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Question paper · UP TGT

UP TGT 2016 — Mathematics (Series A)

8 or 9 March 2019125 questions+3.4 for a correct answer, −0 for a wrong one

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Questions from the U.P. Secondary Education Services Selection Board’s paper of 8 or 9 March 2019. Each question’s answer is under it, hidden until you open it, and the full answer key is at the end. Take the paper as a timed practice test to be scored.

The official booklet is a scan with no text, so ExamBharo typed this paper from it by hand. A typing slip is possible; report one through the contact page.

Mathematics

1. If f(x) = cos²x/(1 + sin²x), then the value of f(π/4) − 3 f′(π/4) is
- (a) 0
- (b) 1
- (c) 3
- (d) 4
Show answer
Answer: (c)
2. If a⃗, b⃗, c⃗ are three non-coplanar vectors, such that [b⃗ × c⃗, c⃗ × a⃗, a⃗ × b⃗] = 2[a⃗, b⃗, c⃗], then the value of [a⃗, b⃗, c⃗] is equal to
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Show answer
Answer: (c)
3. Mohan is father of 3 children with atleast one boy. The probability that he has 2 boys and 1 girl is
- (a) 1/2
- (b) 1/3
- (c) 1/4
- (d) 2/3
Show answer
Answer: (b)
4. If sum of n terms of an arithmetical progression is 5n² − 3n, then its pᵗʰ term is
- (a) 10p + 8
- (b) 10p − 8
- (c) 10p + 3
- (d) 10p − 3
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Answer: (b)
5. If each force F₁⃗ + F₂⃗ and F₁⃗ − F₂⃗ has magnitude √(F₁² + F₂²), then the angle between F₁⃗ and F₂⃗ is
- (a) 90°
- (b) 60°
- (c) 45°
- (d) 0°
Show answer
Answer: (a)
6. Which of the following series converges ? I. Σ_(n=1)^∞ (3 + cos n)/eⁿ II. Σ_(n=1)^∞ cos(1/n)
- (a) I only
- (b) II only
- (c) I and II both
- (d) neither I nor II
Show answer
Answer: (a)
7. If three forces, acting at a point, be in equilibrium, then each force is proportional to the sine of the angle between the other two. This theorem is called
- (a) Law of triangle of forces
- (b) Law of parallelogram of forces
- (c) Lami’s theorem
- (d) Trigonometrical theorem
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Answer: (c)
8. A sum of money doubles itself in 4 years at compound interest. It will amount to 8 times to itself at the same rate of interest in time
- (a) 12 years
- (b) 16 years
- (c) 18 years
- (d) 24 years
Show answer
Answer: (a)
9. Which one of the following is true ?
- (a) mean = 3 median − 2 mode
- (b) median = 3 mode − 2 mean
- (c) mode = 2 median − 3 mean
- (d) mode = 3 median − 2 mean
Show answer
Answer: (d)
10. If the geometric mean of two numbers is 6.0 and arithmetic mean is 6.5, then the difference of squares of these numbers is
- (a) 65
- (b) 120
- (c) 130
- (d) 140
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Answer: (a)
11. The smallest number among the numbers 2²⁵⁰, 3¹⁵⁰, 5¹⁰⁰ and 4²⁰⁰ is
- (a) 4²⁰⁰
- (b) 5¹⁰⁰
- (c) 3¹⁵⁰
- (d) 2²⁵⁰
Show answer
Answer: (b)
12. If a : b = 2/9 : 1/3, b : c = 2/7 : 5/14 and d : c = 7/10 : 3/5, then a : b : c : d = ?
- (a) 2 : 3 : 5 : 7
- (b) 3 : 15 : 7 : 40
- (c) 16 : 24 : 30 : 35
- (d) 18 : 24 : 30 : 49
Show answer
Answer: (c)
13. If ⁹P₅ + 5.⁹P₄ = ¹⁰Pᵣ, then the value of r is
- (a) 2
- (b) 3
- (c) 5
- (d) 7
Show answer
Answer: (c)
14. The value of the determinant |−a² ab ac; ab −b² bc; ac bc −c²| is
- (a) 0
- (b) − (a² + b² + c²)
- (c) 4 a²b²c²
- (d) 2 (ab + bc + ca)
Show answer
Answer: (c)
15. Sum of the series 1/2! + 1/4! + 1/6! + .... is
- (a) (e + 1)²/(2e)
- (b) (e − 1)²/(2e)
- (c) (e² − 1)/(2e)
- (d) (e² + 1)/(2e)
Show answer
Answer: (b)
16. If the mapping f and g are given by f = {(1, 2), (3, 5), (4, 1)} g = {(2, 3), (5, 1), (1, 3)} then gof is
- (a) {(2, 5), (5, 2), (1, 5)}
- (b) {(1, 2), (3, 5), (4, 1)}
- (c) {(1, 3), (3, 1), (4, 3)}
- (d) {(2, 3), (5, 1), (1, 3)}
Show answer
Answer: (c)
