Mathematics (ਗਣਿਤ) - Questions 31 to 60
31. If in the given figure PS/SQ = PT/TR and ∠PST = ∠PRQ, then what type of triangle is ΔPQR?
(1) An isosceles triangle (ਇੱਕ ਸਮ ਦੋ ਭੁਜੀ ਤ੍ਰਿਭੁਜ)
(2) An equilateral triangle (ਇੱਕ ਸਮਭੁਜੀ ਤ੍ਰਿਭੁਜ)
(3) A scalene triangle (ਬਿਖਮ ਭੁਜੀ ਤ੍ਰਿਭੁਜ)
(4) None of these (ਇਹਨਾਂ ਵਿੱਚੋਂ ਕੋਈ ਨਹੀਂ)
Answer: (1)
Explanation: PS/SQ = PT/TR implies ST || QR (Converse of BPT). Thus ∠PST = ∠PQR (Corresponding angles). Since ∠PST = ∠PRQ (given), then ∠PQR = ∠PRQ. Equal angles mean equal opposite sides (PQ = PR), making it an isosceles triangle.
32. The area of the sector whose perimeter is four times its radius 'r' units, is:
ਉਸ ਅਰਧ ਵਿਆਸੀ ਖੰਡ ਦਾ ਖੇਤਰਫਲ, ਜਿਸ ਦਾ ਪਰਿਮਾਪ ਅਰਧ ਵਿਆਸ 'r' ਇਕਾਈ ਦਾ ਚਾਰ ਗੁਣਾ ਹੈ:
(1) r²/4 (2) 2r² (3) r² (4) r²/2
Answer: (3)
Explanation: Perimeter = l + 2r. Given l + 2r = 4r, so arc length l = 2r. Area = (1/2)lr = (1/2)(2r)(r) = r².
33. A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or 5?
(1) 13/25 (2) 21/50 (3) 12/25 (4) 23/50
Answer: (4)
Explanation: Multiples of 3 = 16. Multiples of 5 = 10. Multiples of both (15, 30, 45) = 3. Total = 16 + 10 - 3 = 23. Probability = 23/50.
34. A card is drawn from a pack of 52. The probability that the drawn card is not an ace is:
(1) 12/13 (2) 9/13 (3) 4/13 (4) 1/13
Answer: (1)
Explanation: Total cards = 52. Aces = 4. Non-aces = 48. P(not ace) = 48/52 = 12/13.
35. How many questions were in the test? (Sandeep scored 40 marks with 3 for right/-1 for wrong; would have scored 50 with 4 for right/-2 for wrong).
(1) 10 (2) 20 (3) 15 (4) 18
Answer: (2)
Explanation: 3x - y = 40 and 4x - 2y = 50. Solving gives x=15 (right) and y=5 (wrong). Total = 15 + 5 = 20.
36. If in two triangles ABC and PQR, AB/QR = BC/PR = CA/PQ, then:
(1) ΔPQR ~ ΔCAB (2) ΔPQR ~ ΔABC (3) ΔCBA ~ ΔPQR (4) ΔBCA ~ ΔPQR
Answer: (1)
Explanation: By matching corresponding sides: AB corresponds to QR, BC to RP, and CA to PQ. This aligns the vertices as A→Q, B→R, C→P.
37. cot²A - [1 / (1 - cos²A)] =
(1) 2 (2) -1 (3) 1 (4) 0
Answer: (2)
Explanation: 1 - cos²A = sin²A. Expression becomes cot²A - cosec²A. Since 1 + cot²A = cosec²A, cot²A - cosec²A = -1.
38. If ax + by = a² - b² and bx + ay = 0, then the value of (x + y) is:
(1) a - b (2) b - a (3) a + b (4) 0
Answer: (1)
Explanation: Adding both equations: (a+b)x + (a+b)y = a² - b². (a+b)(x+y) = (a-b)(a+b). Dividing by (a+b) gives x + y = a - b.
39. If the sum of zeros of p(x) = (k² - 14)x² - 2x - 4 is 1, then the value of k is:
(1) ±√18 (2) ±4 (3) ±2 (4) ±9
Answer: (2)
Explanation: Sum of zeros = -b/a = 2 / (k² - 14). Given sum = 1, so k² - 14 = 2 → k² = 16 → k = ±4.
40. Which of the following is rational?
ਹੇਠਾਂ ਦਿੱਤੀਆਂ ਵਿੱਚੋਂ ਕਿਹੜਾ ਪਰਿਮੇਯ ਹੈ?
(1) √6 + √9 (2) √2 + √4 (3) √4 + √9 (4) √3 + √5
Answer: (3)
Explanation: √4 + √9 = 2 + 3 = 5, which is a rational number. Others contain square roots of non-perfect squares.
41. If sin A - cos A = 0, then the value of sin⁶A + cos⁶A is:
(1) 2/3 (2) 1/3 (3) 3/4 (4) 1/4
Answer: (4)
Explanation: sin A = cos A means A = 45°. sin 45° = 1/√2. (1/√2)⁶ + (1/√2)⁶ = 1/8 + 1/8 = 2/8 = 1/4.
