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Question paper · UP PGT

UP PGT 2021 — Mathematics (Series A)

17 or 18 August 2021125 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 17 or 18 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.

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Mathematics

1. Sum of the series 1/(1.2) − 1/(2.3) + 1/(3.4) − 1/(4.5) + ........
- (a) 2 log 2 − 1
- (b) 2 log 2 − 3
- (c) 2 log 2
- (d) None of the above
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Answer: (a)
2. If the horizontal range of a projectile is equal to its gained maximum height, then its angle of projection is
- (a) π/4
- (b) tan⁻¹2
- (c) tan⁻¹4
- (d) π/3
Show answer
Answer: (c)
3. The real part of e^(sin(x+iy)) is
- (a) e^(sinx coshy) [cos(cosx sinhy)]
- (b) e^(sinx coshy) [sin(cosx sinhy)]
- (c) e^(cosx sinhy) [cos(cosx sinhy)]
- (d) e^(cosx sinhy) [sin(cosx sinhy)]
Show answer
Answer: (a)
4. sinh(x + iy) is equal to
- (a) sinx coshy + i coshx siny
- (b) sinhx cosy + i coshx siny
- (c) sinx coshy − i coshx siny
- (d) sinhx cosy − i coshx siny
Show answer
Answer: (b)
5. The equation of a right circular cone with vertex at the origin the axis the z axis and semi vertical angle π/4 is
- (a) x² + z² = y²
- (b) y² + x² = z²
- (c) z² + y² = x²
- (d) xy = z²
Show answer
Answer: (b)
6. lim_(x,y)→(2,1) sin⁻¹(xy − 2) / tan⁻¹(3xy − 6) is equal to
- (a) 1/3
- (b) 1/2
- (c) 1
- (d) 2
Show answer
Answer: (a)
7. The dimension of the vector space C(R) of the complex number over real numbers is
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Show answer
Answer: (b)
8. If the straight line y = mx lies outside the circle x² + y² − 20y + 90 = 0, then the value of m will satisfy
- (a) |m| < 3
- (b) m < 3
- (c) m > 3
- (d) |m| > 3
Show answer
Answer: (a)
9. The sum of the series lim_n→∞ [1/n + 1/(n+1) + 1/(n+2) + .... + 1/(4n)] is equal to
- (a) 0
- (b) 1
- (c) log 3
- (d) log 4
Show answer
Answer: (d)
10. The work done by the force F⃗ = (x² − y² + x)î − (2xy + y)ĵ, displacing a particle in the xy plane from (0, 0) to (1, 1) along the parabola y² = x, is
- (a) 2
- (b) 3
- (c) 1/2
- (d) none of these
Show answer
Answer: (d)
11. The distinct eigen values of the matrix [[1, 1, 0], [1, 1, 0], [0, 0, 0]] are
- (a) 0, 1
- (b) 1, − 1
- (c) 0, 2
- (d) 1, 2
Show answer
Answer: (c)
12. General solution of x²(y − px) = p²y where p = dy/dx is
- (a) y² − c² = 2cx³
- (b) x²(y − cx) = c²y
- (c) xy² = cx⁴ + c²
- (d) y² = cx² + c²
Show answer
Answer: (d)
13. Let (z, o), where a ∘ b ≝ a + b + 1, a, b ∈ z is a commutative group. Let a′ be inverse of a, then a′ is equal to
- (a) − a + 1
- (b) − a − 1
- (c) − a − 2
- (d) − a + 2
Show answer
Answer: (c)
14. If the two roots of the equation x³ − 5x² − 16x + 80 = 0 are 4 and − 4 then the third root of this equation is
- (a) 1
- (b) 2
- (c) 6
- (d) 5
Show answer
Answer: (d)
15. A hemisphere rests in equilibrium on a sphere of equal radius. If the flat surface of the hemisphere rests on the sphere then this equilibrium is