17. In how many ways the letter of the word BALLOON be arranged so that two L do not come together ?
- (a) 1260
- (b) 360
- (c) 900
- (d) 1060
Show answer
Answer: (c)
18. The sequence {sin(nπ/2)/n}_(n=1)^∞ converges to
- (a) 0
- (b) 1
- (c) π
- (d) − 1
Show answer
Answer: (a)
19. The value of (sin75° − sin15°)/(cos75° + cos15°) is
- (a) √3
- (b) √3/2
- (c) 1/√3
- (d) 1/√2
Show answer
Answer: (c)
20. The value of sin(cot⁻¹(tan(cos⁻¹ x)))
- (a) 0
- (b) √(1 − x²)
- (c) π
- (d) x
Show answer
Answer: (d)
21. If A + iB = tan(x + iy), then the value of tan 2x is as
- (a) 2A/(1 + A² + B²)
- (b) 2A/(1 − A² + B²)
- (c) 2A/(1 − A² − B²)
- (d) None of these
Show answer
Answer: (c)
22. If cos(A + B) = 3/5 and sin(A − B) = 5/13, where 0 ≤ A, B ≤ π/4, then tan 2B is equal to
- (a) 11/34
- (b) 21/56
- (c) 33/56
- (d) 1
Show answer
Answer: (c)
23. AB is a chord of a circle and AOC is its diameter such that ∠ACB = 50°, if AT is tangent to the circle at the point A, then ∠BAT is equal to
- (a) 50°
- (b) 60°
- (c) 65°
- (d) None of these
Show answer
Answer: (a)
24. The curve represented by the equations x = 3 (cost + sint) y = 4(cost − sint) is
- (a) A straight line
- (b) A circle
- (c) A hyperbola
- (d) An ellipse
Show answer
Answer: (d)
25. The equation of sphere is x² + y² + z² − x + z − 2 = 0, its radius is
- (a) √5/2
- (b) √(5/2)
- (c) 5/√2
- (d) 5
Show answer
Answer: (b)
26. If the straight line x cosα + y sinα = p is tangent to the ellipse x²/a² + y²/b² = 1, then
- (a) p² = a²b²/4
- (b) p² = a² cos²α − b² sin²α
- (c) p² = a² cos²α + b² sin²α
- (d) None of these
Show answer
Answer: (c)
27. Lines x = ay + b, z = cy + d and x = a′y + b′, z = c′y + d′ are perpendicular, if
- (a) aa′ + cc′ + 1 = 0
- (b) aa′ + cc′ − 1 = 0
- (c) ac + a′c′ − 1 = 0
- (d) ac + a′c′ + 1 = 0
Show answer
Answer: (a)
28. If 2x − y = 5, then (2x − 3)³ − (y + 2)³ is equal to
- (a) 0
- (b) 25
- (c) 40
- (d) 125
Show answer
Answer: (a)
29. The angle of intersection of the curves y = 4 − x² and y = x² is
- (a) tan⁻¹(4√2/7)
- (b) tan⁻¹(2/7)
- (c) tan⁻¹(3√2/7)
- (d) π/2
Show answer
Answer: (a)
30. The area of the region bounded by the curves y² = x and x² = y is
- (a) 1/3
- (b) 2/3
- (c) 1
- (d) 2
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Answer: (a)
31. d/dx (tan(cos⁻¹x))² is equal to
- (a) −2/x³
- (b) 2/x³
- (c) √(1 − x²)/x
- (d) − x/√(1 − x²)
Show answer
Answer: (a)
32. √(3 + 2√2) − √(3 − 2√2) is equal to
- (a) 1
- (b) √3 + 1
- (c) √3 − 1
- (d) 2
Show answer
Answer: (d)
