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Question paper · UP TGT

UP TGT 2021 — Mathematics (Series A)

7 or 8 August 2021125 questions+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 7 or 8 August 2021. 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. A person travels from A to B at a speed of 40 kmph and returns by increasing his speed by 50%. What is his average speed for both the trips ?
- (a) 40 kmph
- (b) 45 kmph
- (c) 48 kmph
- (d) 50 kmph
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Answer: (c)
2. What is the real part of (sin x + i cos x)⁴, i = √−1 ?
- (a) − cos 4x
- (b) − cos 2x
- (c) cos 4x
- (d) cos 2x
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Answer: (c)
3. is equal to
- (a) log (3/2)
- (b) log (9/8)
- (c) log 2
- (d) 1
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Answer: (b)
4. If ω is an imaginary cube root of unity, then value of is
- (a) 0
- (b) 3
- (c) 4
- (d) 1
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Answer: (b)
5. The sides of a triangle are 15 cm, 20 cm and 25 cm respectively, then the radius of in-circle is
- (a) 10 cm
- (b) 12.5 cm
- (c) 5 cm
- (d) 7.5 cm
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Answer: (c)
6. If the arithmetic mean from the following data x: 5, 15, 25, 35, 45 f: 6, 4, 3, k, 2 is 21.5, the value of k will be
- (a) 3
- (b) 3.5
- (c) 4
- (d) 5
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Answer: (d)
7. The value of [(x − y)³ + (y − z)³ + (z − x)³] / [9(x − y)(y − z)(z − x)] is
- (a) 0
- (b) 1/9
- (c) 1/3
- (d) 1
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Answer: (c)
8. Find the equation of the cone with vertex at the origin and direction cosines of its generators satisfy the equation l² + 2m² − 3n² = 0.
- (a) x² + y² − z² = 0
- (b) x² + 2y² − 3z² = 0
- (c) x² − 2y² + 3z² = 0
- (d) xy + yz + zx = 0
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Answer: (b)
9. A balloon is moving upward with uniform acceleration g/8 cm/sec². After half minute a body is dropped from them. The time taken by the body to reach on the ground is
- (a) 10 seconds
- (b) 15 seconds
- (c) 20 seconds
- (d) 24 seconds
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Answer: (b)
10. is equal to
- (a) 1
- (b) e²
- (c) e⁴
- (d) e⁵
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Answer: (d)
11. is equal to
- (a) 1
- (b) 0
- (c) 2/√3
- (d) 1/√3
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Answer: (c)
12. If u⃗ = î × (a⃗ × î) + ĵ × (a⃗ × ĵ) + k̂ × (a⃗ × k̂) then u⃗ is equal to
- (a) 0⃗
- (b) a⃗
- (c) 2a⃗
- (d) 3a⃗
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Answer: (c)
13. If z is a complex number of unit modulus and argument θ, then arg((1 + z)/(1 + z̄)) is equal to
- (a) − θ
- (b) π/2 − θ
- (c) θ
- (d) π − θ
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Answer: (c)
14. The equations of tangents to the ellipse 3x² + 4y² = 12 which are parallel to the line y + 2x = 4 are
- (a) y + 2x = ±4
- (b) y + 2x = ±7
- (c) y + 2x = ±√15
- (d) y + 2x = ±√19
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Answer: (d)
15. Which one of the following is not a measure of central tendency ?
- (a) Mean
- (b) Range
- (c) Median
- (d) Mode
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Answer: (b)
16. sin⁻¹(3/5) − cos⁻¹(12/13) equals to
- (a) sin⁻¹(56/65)
- (b) sin⁻¹(16/65)
- (c) 1
- (d) 0
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Answer: (b)
17. The point on the curve y² = x, the tangent at which makes an angle π/4 with x – axis, is
- (a) (−1/4, 1/2)
- (b) (1/4, 1/2)
- (c) (−1/4, −1/2)
- (d) (1/4, −1/2)
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Answer: (b)
