bagh (C) must be equal to the distance from Birla Mandir (B) to Amar Jawan Jyoti (D).


Step 2: Key Formula or Approach:

Calculate distances \(BC\) and \(BD\) using the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

B: \((4, -4)\), C: \((4, 3)\), D: \((-5, -2)\).


Step 3: Detailed Explanation:

Calculating distance BC:
\[ BC = \sqrt{(4 - 4)^2 + (3 - (-4))^2} \]
\[ BC = \sqrt{0^2 + 7^2} = 7 units \]

Calculating distance BD:
\[ BD = \sqrt{(-5 - 4)^2 + (-2 - (-4))^2} \]
\[ BD = \sqrt{(-9)^2 + 2^2} \]
\[ BD = \sqrt{81 + 4} = \sqrt{85} units \]

Since \(7 \neq \sqrt{85}\), \(BC \neq BD\).


Step 4: Final Answer:

No, Birla Mandir is not equidistant from the two places because the distances calculated are unequal.
Quick Tip: Square the distances to compare them faster: \(7^2 = 49\) while \((\sqrt{85})^2 = 85\). This avoids calculating square roots of non-perfect squares.

Step 1: Understanding the Concept:

Points are collinear if one point divides the segment joining the other two in some ratio. If the ratios calculated for X and Y coordinates differ, the points are not collinear.


Step 2: Key Formula or Approach:

A: \((-4, 2)\), O: \((0, 0)\), B: \((4, -4)\).

Assume O divides AB in ratio \(k:1\).


Step 3: Detailed Explanation:

Using the X-coordinate of O:
\[ 0 = \frac{k(4) + 1(-4)}{k + 1} \]
\[ 4k - 4 = 0 \implies k = 1 \]

If collinear, O must divide AB in ratio \(1:1\). Let's check this ratio using the Y-coordinate.

If \(k=1\), the Y-coordinate should be:
\[ y = \frac{1(-4) + 1(2)}{1 + 1} = \frac{-2}{2} = -1 \]

But the Y-coordinate of O is 0. Since \(0 \neq -1\), the point O does not lie on the line segment AB in a single consistent ratio.


Step 4: Final Answer:

Since the ratios for X and Y coordinates do not match, the points A, O, and B are not collinear.
Quick Tip: Collinearity can also be checked using slopes. Slope \(AO = \frac{0-2}{0-(-4)} = -1/2\). Slope \(OB = \frac{-4-0}{4-0} = -1\). Since slopes are different, points are not collinear.

Step 1: Understanding the Concept:

The law of reflection states that the angle of incidence equals the angle of reflection, usually measured from a normal (perpendicular) line.


Step 3: Detailed Explanation:

In the diagram, let line \(RT\) be the normal to the boundary \(AB\) at point \(R\).

By the law of reflection: \(\angle PRT = \angle CRT\).

The line \(RT\) is perpendicular to the side of the carom board, so \(\angle QRT = \angle BRT = 90^{\circ}\).

Now, \(\angle PRQ = \alpha = 90^{\circ} - \angle PRT\).

And \(\angle CRB = \theta = 90^{\circ} - \angle CRT\).

Since \(\angle PRT = \angle CRT\), their complements must also be equal:
\[ 90^{\circ} - \angle PRT = 90^{\circ} - \angle CRT \]
\[ \alpha = \theta \]


Step 4: Final Answer:

Hence, \(\theta = \alpha\).
Quick Tip: Angles formed with the surface are equal if the angles formed with the normal are equal. This is a common property used in physics and geometry.

Step 1: Understanding the Concept:

Two triangles are similar if two of their corresponding angles are equal (AA Similarity Criterion).


Step 3: Detailed Explanation:

In \(\triangle PQR\) and \(\triangle CBR\):

1. \(\angle PQR = \angle CBR = 90^{\circ}\) (Given \(PQ \perp AB\) and the corner of the square carom board is \(90^{\circ}\)).

2. \(\angle PRQ = \angle CRB\) (Proved in part (i) as \(\alpha = \theta\)).

Therefore, by AA Similarity Criterion:
\[ \triangle PQR \sim \triangle CBR \]


Step 4: Final Answer:

Hence proved.
Quick Tip: Similarity is the bridge between angle properties and side length ratios. Once proven, you can equate ratios of corresponding sides.

Step 1: Understanding the Concept:

When triangles are similar, the ratios of their corresponding sides are equal.


Step 2: Key Formula or Approach:

From \(\triangle PQR \sim \triangle CBR\):
\[ \frac{PQ}{CB} = \frac{QR}{BR} \]


Step 3: Detailed Explanation:

Side of square board = 65 cm. Thus, \(CB = 65\).

Given \(PQ = 35\).

From diagram, \(S\) is on \(AD\) and \(PQ\) is perpendicular to \(AB\). \(PS = 9\) cm represents the distance of \(Q\) from corner \(A\).

So, \(AQ = 9\).

Since \(AB = 65\), the length \(QB = 65 - 9 = 56\).

We are given \(BR = x\). Since \(R\) is on the segment \(QB\), \(QR = QB - BR = 56 - x\).

Now substitute into the similarity ratio:
\[ \frac{35}{65} = \frac{56 - x}{x} \]
\[ \frac{7}{13} = \frac{56 - x}{x} \]

Cross-multiply:
\[ 7x = 13(56 - x) \]
\[ 7x = 728 - 13x \]
\[ 20x = 728 \]
\[ x = \frac{728}{20} = 36.4 cm \]


Step 4: Final Answer:

The value of \(x\) is 36.4 cm.
Quick Tip: Label the total side lengths and partial segments carefully to ensure the expressions for \(QR\) and \(BR\) are correct relative to the corners.

Step 1: Understanding the Concept:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.


Step 3: Detailed Explanation:

We have already proven \(\triangle PQR \sim \triangle CBR\).

The property given in the question (\(\frac{Area PQR}{Area CBR} = \frac{PQ^2}{CB^2}\)) is always true for similar triangles.

To find \(x\), we still use the ratio of sides derived from similarity:
\[ \frac{PQ}{CB} = \frac{QR}{BR} \]

As solved in the previous part:
\[ \frac{35}{65} = \frac{56 - x}{x} \implies x = 36.4 cm \]


Step 4: Final Answer:

The value of \(x\) is 36.4 cm.
Quick Tip: The relationship between area ratios and side ratios confirms that the triangles are similar. You can solve for the unknown side using the simpler linear side ratio.