33. Which one of the following is not true ?
- (a) |a⃗| = |b⃗| ⇒ a⃗ = b⃗
- (b) |a⃗ × b⃗|² = (a⃗)²(b⃗)² − (a⃗ · b⃗)²
- (c) a⃗ × (b⃗ · c⃗) is not defined
- (d) If the adjacent sides of a parallelogram are represented by the vectors a⃗ and b⃗ respectively, then its area is |a⃗ × b⃗|
Show answer
Answer: (a)
34. A particle is projected with velocity u at an inclination α with the horizontal. Then Maximum height (H) attained is
- (a) u² sin²α / g
- (b) 2u² sin²α / g
- (c) u² sin²α / 2g
- (d) u² sin2α / g
Show answer
Answer: (c)
35. If |x+2 2 2; 2 x+2 2; 2 2 x+2| = 0, then values of x satisfying this equation are
- (a) 0, − 2, − 6
- (b) 0, −1, −2
- (c) 0, 0, − 2
- (d) 0, 0, − 6
Show answer
Answer: (d)
36. Series Σ_(n=1)^∞ (((n+1)/n)ⁿ⁺¹ − (n+1)/n)⁻ⁿ is
- (a) divergent
- (b) convergent
- (c) oscillates finitely
- (d) oscillates infinitely
Show answer
Answer: (b)
37. 10² + 11² + 12² + .... + 19² is equal to
- (a) 1580
- (b) 2010
- (c) 2121
- (d) 2185
Show answer
Answer: (d)
38. The resultant force of two forces P and Q is R. If Q is doubled, R is doubled and if Q is reversed, R is again doubled. Then which one is the true relation ?
- (a) P/√2 = Q/√3 = R/√2
- (b) P/1 = Q/2 = R/1
- (c) P/√3 = Q/√2 = R/√2
- (d) P/1 = Q/1 = R/√2
Show answer
Answer: (a)
39. The compound interest of ₹ 8,000 @ 10% per annum for 1.5 years, if the interest is calculated half yearly is
- (a) ₹ 9,261
- (b) ₹ 860
- (c) ₹ 961
- (d) ₹ 1,261
Show answer
Answer: (d)
40. If the mean of 10 observations x₁, x₂, x₃, ..., x₁₀ is 20, then mean of x₁ + 2, x₂ + 4, x₃ + 6, ...., x₁₀ + 20 is
- (a) 24
- (b) 28
- (c) 31
- (d) 32
Show answer
Answer: (c)
41. For showing the growth of population for the last 8 years, the best diagram is
- (a) pie
- (b) histogram
- (c) scatter plot
- (d) simple bar diagram
Show answer
Answer: (d)
42. If non-zero a, b, c are such that a + b + c = 0, then the value of a²/bc + b²/ac + c²/ab is
- (a) 3
- (b) 2
- (c) − 3
- (d) 0
Show answer
Answer: (a)
43. If x + 1/x = √3, then the value of x¹⁸ + x¹² + x⁶ + 1 is
- (a) 0
- (b) 1
- (c) 2
- (d) 3
Show answer
Answer: (a)
44. Factorisation of x³ − y³ is
- (a) (x + y)(x² − xy + y²)
- (b) (x − y)(x² − xy + y²)
- (c) (x − y)(x² − xy − y²)
- (d) (x − y)(x² + xy + y²)
Show answer
Answer: (d)
45. The roots of the equation x³ − 12x² + 39x − 28 = 0 are in A. P., the common difference is
- (a) 2
- (b) 3
- (c) − 2
- (d) 4
Show answer
Answer: (b)
46. If A = [[coshx, sinhx], [−sinhx, coshx]], then trace (A²) is equal to
- (a) 2
- (b) − 2
- (c) cosh 2x
- (d) sinh 2x
Show answer
Answer: (a)
47. If x − 1 and x + 3 are the two factors of x³ + ax + b, then remaining factor is
- (a) x + 2
- (b) x − 2
- (c) x − 3
- (d) x + 1
Show answer
Answer: (b)
48. If A is an open set and B is a closed set, then B − A is
- (a) Open set
- (b) Closed set
- (c) Both open and closed set
- (d) None of these
Show answer
Answer: (b)
49. lim_y→a (sin((y − a)/2) tan(πy/2a)) is equal to
- (a) 0
- (b) 1
- (c) π/a
- (d) − a/π
Show answer
Answer: (d)
50. The value of sin 18° is
- (a) (√5 − 1)/4
- (b) (√5 + 1)/4
- (c) (√5 − 1)/2
- (d) (√5 + 1)/2
Show answer
Answer: (a)
51. If sin²(x + iy) = A + iB, then value of A is
- (a) (1/2)(1 + cos 2x cos h 2y)
- (b) (1/2)(1 − cos 2x cos h 2y)