18. A sum becomes double in 5 years in C.I. When will it become 8 times at the same rate of interest ?
- (a) 10 years
- (b) 15 years
- (c) 7 years
- (d) 20 years
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Answer: (b)
19. What is the principal value of sin⁻¹(sin (2π/3)) ?
- (a) π/4
- (b) π/2
- (c) π/3
- (d) 2π/3
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Answer: (c)
20. is equal to
- (a) 1
- (b) π²/8
- (c) π²/32
- (d) π²/3
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Answer: (c)
21. Sum of the series 2 + 3/2 + 1 + 5/8 + .... ∞ is
- (a) 2
- (b) 4
- (c) 6
- (d) 12
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Answer: (c)
22. is equal to
- (a) 2 sin θ
- (b) 2 cos θ
- (c) sin 2θ
- (d) cos 2θ
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Answer: (b)
23. The distances travelled by a body falling freely from rest in the first, second and third second are in the ratio
- (a) 1 : 4 : 9
- (b) 1 : 3 : 5
- (c) 1 : 2 : 3
- (d) 1 : 4 : 5
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Answer: (b)
24. The number of real solutions of the equation |x|² − 3|x| + 2 = 0 is
- (a) 4
- (b) 1
- (c) 3
- (d) 2
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Answer: (a)
25. In the following figure the value of θ is
- (a) 110°
- (b) 70°
- (c) 60°
- (d) 50°
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Answer: (c)
26. The position vectors of the points A, B, C and D are 3î − 2ĵ − k̂, 2î + 3ĵ − 4k̂, −î + ĵ + 2k̂ and 4î + 5ĵ + λk̂ respectively. If the points A, B, C and D lie on a plane, the value of λ is
- (a) −146/17
- (b) −73/17
- (c) 7
- (d) 13
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Answer: (a)
27. is equal to
- (a) 0
- (b) 1/3
- (c) 2/3
- (d) Does not exist
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Answer: (b)
28. Three numbers are chosen at random from numbers 1 to 15. The probability that they are consecutive is
- (a) 1/15
- (b) 2/25
- (c) 1/35
- (d) 1/30
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Answer: (c)
29. The points with position vectors 60î + 3ĵ, 40î − 8ĵ, aî − 52ĵ are collinear if a is equal to
- (a) − 40
- (b) 40
- (c) 20
- (d) 25
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Answer: (a)
30. If the mean of x, x + 2, x + 4, x + 6, x + 8 is 11, then what is the mean of the last three observations ?
- (a) 8
- (b) 10
- (c) 12
- (d) 13
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Answer: (d)
31. ((1 + i)/√(2))⁸ + ((1 − i)/√(2))⁸ is equal to
- (a) 1
- (b) 2
- (c) 4
- (d) 8
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Answer: (b)
32. is equal to
- (a) 2
- (b) 6
- (c) 0
- (d) −2
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Answer: (b)
33. The speeds of three cars are in the ratio 2 : 3 : 4. The ratios of the times taken by these cars in travelling 100 km distance will be
- (a) 2 : 3 : 4
- (b) 4 : 3 : 2
- (c) 4 : 3 : 6
- (d) 6 : 4 : 3
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Answer: (d)
34. In the equation value of x is
- (a) (a + b)/(1 + ab)
- (b) (a − b)/(1 + ab)
- (c) (a − b)/(1 − ab)
- (d) None of the above
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Answer: (b)
35. Integral of is
- (a) log(sin x) + C
- (b) log(cos x) + C
- (c) log(log tan x) + C
- (d) log(log sin x) + C
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Answer: (d)
36. If then AB is equal to
- (a)
- (b)
- (c)
- (d) None of the above
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Answer: (a)
37. In the figure ∠A is equal to
- (a) 70°
- (b) 45°
- (c) 80°
- (d) 60°
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Answer: (a)
