✅ ANSWER: A (960)

📐 EXPLANATION:

LCM Calculation: Prime Factorization

✅ ANSWER: D (neither prime nor composite)

📐 EXPLANATION: A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 has only one factor (itself), so it is neither prime nor composite.

✅ ANSWER: B (5)

📐 EXPLANATION: Any power of 5 ends with digit 5.
Check: \(5^1=5\),
\(5^2=25\),
\(5^3=125\), etc.
All end with 5.

✅ ANSWER: C (2)

📐 EXPLANATION: The zeroes of a function are the x-coordinates where the graph intersects the x-axis. From the given graph (parabola), it crosses the x-axis at two distinct points, so there are 2 distinct zeroes.

✅ ANSWER: B (\(\frac{2}{3}\))

📐 EXPLANATION: For quadratic \(px^2 – 2x + 3p\):
Sum of zeroes \(\alpha + \beta = \frac{-(-2)}{p} = \frac{2}{p}\)
Product of zeroes \(\alpha \beta = \frac{3p}{p} = 3\)
Given \(\alpha + \beta = \alpha \beta\) ⇒ \(\frac{2}{p} = 3\) ⇒ \(p = \frac{2}{3}\)

✅ ANSWER: D (\(\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}\))

📐 EXPLANATION: For a pair of linear equations to be consistent and dependent (coincident lines), the ratios of coefficients must be equal: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

✅ ANSWER: D (\(1^{2},3^{2},5^{2},7^{2},\dots\))

📐 EXPLANATION: Check common difference:
(A) \(d = \frac{5}{2}-2 = \frac{1}{2}\) constant → AP
(B) \(d = -3.2-(-1.2) = -2\) constant → AP
(C) \(\sqrt{8}-\sqrt{2}\)\( = 2\sqrt{2}-\sqrt{2} \)\(= \sqrt{2}\), \(\sqrt{18}-\sqrt{8} \)\(= 3\sqrt{2}-2\sqrt{2}\)\( = \sqrt{2}\) constant → AP
(D) \(1,9,25,49\) differences: 8,16,24 → not constant → not AP

✅ ANSWER: ANSWER: C (QR:QP)

📐 EXPLANATION:

Triangle Similarity Problem

Step 1: Establish Similarity

In \(\triangle ABC\) and \(\triangle QRP\), we are given:

• \(\angle A = \angle Q\)
• \(\angle B = \angle R\)

According to the AA (Angle-Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Therefore:

$$\triangle ABC \sim \triangle QRP$$

Step 2: Determine Corresponding Sides

When two triangles are similar, the ratios of their corresponding sides are equal. Based on the similarity \(\triangle ABC \sim \triangle QRP\), the corresponding sides are:

• \(AB\) corresponds to \(QR\)
• \(BC\) corresponds to \(RP\)
• \(AC\) corresponds to \(QP\)

Setting up the ratio for the sides mentioned in the question:

$$\frac{AB}{QR} = \frac{AC}{QP}$$

Step 3: Solve for the Required Ratio

To find what \(AB : AC\) is equal to, we rearrange the proportion:

$$\frac{AB}{AC} = \frac{QR}{QP}$$

In ratio form, this is \(AB : AC = QR : QP\).

✅ CORRECT ANSWER: C (QR:QP)

✅ ANSWER: A (3a)

📐 EXPLANATION: Distance from x-axis = |y-coordinate| = |3a| = 3a (distance is always positive).

✅ ANSWER: B (\(\frac{3}{4}\))

📐 EXPLANATION: Using identity \(\sin^2 A + \cos^2 A = 1\): \(\sin^2 A = 1 – \left(\frac{4}{5}\right)^2\)\( = 1 – \frac{16}{25}\)\( = \frac{9}{25}\) \(\sin A = \frac{3}{5}\) (positive, assuming acute angle) \(\tan A = \frac{\sin A}{\cos A} = \frac{3/5}{4/5} = \frac{3}{4}\)

✅ ANSWER: B (\(\frac{4}{\sqrt{3}}\))

📐 EXPLANATION:
2 sin A = 1 ⇒ sin A = \(\frac{1}{2}\) ⇒ A = 30° (or 150°, but take acute) cos 30° = \(\frac{\sqrt{3}}{2}\)
tan 30° = \(\frac{1}{\sqrt{3}}\),
cot 30° = \(\sqrt{3}\)
tan A + cot A
= \(\frac{1}{\sqrt{3}} + \sqrt{3}
= \frac{1 + 3}{\sqrt{3}}
= \frac{4}{\sqrt{3}}\)

✅ ANSWER: C (60)

📐 EXPLANATION:

Height of a Vertical Tower

Step 1: Identify Given Information

The problem describes a right-angled triangle formed by:

• The vertical tower (the opposite side, $$h$$).
• The distance from the foot of the tower (the adjacent side), which is $$60 \text{ m}$$.
• The angle of elevation to the top of the tower, which is $$45^\circ$$.