- (c) (1/2)(sin 2x sin h 2y)
- (d) − (1/2)(sin 2x sin h 2y)
Show answer
Answer: (b)
52. The general solution of sinx + cosx = 1 is given by
- (a) x = 2nπ
- (b) x = 2nπ + π/2
- (c) x = nπ + (−1)ⁿ π/4 − π/4
- (d) None of these
Show answer
Answer: (c)
53. In an equilateral triangle, the ratio of the radius of circumcircle to that of incircle is
- (a) 3 : 1
- (b) 5 : 2
- (c) 3 : 2
- (d) 2 : 1
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Answer: (d)
54. The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
- (a) 60°
- (b) 70°
- (c) 55°
- (d) 110°
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Answer: (b)
55. The equation of a normal to the parabola y² = 4x which passes through the point (6, 0) is
- (a) y + 2x = 12
- (b) y − 2x = 12
- (c) y + 2x = 6
- (d) y − 2x = 6
Show answer
Answer: (a)
56. The equation 6x² − 5xy − 6y² + 14x + 5y + 4 = 0 represents
- (a) A circle
- (b) A hyperbola
- (c) An ellipse
- (d) A pair of perpendicular straight lines
Show answer
Answer: (d)
57. The conic x² + xy + y² + x + y = 1 is
- (a) an ellipse
- (b) a hyperbola
- (c) a parabola
- (d) a pair of straight lines
Show answer
Answer: (a)
58. If y-axis is generator line of the cone ax² + by² + cz² + 2fyz + 2gzx + 2hxy = 0, then value of b is equal to
- (a) 1
- (b) − 1
- (c) 0
- (d) None of these
Show answer
Answer: (c)
59. The second degree equation 2x² + 2y² − 2x − 6y + 5 = 0 represents
- (a) a circle
- (b) an ellipse
- (c) a point
- (d) none of the above
Show answer
Answer: (c)
60. The equation of the normal at the point (1, 1) on the curve 2y + x² = 3 is
- (a) x + y = 0
- (b) x + y + 1 = 0
- (c) x − y = 0
- (d) x − y = 1
Show answer
Answer: (c)
61. The value of ∫_(π/4)^(π/3) dx/(sin²x cos²x) is
- (a) 1
- (b) 0
- (c) 2/√3
- (d) 1/√3
Show answer
Answer: (c)
62. ∫₋₁² x|x| dx is equal to
- (a) 0
- (b) 2/3
- (c) 5/3
- (d) 7/3
Show answer
Answer: (d)
63. H.C.F. of x³ − x² + 4x − 4 and x⁶ − 1 is
- (a) x + 1
- (b) x − 1
- (c) x² − 1
- (d) x³ − 1
Show answer
Answer: (b)
64. The value of ‘a’ such that the vectors 2î − ĵ + k̂, î + 2ĵ − 3k̂ and 3î + aĵ + 5k̂ are coplanar, is
- (a) 1
- (b) − 2
- (c) 4
- (d) − 4
Show answer
Answer: (d)
65. A particle is projected in vertically upward direction. If after t₁ and t₂ time, its height is h, then h is equal to
- (a) (1/2) g t₁ t₂
- (b) g √(t₁ t₂)
- (c) 2g (t₁ + t₂)
- (d) none of these
Show answer
Answer: (a)
66. If A = [[1, 3+x, 2], [1−x, 2, y+1], [2, 5−y, 3]] is a symmetric matrix, then 3x + y is equal to
- (a) −1
- (b) 0
- (c) 1
- (d) none of these
Show answer
Answer: (a)
67. If in any projectile maximum height is equal to horizontal range, then the angle of projection is
- (a) tan⁻¹(1/4)
- (b) tan⁻¹(1/2)
- (c) tan⁻¹ 2
- (d) tan⁻¹ 4
Show answer
Answer: (d)
68. If ω is a complex cube root of unit, then 1 + ω + ω² + ... + ω¹⁰⁰ is equal to
- (a) 0
- (b) 1 + ω
- (c) 1 − ω
- (d) ω
Show answer
Answer: (b)
69. The centre of gravity of a semi-circular disc of radius ‘a’ is
- (a) (2a/π, 0)
- (b) (3a/4π, 0)
- (c) (4a/3π, 0)
- (d) (a/2, a/2)
Show answer
Answer: (c)
70. When a train travels with speed (3/5)ᵗʰ of its normal speed, it reaches its destination 4 hours late. What is the time taken by the train in travelling this distance with its normal speed ?