- (a) Stable
- (b) Unstable
- (c) Neutral
- (d) None of the above
Show answer
Answer: (a)
16. The line y = mx + 1 is a tangent to the parabola y² = 4x, if
- (a) m = 1
- (b) m = 2
- (c) m = − 1
- (d) m = − 2
Show answer
Answer: (a)
17. ∫₀^∞ e^(−x²) dx is equal to
- (a) (1/2)√π
- (b) π/2
- (c) π/√2
- (d) π
Show answer
Answer: (a)
18. The moment of inertia of a circular ring of radius a and mass M about an axis through the centre perpendicular its plane is
- (a) (1/2)Ma²
- (b) Ma²
- (c) (2/3)Ma²
- (d) (4/3)Ma²
Show answer
Answer: (b)
19. The eigen values of the matrix A = [[5, 4], [1, 2]] are
- (a) 6, 0
- (b) 3, 2
- (c) 6, 1
- (d) 1, 2
Show answer
Answer: (c)
20. If f(x) = (x + 2)/(x + 3) and y = f⁻¹(x), then dy/dx is equal to
- (a) 2/(x + 3)²
- (b) 1/(x − 1)²
- (c) (x − 2)/(x − 3)
- (d) 1/(x + 1)²
Show answer
Answer: (b)
21. The circles (x − 1)² + (y − 3)² = r² and x² + y² − 8x + 2y + 8 = 0 intersect at two distinct points. Which of the following is correct ?
- (a) r = 1
- (b) 1 < r < 1
- (c) r = 2
- (d) 2 < r < 8
Show answer
Answer: (d)
22. If y = sin(log x), then which of the following is correct ?
- (a) d²y/dx² + xy = 0
- (b) d²y/dx² + y = 0
- (c) x² d²y/dx² + x dy/dx + y = 0
- (d) x² d²y/dx² − x dy/dx + y = 0
Show answer
Answer: (c)
23. If B is a matrix such that B² = B and A = I − B, then which of the following is not correct ?
- (a) A² = A
- (b) A² = I
- (c) AB = 0
- (d) BA = 0
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Answer: (b)
24. The orthogonal trajectories to the family of straight lines y = k(x − 1), k ∈ ℝ, are given by
- (a) (x − 1)² + y² = c²
- (b) (x − 1)² + (y − 1)² = c²
- (c) ky + x − 1 = 0
- (d) x² + y² = c²
Show answer
Answer: (a)
25. If A = f(x) = [[cosx, sinx, 0], [−sinx, cosx, 0], [0, 0, 1]] then A⁻¹ is
- (a) f(x)
- (b) − f(x)
- (c) f(−x)
- (d) −f(−x)
Show answer
Answer: (c)
26. A and B throw a dice. The probability that A’s throw is greater than B’s throw in numbers is
- (a) 1/2
- (b) 5/6
- (c) 5/12
- (d) 7/12
Show answer
Answer: (c)
27. Asymptotes of the curve x²y² = a²(x² + y²) are
- (a) x = 0, y = 0
- (b) x = ± a, y = 0
- (c) x = 0, y = ± a
- (d) x = ± a, y = ± a
Show answer
Answer: (d)
28. The Cartesian equation of the common catenary is
- (a) y² = c² + x²
- (b) y = c cosh(x/c)
- (c) y = c secx
- (d) y = c tan hx
Show answer
Answer: (b)
29. A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition, the probability that the number formed is divisible by 4, is
- (a) 1/4
- (b) 2/5
- (c) 3/5
- (d) 1/5
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Answer: (d)
30. If sin(θ + iφ) = tan α + i sec α then cos 2θ cosh 2φ is equal to
- (a) 3
- (b) 2
- (c) 6
- (d) 4
Show answer
Answer: (a)
31. If f(x) = ax + b and f(f(f(x))) = 8x + 21 and if a, b are real numbers then a + b is equal to
- (a) 2
- (b) 3
- (c) 5
- (d) 7
Show answer
Answer: (c)
32. If the vector F⃗ = (x + 3y)î + (y − 2z)ĵ + (x − az)k̂ is solenoidal then a is equal to
- (a) 1
- (b) −1
- (c) 2
- (d) −2
Show answer
Answer: (c)
33. The radius of sphere x² + y² + z² + x + y + z = 4 is