38. Find the value of ‘a’ such that the vectors 2î − ĵ + k̂, î + 2ĵ − 3k̂ and 3î + aĵ + 5k̂ are coplanar.
- (a) 1
- (b) 2
- (c) 4
- (d) − 4
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Answer: (d)
39. If x₁ and x₂ are the roots of the equation then the value of x₁² + x₂² is
- (a) 1
- (b) 3
- (c) 5
- (d) 0
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Answer: (c)
40. In the following figure AD is tangent of the circle and ABC is secant. If AB = 4 cm and BC = 5 cm, then the length of tangent AD is
- (a) 3 cm
- (b) 4 cm
- (c) 5 cm
- (d) 6 cm
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Answer: (d)
41. A particle is projected with a velocity u so that its range on a horizontal plane is twice the greatest height attained, the range is
- (a) 2u²/3g
- (b) u²/2g
- (c) 3u²/2g
- (d) 4u²/5g
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Answer: (d)
42. If f(x) = tan⁻¹x and g(x) = sin x, then (gof) (x) is equal to
- (a) x√(1 − x²)
- (b) x/√(1 + x²)
- (c) √(1 + x²)/x
- (d) 1
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Answer: (b)
43. Solution of the differential equation log(dy/dx) = ax + by is
- (a) beᵃˣ + ae⁻ᵇʸ + C = 0
- (b) eᵃˣ/a − e⁻ᵇʸ/b + C = 0
- (c) eᵃˣ + e⁻ᵇʸ + C = 0
- (d) eᵃˣ⁺ᵇʸ = C(ax + by)
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Answer: (a)
44. If â and b̂ are unit vectors such that â + 2b̂ and 5â − 4b̂ are perpendicular to each other, then the angle between â and b̂ is
- (a) π/4
- (b) cos⁻¹(1/3)
- (c) cos⁻¹(2/7)
- (d) π/3
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Answer: (d)
45. If the mean of the observations 6, 8, 5, 7, x and 4 is 7, then the median of these observations is
- (a) 6.5
- (b) 6
- (c) 5.5
- (d) 7
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Answer: (a)
46. If x₁ and x₂ are roots of x² − 2x + 4 = 0, then the value of x₁³ + x₂³ is equal to
- (a) 16
- (b) 8
- (c) − 16
- (d) 4
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Answer: (c)
47. If y = log₅x + logₓ5, then what will be the value of dy/dx at x = 5 ?
- (a) 5
- (b) 1/5
- (c) 0
- (d) 1
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Answer: (c)
48. If 3 men working 3 hours per day make 3 toys in 3 days, then the number of toys which can be made by 6 men working 6 hours per day in 6 days is
- (a) 6
- (b) 12
- (c) 24
- (d) 48
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Answer: (c)
49. If A + iB = tan (x + iy), then tan 2x is equal to
- (a) 2A/(1 + A² + B²)
- (b) 2A/(1 − A² + B²)
- (c) 2A/(1 − A² − B²)
- (d) None of these
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Answer: (c)
50. For all real x, the minimum value of (1 − x + x²)/(1 + x + x²) is
- (a) 0
- (b) 1/3
- (c) 1
- (d) 3
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Answer: (b)
51. The term independent of x in the expansion of (2x + 1/(3x))⁶ is
- (a) 160/9
- (b) 80/9
- (c) 160/27
- (d) 80/3
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Answer: (c)
52. Two parallel chords of a circle of radius 5 cm are in opposite side of the centre and if the chords are at distances 3 cm and 4 cm from the centre, then the difference of the lengths of the chords is
- (a) 10 cm
- (b) 5 cm
- (c) 2 cm
- (d) 1 cm
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Answer: (c)
53. An aeroplane is flying at height h with horizontal velocity u. The velocity of a dropped packet on reaching the ground will be
- (a) √(u² + 2gh)
- (b) √(2gh)
- (c) 2gh
- (d) √(u² − 2gh)
Show answer
Answer: (a)
54. Function f(x) = (x + 1)ᶜᵒᵗ ˣ will be continuous at x = 0 if the value of f(0) is
- (a) 1/e
- (b) 0
- (c) e
- (d) 1
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Answer: (c)