- (a) 3 hours
- (b) 6 hours
- (c) 4 hours
- (d) 5 hours
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Answer: (b)
71. A 75 meter long train is moving at 20 km/hr. It will cross a man standing on the platform in time
- (a) 12 seconds
- (b) 14 seconds
- (c) 13.5 seconds
- (d) 15.5 seconds
Show answer
Answer: (c)
72. Three dice are thrown randomly. The probability of coming 3 in at least one die is
- (a) 180/216
- (b) 91/216
- (c) 5/216
- (d) 125/216
Show answer
Answer: (b)
73. If the algebraic sum of the deviations of 10 observations measured from 15 is 7, then the mean is
- (a) 105
- (b) 70
- (c) 15.7
- (d) 16.7
Show answer
Answer: (c)
74. The equations 2x − ky + 7 = 0 and 6x − 12y + 15 = 0 have no solution for
- (a) k = − 4
- (b) k = 4
- (c) k = 1
- (d) k = − 1
Show answer
Answer: (b)
75. If a + b + c = 5 and ab + bc + ca = 10, then the value of a³ + b³ + c³ − 3abc is
- (a) − 25
- (b) 25
- (c) 0
- (d) 75
Show answer
Answer: (a)
76. If logₓ4 + logₓ16 + logₓ64 = 12, the value of x is
- (a) 2
- (b) 4
- (c) 5
- (d) 10
Show answer
Answer: (a)
77. If a, b, c are in A. P. and x, y, z are in G. P., then the value of x^(b−c) y^(c−a) z^(a−b) is
- (a) 0
- (b) 1
- (c) 2
- (d) − 1
Show answer
Answer: (b)
78. 1/log₂x + 1/log₃x + 1/log₄x + .... + 1/log₅₀x, x ≠ 1, is equal to
- (a) 50/log_(50!) x
- (b) 49/log_(49!) x
- (c) 1/log_(50!) x
- (d) 1/log_(49!) x
Show answer
Answer: (c)
79. If G is a group of even order, then it has an element a ≠ e, satisfying
- (a) a² = e
- (b) a³ = e
- (c) a⁵ = e
- (d) a⁷ = e
Show answer
Answer: (a)
80. lim_x→1 (1 − √x)/cos⁻¹x is equal to
- (a) 0
- (b) 1/2
- (c) 1/4
- (d) 1
Show answer
Answer: (a)
81. The value of Σ_(r=1)^9 sin²(rπ/18) is
- (a) 1
- (b) 0
- (c) 5
- (d) π
Show answer
Answer: (c)
82. In the equation cos⁻¹((1 − a²)/(1 + a²)) − cos⁻¹((1 − b²)/(1 + b²)) = 2tan⁻¹x, value of x is
- (a) (a + b)/(1 + ab)
- (b) (a − b)/(1 + ab)
- (c) (a − b)/(1 − ab)
- (d) None of the above
Show answer
Answer: (b)
83. The polar form of − 1 − √(−3) is
- (a) 2 (cos 2π/3 − i sin 2π/3)
- (b) 2 (cos π/3 + i sin π/3)
- (c) 2 (cos 2π/3 + i sin 2π/3)
- (d) 2 (cos π/3 − i sin π/3)
Show answer
Answer: (a)
84. If the sides of a right angled triangle are a, ar, ar² (r < 1), then r² is equal to
- (a) (√5 − 1)/2
- (b) (√5 + 1)/2
- (c) √5 − 1
- (d) √5 + 1
Show answer
Answer: (a)
