1. If the sum of the zeroes of the quadratic polynomial 3x²–5x+p is equal to the product of its zeroes, then the value of p is:
Solution
Correct: A
For ax²+bx+c, sum of zeroes = –b/a = 5/3, product = c/a = p/3. Given 5/3 = p/3 → p = 5.
2. The 10th term of the AP: 5, 9, 13, … is
Solution
Correct: A
a=5, d=4. a₁₀ = 5 + 9×4 = 5 + 36 = 41.
3. For what value of k will the system 2x+3y=7 and 4x+ky=14 have infinitely many solutions?
Solution
Correct: A
For infinitely many solutions, 2/4 = 3/k = 7/14 → 1/2 = 3/k → k = 6.
4. If the distance between the points (2, k) and (4, 5) is √8, then k equals
Solution
Correct: A
√[(4–2)²+(5–k)²]=√8 → 4+(5–k)²=8 → (5–k)²=4 → 5–k=±2 → k=3 or 7.
5. A sphere of radius 3 cm is melted and recast into a cylinder of same radius. The height of the cylinder is
Solution
Correct: A
Volume sphere = 4/3 π(3)³ = 36π. Volume cylinder = π(3)²h = 9πh. Equating, 9πh = 36π → h = 4 cm.
6. If sin θ = 3/5 and θ is acute, then cos θ equals
Solution
Correct: A
sin²θ + cos²θ = 1 → (3/5)² + cos²θ = 1 → cos²θ = 16/25 → cos θ = 4/5.
7. The probability of getting a prime number when a fair die is tossed once is
Solution
Correct: A
Primes on die: 2, 3, 5. Favourable = 3, total = 6. Probability = 3/6 = 1/2.
8. The quadratic equation x² + 4x + k = 0 has real roots if k satisfies
Solution
Correct: A
Discriminant D = 16–4k ≥ 0 → 4k ≤ 16 → k ≤ 4.
9. If the area of a triangle with vertices (0, 0), (6, 0) and (0, y) is 15 sq units, then y equals
Solution
Correct: A
Area = 1/2 |x₁(y₂–y₃)+...| = 1/2 |6y| = 15 → 3|y| = 15 → |y| = 5 → y = 5.
10. If the mean of 10 numbers is 15 and each number is multiplied by 3, the new mean is
Solution
Correct: A
New mean = 3 × old mean = 3 × 15 = 45.
11. The largest number that divides 70 and 125 leaving remainders 5 and 8 respectively is
Solution
Correct: A
Required number = HCF(70–5, 125–8) = HCF(65, 117) = 13.
12. If the roots of 3x²–10x+k=0 are reciprocals of each other, then k equals
Solution
Correct: A
Product of roots = k/3. If roots are reciprocals, product = 1. Thus k/3 = 1 → k = 3.
13. A 15 m ladder leaning against a wall makes 60° with the ground. The height it reaches on the wall is
Solution
Correct: A
height = 15 sin 60° = 15 × √3/2 = 7.5√3 m.
14. The ratio of the volume of a cone to the volume of a cylinder of same radius and height is
Solution
Correct: A
Vol cone = 1/3 πr²h, Vol cylinder = πr²h. Ratio = 1:3.
15. If the mode of the data 5, 6, 6, 8, x, 5, 7, 6 is 6, then x cannot be
Solution
Correct: B
Mode is 6, so 6 must occur most. If x = 5, then 5 occurs thrice and 6 occurs thrice, so mode not uniquely 6. Hence x cannot be 5.
16. The sum of first n odd natural numbers is 144. Then n equals
Solution
Correct: A
Sum = n² = 144 → n = 12.
17. If the angle of elevation of the sun is 45°, the length of the shadow of a 12 m high pole is
Solution
Correct: A
tan 45° = 12/shadow → 1 = 12/shadow → shadow = 12 m.
18. A card is drawn from a well-shuffled 52-card deck. The probability that it is a face card (Jack, Queen, King) is
Solution
Correct: A
Face cards = 12. Probability = 12/52 = 3/13.
19. For what value of p does the pair of linear equations 3x–4y=7 and px+2y=5 have a unique solution?
Solution
Correct: A
Unique if 3/p ≠ –4/2 → 3/p ≠ –2 → p ≠ –6.
20. The area of a circular ring with outer radius 5 cm and inner radius 3 cm is
Solution
Correct: A
Area = π(5²–3²) = π(25–9) = 16π cm².
21. If the polynomial x²–kx+6 has one zero 3, then the other zero is
Solution
Correct: A
Product of zeroes = 6. One zero = 3, so other = 6/3 = 2.
22. A frustum of a cone has height 6 cm and radii 3 cm and 5 cm. Its volume is
Solution
Correct: A
V = πh/3(R²+Rr+r²) = π×6/3(25+15+9) = 2π×49 = 98π cm³.
23. The 7th term from the end of the AP 7, 10, 13, …, 52 is
Solution
Correct: A
AP has 16 terms (a=7, d=3, a₁₆=52). 7th from end = 16–6 = 10th term = 7+9×3 = 34.
24. If tan A = 2, then the value of (sin A + cos A)/(sin A – cos A) is
Solution
Correct: A
Divide numerator and denominator by cos A: (tan A + 1)/(tan A – 1) = (2+1)/(2–1) = 3.
25. A two-digit number is 4 times the sum of its digits. If 9 is added to it, the digits reverse. The number is
Solution
Correct: A
Let number = 10a+b. 10a+b = 4(a+b) → 6a = 3b → b = 2a. Also 10a+b+9 = 10b+a → 9a–9b = –9 → a–b = –1. Solving gives a = 1, b = 2. Number = 12.
26. The median of the data 8, 6, 10, 12, 14, 16 is
Solution
Correct: A
Ordered data: 6, 8, 10, 12, 14, 16. Median = average of 3rd and 4th = (10+12)/2 = 11.
27. A bag contains 3 red and 5 black balls. The probability of drawing a red or black ball is
Solution
Correct: A
All balls are either red or black, so probability = 1.
28. If the circumference of a circle equals its area numerically, its radius is
Solution
Correct: A
2πr = πr² → r = 2.
29. The value of k for which the points (k, 2), (3, 4) and (5, 6) are collinear is
Solution
Correct: A
Area of triangle formed by three points must be zero. Solving determinant gives k = 1.
30. If x = 1 + √2, then the value of x + 1/x is
Solution
Correct: B
1/x = 1/(1+√2) = (√2–1)/((1+√2)(√2–1)) = √2–1. Thus x + 1/x = (1+√2) + (√2–1) = 2√2.
Discussion & Comments