55. If the equation x² + y² + (c − 2)xy − 4x − 4y + c² = 0 represents a circle, then its radius is
- (a) 1
- (b) 2
- (c) 3
- (d) 5
Show answer
Answer: (b)
56. If the radius of a hemisphere is a, then its centre of gravity from its centre is at a distance
- (a) 3a/8
- (b) 5a/8
- (c) a/2
- (d) 2a/5
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Answer: (a)
57. The sequence ⟨log (1/n)⟩ is
- (a) Convergent
- (b) Divergent to ∞
- (c) Divergent to − ∞
- (d) None of the above
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Answer: (c)
58. Two consecutive sides of a parallelogram are 4x + 5y = 0 and 7x + 2y = 0. If the equation of one diagonal is 11x + 7y = 9, the equation of other diagonal is
- (a) x + y = 0
- (b) 7x − 11y = 0
- (c) 7x + 11y = 0
- (d) x − y = 0
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Answer: (d)
59. The multiplicative group {1, − 1, i, − i} is a cyclic group, its generators are
- (a) 1 and i
- (b) 1 and − 1
- (c) i and − i
- (d) only i
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Answer: (c)
60. The mean weight of 9 items is 15 kg. If one more item is added then the mean weight becomes 16 kg, then the weight of tenth item will be
- (a) 20 kg
- (b) 15 kg
- (c) 25 kg
- (d) 30 kg
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Answer: (c)
61. The smallest positive integer n for which ((1 + i)/(1 − i))ⁿ = 1, is
- (a) 8
- (b) 12
- (c) 4
- (d) 16
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Answer: (c)
62. ∫ eˣ(x − 1)/(x + 1)³ dx equals to
- (a) eˣ/(x + 1)² + C
- (b) eˣ/(x + 1) + C
- (c) −eˣ/(x + 1)² + C
- (d) eˣ/(x − 1)² + C
Show answer
Answer: (a)
63. If (b − c)², (c − a)², (a − b)² are in A.P., then 1/(b − c), 1/(c − a), 1/(a − b) will be in
- (a) H.P.
- (b) G.P.
- (c) A.P.
- (d) None of the above
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Answer: (c)
64. If i = √−1, then (1 + i)⁴ − (1 − i)⁴ is equal to
- (a) − 4i
- (b) 4i
- (c) 0
- (d) 2
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Answer: (c)
65. The general equation to the cone of second degree which passes through the axes, is
- (a) x²/a + y²/b + z²/c = 0
- (b) fyz + gzx + hxy = 0
- (c) ax² + by² + cz² = 0
- (d) √(fx) + √(gy) + √(hz) = 0
Show answer
Answer: (b)
66. If one zero of the third degree polynomial x³ + ax² + bx + c is − 1, then the product of other two zeroes is
- (a) b − a + 1
- (b) b − a − 1
- (c) a − b + 1
- (d) a + b + 1
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Answer: (a)
67. The equation of the sphere on the join of (2, −3, 1) and (3, −1, 2) as diameter is
- (a) x² + y² + z² − 5x + 4y + 3z − 11 = 0
- (b) x² + y² + z² + 5x − 4y + 3z − 11 = 0
- (c) x² + y² + z² + 11 = 0
- (d) x² + y² + z² − 5x + 4y − 3z + 11 = 0
Show answer
Answer: (d)
68. Three forces P, 2P and 3P are acting along sides AB, BC and CA of equilateral Δ ABC. Magnitude of resultant force is
- (a) P√3
- (b) P√2
- (c) Zero
- (d) 2P/3
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Answer: (a)
69. In a triangle ABC if cos A = sin B/(2 sin C), then triangle is
- (a) Isosceles
- (b) Equilateral
- (c) Right angled
- (d) None of the above
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Answer: (a)
70. The area of the region bounded by the curves y² = x and x² = y is
- (a) 1
- (b) 2
- (c) 1/3
- (d) 2/3
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Answer: (c)
71. If x² + 1/x² = 11, then value of x³ − 1/x³ may be equal to
- (a) 22
- (b) 25
- (c) 33
- (d) 36
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Answer: (d)
72. The order of the element (− i) of the multiplicative group G = {1, − 1, i, − i} is
- (a) 4
- (b) 3
- (c) 2