85. If f(x) = cos hx + sin hx, then which one of the following is correct ?
- (a) f(x) f(y) = f(x) + f(y)
- (b) f(x) f(y) = f(xy)
- (c) f(x) f(y) = f(x + y)
- (d) f(x) f(y) = f(x/y)
Show answer
Answer: (c)
86. The equation of the base of an equilateral triangle is x + y = 1 and its vertex is (1, − 1), length of its side is
- (a) 1
- (b) 1/√2
- (c) √(2/3)
- (d) 1/√3
Show answer
Answer: (c)
87. If a straight line makes angles α, β and γ with the co-ordinate axes, then sin²α + sin²β + sin²γ is equal to
- (a) 0
- (b) 1
- (c) 2
- (d) None of these
Show answer
Answer: (c)
88. If the equation 3x² + 7xy + 2y² + 5x + 5y + k = 0 represents a pair of straight lines, then the value of k is
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Show answer
Answer: (b)
89. The number of spheres that can be made to pass through the three given points (1, 0, 0), (0, 1, 0) and (0, 0, 1) is
- (a) 1
- (b) 2
- (c) 3
- (d) Infinite
Show answer
Answer: (d)
90. The length of the common chord of two circles of radii 15 cm and 20 cm, whose centres are 25 cm apart, is (in cm)
- (a) 10
- (b) 12
- (c) 20
- (d) 24
Show answer
Answer: (d)
91. lim_n→∞ (1 − 1/(2n))ⁿ⁺¹ is equal to
- (a) √e
- (b) 1/√e
- (c) e
- (d) 1/e
Show answer
Answer: (b)
92. The derivative of the function y = 3 |x| + 1 at the point x = 0 is
- (a) 3
- (b) − 3
- (c) 0
- (d) not existing
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Answer: (d)
93. The area enclosed between the curves y = sinx, y = cosx, 0 ≤ x ≤ π/2 is
- (a) √2 − 1
- (b) √2 + 1
- (c) 2(√2 − 1)
- (d) 2(√2 + 1)
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Answer: (c)
94. If y(x) is a solution of the differential equation dy/dx + 4xy = x³, y(0) = 0, then lim_x→0 y(x) is
- (a) 0
- (b) − 2
- (c) 1
- (d) not existing
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Answer: (a)
95. Two forces F₁⃗ = î − ĵ + k̂ and F₂⃗ = 4î + 2ĵ + 3k̂ act on a particle and displace it from the point (0, 1, 2) to (1, − 2, 3), then the total work done is
- (a) 2 unit
- (b) 6 unit
- (c) 4 unit
- (d) 8 unit
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Answer: (b)
96. The moment of the force 3î + 4ĵ − 5k̂ acting through the point î − 2ĵ about the point î − ĵ + k̂ is
- (a) 6î + 2ĵ − 5k̂
- (b) 8î + 2ĵ − k̂
- (c) 9î + 3ĵ − 8k̂
- (d) 9î − 3ĵ + 3k̂
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Answer: (d)
97. Value of the integral ∫₋₁¹ log((2 − x)/(2 + x)) dx is
- (a) 2
- (b) 1
- (c) 1/2
- (d) 0
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Answer: (d)
98. If the foot of the perpendicular drawn from (−2, 1, 0) on a plane is (1, −2, 1), then the equation of the plane is
- (a) 3x + 3y + z = 10
- (b) 3x + 3y − z = 10
- (c) 3x − 3y + z = 10
- (d) 3x − 3y − z = 10
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Answer: (c)
99. If â, b̂ and ĉ are unit vectors and |â + b̂|² = |b̂ + ĉ|² = |ĉ + â|² = 8, then |2â + b̂ + ĉ| is equal to
- (a) 2
- (b) 4
- (c) 6
- (d) none of the above
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Answer: (c)
100. If f′(x) = g(x) and g′(x) = f(x²), then f″(x²) is equal to
- (a) g(x²)
- (b) f(x⁴)
- (c) f(x³)
- (d) g(x⁴)
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Answer: (b)
101. A and B work together and complete a given work in x days. If A alone completes this work in x + 2 days and B alone completes this work in x + 8 days, then the value of x is
- (a) 3
- (b) 4
- (c) 5
- (d) 2
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Answer: (b)
102. A person travels from A to B at an average speed of x km/hr and returns from B to A at an average speed of y km/hr. His average speed during the total journey, is
- (a) (x + y)/(2xy)
- (b) 2xy/(x + y)
- (c) 2/(x + y)
- (d) 1/x + 1/y
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Answer: (b)
103. How much will ₹ 25,000 amount to in 2 years at compound interest, if the rate of interest for successive years be 4% and 5% per year ?