42. If the sum of the n terms of an AP is Sₙ = 3n² + 4n, then the common difference is:
(1) 5 (2) 6 (3) 7 (4) 8
Answer: (2)
Explanation: S₁ = 7, S₂ = 20. First term a = 7. Second term = 13. d = 13 - 7 = 6.
43. If the equation x² + 4x + k = 0 has real and distinct roots, then:
(1) k < 4 (2) k > 4 (3) k ≥ 4 (4) k ≤ 4
Answer: (1)
Explanation: Discriminant > 0 → 16 - 4k > 0 → k < 4.
44. A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has no solution if:
(1) a₁/a₂ ≠ b₁/b₂
(2) a₁/a₂ = b₁/b₂ ≠ c₁/c₂
(3) a₁/a₂ = b₁/b₂ = c₁/c₂
(4) None of these
Answer: (2)
Explanation: This condition represents parallel lines which never intersect.
45. If a and b are zeros of the polynomial p(x) = 4x² + 3x + 7, then 1/a + 1/b is equal to:
(1) 7/3 (2) -7/3 (3) 3/7 (4) -3/7
Answer: (4)
Explanation: 1/a + 1/b = (a+b)/ab = (-3/4)/(7/4) = -3/7.
46. The upper limit of the median class for the given frequency distribution is:
(1) 17 (2) 17.5 (3) 18 (4) 18.5
Answer: (2)
Explanation: N/2 = 28.5. Median class is 12-17. True upper limit is 17.5.
47. If for a data, Mean : Median = 9 : 8, then Median : Mode =
(1) 8:9 (2) 7:6 (3) 4:3 (4) 5:4
Answer: (3)
Explanation: Mode = 3 Median - 2 Mean → 6. Median : Mode = 8 : 6 = 4 : 3.
48. The maximum volume of a cone that can be carved out of a solid hemisphere of radius r is:
(1) πr³/3 (2) 3πr² (3) 3πr³ (4) πr²/3
Answer: (1)
Explanation: Volume = (1/3)πr²h = (1/3)πr³.
49. A solid is hemispherical at bottom and conical above. If surface areas are equal, the ratio of radius to height of conical part is:
(1) 1:√2 (2) √2:1 (3) 1:√3 (4) √3:1
Answer: (3)
Explanation: r:h = 1:√3.
50. In figure, the area of the segment PAQ is:
(1) a²/4(π+2) (2) a²/4(π-2) (3) a²/4(π-1) (4) a²/4(π+1)
Answer: (2)
Explanation: Area = (a²/4)(π - 2).
51. If the height of a pole is √3 times its shadow, the angle of elevation of the sun is:
(1) 30° (2) 45° (3) 60° (4) 75°
Answer: (3)
Explanation: tan θ = √3 → θ = 60°.
52. If tan² 45° - cos² 30° = x sin 45° cos 45°, then x =
(1) 2 (2) -2 (3) -1/2 (4) 1/2
Answer: (4)
Explanation: 1 - 3/4 = x(1/2) → x = 1/2.
53. If quadrilateral PQRS circumscribes a circle, then:
(1) x=95°, y=95° (2) x=100°, y=90° (3) x=100°, y=85° (4) x=85°, y=90°
Answer: (3)
Explanation: x = 100°, y = 85°.
54. If two tangents inclined at 60° are drawn to a circle of radius 3 cm, then length of each tangent is:
(1) 3√3/2 (2) 3√3 (3) 3 (4) 6
Answer: (2)
Explanation: Tangent = 3√3 cm.
55. If point P(2,1) lies on the line joining A(4,2) and B(8,4), then:
(1) AP = 1/2 AB (2) AP = BP (3) PB = 1/3 AB (4) AP = 1/3 AB
Answer: (1)
Explanation: AP is half of AB.
56. The perimeter of the triangle with vertices (0,4), (0,0) and (3,0) is:
(1) 20 (2) 12 (3) 6 (4) 16
Answer: (2)
Explanation: Perimeter = 12.
57. If 18, a, b, -3 are in AP, then a + b is:
(1) 19 (2) 7 (3) 11 (4) 15
Answer: (4)
Explanation: 18 + (-3) = a + b = 15.
58. If one root of ax² + bx + c = 0 is three times the other, then:
(1) b² = 16ac (2) b² = 3ac (3) 3b² = 16ac (4) 16b² = 3ac
Answer: (3)
Explanation: 3b² = 16ac.
59. The zeros of the quadratic polynomial p(x) = x² + 99x + 127 are:
(1) both positive (2) both negative (3) one pos, one neg (4) both equal
Answer: (2)
Explanation: Both roots are negative.
60. If HCF(x, 8) = 4 and LCM(x, 8) = 24, then x is:
(1) 8 (2) 10 (3) 12 (4) 14
Answer: (3)
Explanation: x = 12.
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