- (d) 1
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Answer: (a)
73. In ΔABC, a = 2b and |A − B| = π/3, then ∠C is
- (a) π/2
- (b) π/3
- (c) π/6
- (d) π/4
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Answer: (b)
74. The area cut off the parabola 4y = 3x² by the straight line 2y = 3x + 12 is
- (a) 15 sq. units
- (b) 18 sq. units
- (c) 24 sq. units
- (d) 27 sq. units
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Answer: (d)
75. If lim (x → a) (aˣ − xᵃ)/(xˣ − aᵃ) = −1, then
- (a) a = 1
- (b) a = 0
- (c) a = e
- (d) none of these
Show answer
Answer: (a)
76. If y = xˣ, then dy/dx will be
- (a) xˣ log x
- (b) xˣ (1 − log x)
- (c) xˣ (1 + log x)
- (d) xˣ + log x
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Answer: (c)
77. The work done in moving an object along a straight line from (3, 2, −1) to (2, −1, 4) in a force field given by F⃗ = 4î − 3ĵ + 2k̂, is
- (a) 15 units
- (b) 18 units
- (c) 20 units
- (d) 24 units
Show answer
Answer: (a)
78. Equation of a circle, with center (0, 0) and 2x + 3y = 5 as a tangent, is
- (a) x² + y² = 5
- (b) x² + y² + 2x = 25
- (c) x² + y² + y = 25
- (d) 13x² + 13y² = 25
Show answer
Answer: (d)
79. Sequence log x, log (x²/y), log (x³/y²), .... is
- (a) a G.P.
- (b) an A.P.
- (c) a H.P.
- (d) both G.P. and H.P.
Show answer
Answer: (b)
80. If the probability for A to fail in an examination is 0.20 and that of B to fail is 0.30, then the probability that either A or B fails is
- (a) 0.50
- (b) 0.54
- (c) 0.44
- (d) 0.45
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Answer: (c)
81. In a group of 300 people, 150 speak Hindi and 200 can speak English. How many can speak both Hindi and English ?
- (a) 40
- (b) 50
- (c) 60
- (d) 85
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Answer: (b)
82. In a ΔABC, b = 5 cm, a = 2 cm and sin A = 3/7. How many such triangles are possible ?
- (a) 0
- (b) 1
- (c) 2
- (d) 3
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Answer: (a)
83. If for a moderately symmetrical distribution mean deviation is 12, then the value of standard deviation is
- (a) 15
- (b) 12
- (c) 24
- (d) None of the above
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Answer: (a)
84. 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)
85. In the given figure O is the centre of the circle and ∠AOB = 70° then 70° − x° is equal to
- (a) 70°
- (b) 35°
- (c) 15°
- (d) 25°
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Answer: (b)
86. If α, β, γ are the roots of the equation x³ + px² + qx + r = 0, then α²β² + α²γ² + β²γ² is equal to
- (a) q² − 2pr
- (b) q² + 2pr
- (c) q² + p²
- (d) q² − p²
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Answer: (a)
87. A particle is projected in vertically upward direction. If after t₁ and t₂ seconds, its height is h, then h is equal to
- (a) (1/2)gt₁t₂
- (b) g√(t₁t₂)
- (c) 2g (t₁ + t₂)
- (d) None of the above
Show answer
Answer: (a)
88. If points (a, 0), (0, b) and (1, 1) are collinear, then 1/a + 1/b is equal to
- (a) 1
- (b) √2
- (c) −1
- (d) 2
Show answer
Answer: (a)
89. The system of equations x + z = 3 ax + y = 2 3x + y + bz = 7 has a unique solution if (a, b) is equal to
- (a) (1, 2)
- (b) (2, 1)
- (c) (1, 1)
- (d) (− 1, 4)
Show answer
Answer: (c)
90. The greatest resultant which two forces can have is P and the least is Q. If they act at an angle α, then the resultant is of magnitude
- (a) √(P² + Q²)
- (b) √(P² − Q²)
- (c) √(P² cos²(α/2) + Q² sin²(α/2))
- (d) √(P² cos² α + Q² sin² α)
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Answer: (c)
91. The equation x²/(1 − r) − y²/(1 + r) = 1, |r| < 1 represents
- (a) an ellipse
- (b) a hyperbola