- (a) ₹ 25,300
- (b) ₹ 26,300
- (c) ₹ 27,300
- (d) ₹ 28,300
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Answer: (c)
104. If the median of the distribution 10, 12, 13, 16, x, 20, 25, 30 is 18, then the value of x is
- (a) 24
- (b) 22
- (c) 23
- (d) 20
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Answer: (d)
105. If x + 9/x = 6, then the value of x² + 9/x² is
- (a) 8
- (b) 12
- (c) 16
- (d) 10
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Answer: (d)
106. Values of x in the equation 1/(x − 3) − 1/(x + 5) = 1/6 are
- (a) 7, 9
- (b) − 7, 9
- (c) − 7, − 9
- (d) 7, − 9
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Answer: (d)
107. In the expansion of (3x − 2/x²)¹⁵ which term is free from x ?
- (a) 4ᵗʰ
- (b) 5ᵗʰ
- (c) 6ᵗʰ
- (d) 7ᵗʰ
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Answer: (c)
108. If A = [[1, −1], [−1, 1]] and B = [[1, 1], [1, 1]], then the value of AB is
- (a) [[1, −1], [−1, 1]]
- (b) [[0, 1], [1, 0]]
- (c) [[−1, 0], [0, −1]]
- (d) [[0, 0], [0, 0]]
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Answer: (d)
109. If |x| < 1, then x²/2 + 2x³/3 + 3x⁴/4 + ..... is equal to
- (a) x/(1 − x) + logₑ(1 − x)
- (b) x/(1 + x) + logₑ(1 + x)
- (c) x/(1 − x) + logₑ(1 + x)
- (d) x/(1 + x) + logₑ(1 − x)
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Answer: (a)
110. Given : Statement A : All cyclic groups are abelian group. Statement B : The order of the cyclic group is same as the order of its generator.
- (a) A and B are false
- (b) A is true, B is false
- (c) B is true, A is false
- (d) A and B both are true
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Answer: (d)
111. If f(x) = (x² − 3x + 2)/(x² − 2x), x ≠ 2 is defined and function f(x) be continuous at x = 2, then the value of f(2) is
- (a) 0
- (b) 1/2
- (c) 1
- (d) 3/4
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Answer: (b)
112. The function f(x) is defined by f(x) = |x|/x , x ≠ 0 ; 0 , x = 0 then, at x = 0 it is
- (a) Continuous
- (b) Discontinuous at x = 0 and has discontinuity of first kind
- (c) Discontinuous at x = 0 and has removable discontinuity
- (d) Discontinuous at x = 0 and has discontinuity of second kind
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Answer: (b)
113. The value of cot(cosec⁻¹(5/3) + tan⁻¹(2/3)) is
- (a) 5/17
- (b) 3/17
- (c) 4/17
- (d) 6/17
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Answer: (d)
114. If r₁, r₂, r₃ are ex-radii and r the in-radius of triangle ABC and r₁ = r + r₂ + r₃, then triangle ABC is
- (a) Isoscales triangle
- (b) Equilateral triangle
- (c) Right angled triangle
- (d) None of these
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Answer: (c)
115. If sin θ cos θ = 1/2, then value of sin⁶θ + cos⁶θ is
- (a) 1/2
- (b) 3/2
- (c) 1
- (d) 1/4
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Answer: (d)
116. If ω is cube root of unity, then (3 + ω + 3ω²)⁶ is equal to
- (a) 32
- (b) 64
- (c) 128
- (d) 16
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Answer: (b)
117. If AB is diameter of the circle x² + y² + 2x + 4y − 3 = 0. If co-ordinates of A are (1, 0), then co-ordinates of B are
- (a) (− 3, 1)
- (b) (− 3, 2)
- (c) (− 3, 3)
- (d) (− 3, − 4)
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Answer: (d)