Step 2: Use Trigonometric Ratios

We use the tangent ratio, which relates the opposite side to the adjacent side:

$$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

Substitute the given values:

$$\tan(45^\circ) = \frac{h}{60}$$

Step 3: Solve for Height

Since $$\tan(45^\circ) = 1$$, the equation becomes:

$$1 = \frac{h}{60}$$
$$h = 60 \text{ m}$$

Answer:

The height of the tower is $$60 \text{ m}$$.

✅ ANSWER: A (30°)

📐 EXPLANATION:

In triangle OAB,
OA = OB (radii),
so it’s isosceles. ∠OAB = 15°
⇒ ∠OBA = 15°
∠AOB = 180° – 15° – 15° = 150°
In quadrilateral OAPB, ∠OAP = ∠OBP = 90° (radius ⊥ tangent)
∠APB = 360° – 90° – 90° – 150° = 30°

✅ ANSWER: B (50°)

📐 EXPLANATION: In quadrilateral OAPB, ∠OAP = ∠OBP = 90° (radius ⊥ tangent) ∠AOB = 130° ∠APB = 360° – 90° – 90° – 130° = 50°

✅ ANSWER: C (\(\frac{\pi r^2}{6} – \frac{\sqrt{3}}{4}r^2\))

📐 EXPLANATION: Area of sector with angle θ = \(\frac{\theta}{360°} \times \pi r^2\) For θ = 60°, sector area = \(\frac{60}{360} \pi r^2 = \frac{\pi r^2}{6}\) Area of equilateral triangle formed by radii and chord = \(\frac{\sqrt{3}}{4}r^2\) Area of segment = sector area – triangle area = \(\frac{\pi r^2}{6} – \frac{\sqrt{3}}{4}r^2\)

✅ ANSWER: D (\(\frac{364}{3} \pi\))

📐 EXPLANATION: Outer radius R = 6 cm, inner radius r = 6 – 1 = 5 cm (thickness 1 cm) Volume of hemisphere = \(\frac{2}{3}\pi r^3\) Volume of steel = outer volume – inner volume \( = \frac{2}{3}\pi (6^3) – \frac{2}{3}\pi (5^3)\) \(= \frac{2}{3}\pi (216 – 125)\) \(= \frac{2}{3}\pi \times 91 = \frac{182}{3}\pi\) cm³ Wait, 2/3 × 91 = 182/3, so answer is \(\frac{182}{3}\pi\) → option B.

✅ CORRECT ANSWER: B (\(\frac{182}{3} \pi\))

✅ ANSWER: C (44.2)

📐 EXPLANATION: Using empirical formula: Mode = 3(Median) – 2(Mean) Mode = 3(43.4) – 2(43) = 130.2 – 86 = 44.2

✅ ANSWER: A (\(\frac{15}{25}\))

📐 EXPLANATION: Numbers from 1 to 25: total 25 numbers. Composite numbers (not prime and not 1): 4,6,8,9,10,12,14,15,16,18,20,21,22,24,25 → 15 numbers. Probability = \(\frac{15}{25}\)

✅ ANSWER: (d) A is false but R is true.

📐 EXPLANATION: Reason (R) is the correct formula for surface area of cuboid: \(2(lb + bh + hl)\) (missing factor 2 in given statement). The given R says \((lb + bh + hl)\) without the factor 2, which is incorrect. So R is false. Assertion (A): Two cubes of side 4 cm joined end-to-end form a cuboid of dimensions 8 cm × 4 cm × 4 cm. Surface area = \(2(8×4 + 4×4 + 4×8) \)\(= 2(32 + 16 + 32) \)\(= 2×80 = 160\) cm². So A is true. But since R is false, correct option is (c) A true, R false.

✅ CORRECT ANSWER: (c) A is true but R is false.

✅ ANSWER: (d) A is false but R is true.

📐 EXPLANATION: Reason (R) is correct: sum of first n natural numbers = \(\frac{n(n+1)}{2}\). Assertion (A): Mean = sum/n = \(\frac{n(n+1)}{2n} = \frac{n+1}{2}\), not \(\frac{n-1}{2}\). So A is false. Thus, A false, R true → option (d).