- (c) a circle
- (d) none of the above
Show answer
Answer: (b)
92. The function defined by at x = 0 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)
93. A force represented by î + 2ĵ + k̂ passes through a point (2, 3, 1). The moment of the force about point (1, 2, − 1) is
- (a) î − 2ĵ + k̂
- (b) 3î − ĵ − k̂
- (c) − 3î + ĵ + k̂
- (d) 3î + ĵ − k̂
Show answer
Answer: (c)
94. The tangent to the curve x = 2at²/(1 + t²), y = 2at³/(1 + t²) at the point for which t = 1/2, is
- (a) 13x − 16y = 2a
- (b) 13x + 16y = 2a
- (c) 13x − 8y = 4a
- (d) 13x + 8y = 4a
Show answer
Answer: (a)
95. If x + 1/x = 2 cos (π/5), then x⁵ + 1/x⁵ is equal to
- (a) − 2
- (b) 0
- (c) 1
- (d) 5
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Answer: (a)
96. A, B and C undertake to do a work for ₹ 500. A and B together do 2/5 of the work and rest is done by C alone. How much amount should C get ?
- (a) ₹ 300
- (b) ₹ 360
- (c) ₹ 200
- (d) ₹ 180
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Answer: (a)
97. A body is projected with velocity 40 metre per second inclined at an angle 60° with horizontal. The velocity of the body when it is at maximum height is
- (a) 10 m/s
- (b) 20 m/s
- (c) 15 m/s
- (d) 5 m/s
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Answer: (b)
98. If 1, ω, ω² are the cube roots of unity, then the value of (1 + ω²)(1 + ω⁴)(1 + ω⁸)(1 + ω¹⁶) is
- (a) −1
- (b) 0
- (c) 1
- (d) 2
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Answer: (c)
99. For any two sets A and B, A − (A − B) equals
- (a) A − B
- (b) B
- (c) A ∪ B
- (d) A ∩ B
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Answer: (d)
100. The distance of centre of gravity of uniform semicircular lamina of radius a, from the centre is
- (a) a/2
- (b) 2a/3π
- (c) a/3
- (d) 4a/3π
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Answer: (d)
101. The intercept on the line y = x by the circle x² + y² − 2x = 0 is AB. Equation of circle with AB as diameter is
- (a) x² + y² = 1
- (b) x² + y² = 2
- (c) x² + y² − x − y = 0
- (d) x² + y² − 3x − 3y + 4 = 0
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Answer: (c)
102. If A and B are two matrices such that AB = B and BA = A, then A² + B² is equal to
- (a) 2AB
- (b) 2BA
- (c) A + B
- (d) AB
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Answer: (c)
103. A stone thrown up is caught by the thrower after 6 sec. What will be the velocity of stone after 2 sec of its throwing ? (g = 9.8 m/s²)
- (a) 9.8 m/s
- (b) 10.2 m/s
- (c) 19.6 m/s
- (d) 20.0 m/s
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Answer: (a)
104. The equation of the directrix of the parabola y² + 4x + 4y + 2 = 0 is
- (a) x = −1
- (b) x = 1
- (c) x = −3/2
- (d) x = 3/2
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Answer: (d)
105. lim (y → a) (sin ((y − a)/2) · tan (πy/2a)) is equal to
- (a) 0
- (b) 1
- (c) π/a
- (d) −a/π
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Answer: (d)
106. The angle of intersection of the curves y = 4 − x² and y = x² is
- (a) tan⁻¹(2/7)
- (b) tan⁻¹(4√2/7)
- (c) tan⁻¹(3√2/7)
- (d) π/2
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Answer: (b)
107. For any complex number z, the minimum value of |z + 1| + |z − 2| is
- (a) −1
- (b) 1
- (c) 2
- (d) 3
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Answer: (d)
108. The number of generators of a cyclic group of order 10 is
- (a) 2
- (b) 3
- (c) 4
- (d) 5
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Answer: (c)
109. Distance between the straight lines x + 2y = 5 and 2x + 4y = 11 is
- (a) 1/√5
- (b) 1/(2√5)
- (c) √5
- (d) √5/2
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Answer: (b)