118. The eccentricity of the ellipse, whose latus rectum is one half of its minor axis is
- (a) a/√3
- (b) √3/2
- (c) √(3/2)
- (d) √2/3
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Answer: (b)
119. The angle between the straight lines represented by the equation y² − xy − 6x² = 0 is
- (a) 30°
- (b) 60°
- (c) 45°
- (d) 65°
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Answer: (c)
120. In the conic l/r = 1 + e cosθ, the sum of the reciprocals of the segments of any focal chord is
- (a) 1/l
- (b) 3/l
- (c) 4/l
- (d) 2/l
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Answer: (d)
121. The co-ordinates of the mid points of sides of a triangle are (5, 2), (3, 3) and (2, 2). Co-ordinates of its centroid is
- (a) (7/3, 10/3)
- (b) (10/3, 7/3)
- (c) (7/3, 2/3)
- (d) (2/3, 7/3)
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Answer: (b)
122. In a geometric progression, first term is 7, the last term is 448 and the sum is 889. The common ratio of the geometric progression is
- (a) 3/2
- (b) 2
- (c) 3
- (d) 3.5
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Answer: (b)
123. The value of ∫₀^(π/2) sinx dx/(sinx + cosx) is
- (a) 1
- (b) π/2
- (c) π/4
- (d) π/8
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Answer: (c)
124. The solution of the differential equation (x + 2y³) dy/dx = y is
- (a) x = y³ − cy²
- (b) x = y³ + cy
- (c) y = x³ + cx
- (d) y = x³ − cx²
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Answer: (b)
125. Solution of the differential equation (1 + 3x) dy − (1 − 3y) dx = 0, y(1) = 0 is
- (a) x + y + 3xy = 1
- (b) x − y + 3xy = 1
- (c) x − y − 3xy = 1
- (d) x + y − 3xy = 1
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Answer: (c)

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Answer key

Answer key from the U.P. Secondary Education Services Selection Board’s official answer key for this sitting. “—” means the key gave none: the question was deleted.

Mathematics

Q 1: (c)
Q 2: (c)
Q 3: (b)
Q 4: (b)
Q 5: (a)
Q 6: (a)
Q 7: (c)
Q 8: (a)
Q 9: (d)
Q 10: (a)
Q 11: (b)
Q 12: (c)
Q 13: (c)
Q 14: (c)
Q 15: (b)
Q 16: (c)
Q 17: (c)
Q 18: (a)
Q 19: (c)
Q 20: (d)
Q 21: (c)
Q 22: (c)
Q 23: (a)
Q 24: (d)
Q 25: (b)
Q 26: (c)
Q 27: (a)
Q 28: (a)
Q 29: (a)
Q 30: (a)
Q 31: (a)
Q 32: (d)
Q 33: (a)
Q 34: (c)
Q 35: (d)
Q 36: (b)
Q 37: (d)
Q 38: (a)
Q 39: (d)
Q 40: (c)
Q 41: (d)
Q 42: (a)
Q 43: (a)
Q 44: (d)
Q 45: (b)
Q 46: (a)
Q 47: (b)
Q 48: (b)
Q 49: (d)
Q 50: (a)
Q 51: (b)
Q 52: (c)
Q 53: (d)
Q 54: (b)
Q 55: (a)
Q 56: (d)
Q 57: (a)
Q 58: (c)
Q 59: (c)
Q 60: (c)
Q 61: (c)
Q 62: (d)
Q 63: (b)
Q 64: (d)
Q 65: (a)
Q 66: (a)
Q 67: (d)
Q 68: (b)
Q 69: (c)
Q 70: (b)
Q 71: (c)
Q 72: (b)
Q 73: (c)
Q 74: (b)
Q 75: (a)
Q 76: (a)
Q 77: (b)
Q 78: (c)
Q 79: (a)
Q 80: (a)
Q 81: (c)
Q 82: (b)
Q 83: (a)
Q 84: (a)
Q 85: (c)
Q 86: (c)
Q 87: (c)
Q 88: (b)
Q 89: (d)
Q 90: (d)
Q 91: (b)
Q 92: (d)
Q 93: (c)
Q 94: (a)
Q 95: (b)
Q 96: (d)
Q 97: (d)
Q 98: (c)
Q 99: (c)
Q 100: (b)
Q 101: (b)
Q 102: (b)
Q 103: (c)
Q 104: (d)
Q 105: (d)
Q 106: (d)
Q 107: (c)
Q 108: (d)
Q 109: (a)
Q 110: (d)
Q 111: (b)
Q 112: (b)
Q 113: (d)
Q 114: (c)
Q 115: (d)
Q 116: (b)
Q 117: (d)
Q 118: (b)
Q 119: (c)
Q 120: (d)
Q 121: (b)
Q 122: (b)
Q 123: (c)
Q 124: (b)
Q 125: (c)