- (a) 3
- (b) 4
- (c) √19/4
- (d) √19/2
Show answer
Answer: (d)
34. The equation of the sphere passing through the origin and making intercepts 1, 3, 5 with the three coordinate axes is
- (a) x² + y² + z² + x + 3y + 5z = 0
- (b) x² + y² + z² − x + 3y − 5z = 0
- (c) x² + y² + z² + x − 3y + 5z = 0
- (d) x² + y² + z² − x − 3y − 5z = 0
Show answer
Answer: (d)
35. If each element of a 3 × 3 matrix A is multiplied by 3 then the determinant of the newly formed matrix is
- (a) 3 |A|
- (b) 9 |A|
- (c) (|A|)³
- (d) 27 |A|
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Answer: (d)
36. The general solution of the partial differential equation ∂²z/∂x∂y = x + y is of the form z =
- (a) (1/2) xy (x − y) + F(x) + G(y)
- (b) (1/2) xy (x + y) + F(x) + G(y)
- (c) (1/2) xy (x − y) + F(x) G(y)
- (d) (1/2) xy (x + y) + F(x) G(y)
Show answer
Answer: (b)
37. Let G be a cyclic group of order 6. Then, the number of elements g ∈ G, such that G = <g> is
- (a) 2
- (b) 3
- (c) 4
- (d) 5
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Answer: (a)
38. Let ‘a’ be an element of a group and O(a) = 30, O(a¹⁸) is equal to
- (a) 2
- (b) 5
- (c) 6
- (d) 10
Show answer
Answer: (b)
39. If the vectors xî − 3ĵ + 7k̂ and î − yĵ − zk̂ are colinear then the value of xy²/z is equal to
- (a) 9/7
- (b) 6/7
- (c) −6/7
- (d) none of the above
Show answer
Answer: (d)
40. If complex numbers a₁, a₂, a₃, ... are in G.P. having common ratio r such that Σ_(k=1)^n a_(2k−1) = Σ_(k=1)^n a_(2k+2) ≠ 0 then number of possible values of r is
- (a) 1
- (b) 2
- (c) 3
- (d) 4
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Answer: (c)
41. Real part of sin²(x + iy) is
- (a) (1/2)[1 + cos 2x cosh 2y]
- (b) (1/2)[1 − cos 2x cosh 2y]
- (c) (1/2)[1 + sin 2x sinh 2y]
- (d) (1/2)[1 − sin 2x cosh 2y]
Show answer
Answer: (b)
42. The distance between the foci of a hyperbola is 16 and its eccentricity is √2 the equation of hyperbola is
- (a) x² − y² = 32
- (b) 2x² − y² = 16
- (c) x² − 2y² = 32
- (d) x² − y² = 8
Show answer
Answer: (a)
43. The value of 5² + 6² + 7² + .... + 20² is
- (a) 2040
- (b) 2540
- (c) 2840
- (d) 3840
Show answer
Answer: (c)
44. If V is a n-dimensional vector space and T is a linear transformation on V such that rank and nullity of T are identical then
- (a) n is even
- (b) n is odd
- (c) some times even some times odd
- (d) none of the above
Show answer
Answer: (a)
45. The function f(x) = (1 − cos x)/x², x ≠ 0 can be made continuous at x = 0 by defining f(0) to be equal to
- (a) 1
- (b) 1/2
- (c) 0
- (d) 2
Show answer
Answer: (b)
46. If f(a − x) = f(x) then ∫₀^a x f(x) dx is equal to
- (a) (a/2) ∫₀^a f(x) dx
- (b) a ∫₀^a f(x) dx
- (c) 0
- (d) 2 ∫₀^a f(x) dx
Show answer
Answer: (a)
47. If f(x) = (ae^(bx) + be^(ax))/(a + b), then f″(0) equals
- (a) 0
- (b) ab
- (c) a + b
- (d) ab(a + b)
Show answer
Answer: (b)
48. If A⃗ and B⃗ are vectors such that |A⃗| = |B⃗| = 5 and A⃗ × B⃗ = 4î − 3k̂, then A⃗.B⃗ is equal to
- (a) 5√6
- (b) 5√2
- (c) 10√2
- (d) 10√6
Show answer
Answer: (d)