110. If three forces acting in one plane upon a rigid body, keep the body in equilibrium, then
- (a) They must meet in a point
- (b) They must be parallel
- (c) They must either meet in a point or be parallel
- (d) They must be perpendicular
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Answer: (c)
111. The value of n, for which (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the harmonic mean of a and b, is
- (a) − 1
- (b) 0
- (c) 1
- (d) 1/2
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Answer: (a)
112. If x is a real number, then the function f(x) = x³ + x² + x + 1 has
- (a) maximum value at x = 0
- (b) minimum value at x = 1
- (c) maximum value at x = − 1
- (d) neither maximum nor minimum value
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Answer: (d)
113. The minimum value of 4 cos θ + 3 is
- (a) −3
- (b) −1
- (c) 0
- (d) 1
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Answer: (b)
114. Two finite sets have m and n elements respectively. The number of subsets of the first set is greater than the number of the subsets of the second by 56. Then the value of m² + n² is equal to
- (a) 40
- (b) 38
- (c) 42
- (d) None of the above
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Answer: (d)
115. Ram can throw a ball vertically to a maximum height of 40 metres, then he can throw the ball in horizontal direction to which maximum distance ?
- (a) 40√2 metre
- (b) 20√2 metre
- (c) 20 metre
- (d) 80 metre
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Answer: (d)
116. How many words can be formed with the letters of the word ‘POSTMAN’, if every word begins with T and ends with M ?
- (a) 60
- (b) 120
- (c) 360
- (d) 720
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Answer: (b)
117. If f(x + 1/x) = x³ + 1/x³, then f(√3) is equal to
- (a) 0
- (b) 1
- (c) √3
- (d) 3√3
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Answer: (a)
118. The order of the differential equation y − x(dy/dx)³ = a(y² + (dy/dx)²) is
- (a) 3
- (b) 1
- (c) 2
- (d) 4
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Answer: (b)
119. If the foci of the ellipse x²/16 + y²/b² = 1 and the hyperbola x²/144 − y²/81 = 1/25 coincide, then the value of b² is
- (a) 1
- (b) 5
- (c) 7
- (d) 9
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Answer: (c)
120. If A⃗ and B⃗ are two vectors, then angle between vectors A⃗ + B⃗ and A⃗ × B⃗ is
- (a) 0°
- (b) 60°
- (c) 90°
- (d) 120°
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Answer: (c)
121. Which of the following is an open set ?
- (a) S = {1, 1/2, 1/3, ....}
- (b) N
- (c) Z
- (d) S = {x ∈ R : 1 < x < 3}
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Answer: (d)
122. If a⃗, b⃗ and c⃗ are three vectors in such a way that a⃗ + b⃗ + c⃗ = 0⃗ and |a⃗| = 1, |b⃗| = 4, |c⃗| = 2, then the value of 2(a⃗.b⃗ + b⃗.c⃗ + c⃗.a⃗) is equal to
- (a) 21
- (b) −21
- (c) 19
- (d) −21/2
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Answer: (b)
123. The co-ordinates of a moving point are (2t² + 4, 4t + 6). Its locus will be
- (a) Circle
- (b) Straight line
- (c) Parabola
- (d) None of the above
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Answer: (c)
124. For all x ∈ (−1, 1), tan h⁻¹x is equal to
- (a) log(x + √(x² + 1))
- (b) log(x − √(x² + 1))
- (c) (1/2) log ((1 + x)/(1 − x))
- (d) (1/2) log ((x + 1)/(x − 1))
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Answer: (c)
125. The number of generators of the cyclic group G of order 8 is
- (a) 1
- (b) 2
- (c) 3
- (d) 4
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Answer: (d)

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