49. If the line ax + by + c = 0 touches the parabola y² = x, then which of the following is correct ?
- (a) abc = 1
- (b) b² = 4ac
- (c) a² = 4bc
- (d) c² = 4ab
Show answer
Answer: (b)
50. The solution of PDE x ∂u/∂x + y ∂u/∂y = 0 is of the form u =
- (a) f(x + y)
- (b) f(x − y)
- (c) f(y/x)
- (d) f(x y)
Show answer
Answer: (c)
51. Let G be a group of order 30 and let A, B be normal subgroups of orders 2 and 5 respectively. Then O(G/AB) is
- (a) 2
- (b) 3
- (c) 5
- (d) 10
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Answer: (b)
52. If α and β are roots of the equation x² + x + 1 = 0, then the equation whose roots are α⁷ and β⁴ is
- (a) x² − x − 1 = 0
- (b) x² − x + 1 = 0
- (c) x² + x − 1 = 0
- (d) x² + x + 1 = 0
Show answer
Answer: (d)
53. Sum of the series 1/2! + (1+2)/3! + (1+2+3)/4! + (1+2+3+4)/5! + .... is equal to
- (a) 2e
- (b) e
- (c) e − 1
- (d) e/2
Show answer
Answer: (d)
54. Value of ∬_S F⃗.n̂ ds, where F⃗ = 4xzî − y²ĵ + yzk̂ and S is the surface of the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 is
- (a) 1
- (b) 3/2
- (c) 3
- (d) 5/2
Show answer
Answer: (b)
55. If the nullity of the matrix [[k, 1, 2], [1, −1, −2], [1, 1, 4]] is 1, then the value of k is
- (a) 0
- (b) 1
- (c) 2
- (d) −1
Show answer
Answer: (d)
56. The smallest value of positive integer n, for which (1 + i)ⁿ = (1 − i)ⁿ, is
- (a) 2
- (b) 4
- (c) 6
- (d) 8
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Answer: (b)
57. The area inside the cardioid r = a(1 + cosθ) and outside the circle r = a is
- (a) a²(π + 2)
- (b) a²(π/4 + 2)
- (c) a²(π − 2)
- (d) none of these
Show answer
Answer: (b)
58. Let â and b̂ be two unit vectors and θ be the angle between them. Which of the following will be value of cos(θ/2) ?
- (a) |â + b̂|/4
- (b) |â − b̂|/4
- (c) |â + b̂|/2
- (d) |â − b̂|/2
Show answer
Answer: (c)
59. If A and B are two sets such that n(A) = 4, n(B) = 3 then the maximum value of n(A∩B) is
- (a) 0
- (b) 1
- (c) 4
- (d) 3
Show answer
Answer: (d)
60. The differential equation of y = ae^(−bx) (a and b are parameters) is
- (a) y (dy/dx) = (d²y/dx²)²
- (b) y (d²y/dx²) = (dy/dx)²
- (c) y (dy/dx)² = d²y/dx²
- (d) y (d²y/dx²)² = dy/dx
Show answer
Answer: (b)
61. The sum of |1 4 7; 2 5 8; 3 6 9| and |4 4 7; 5 5 8; 7 6 9| will be
- (a) zero
- (b) |5 8 14; 7 10 16; 10 12 18|
- (c) |5 4 7; 7 5 8; 10 6 9|
- (d) |1 8 14; 2 10 16; 3 12 18|
Show answer
Answer: (c)
62. The coefficient of xⁿ in the expansion of (1 − 2x + 3x² − 4x³ + .....)⁻ⁿ
- (a) (−1)ⁿ⁺¹ n
- (b) (2n)!/(n!)²
- (c) (2n)!/((n+1)! (n−1)!)
- (d) None of the above
Show answer
Answer: (b)
63. In the expansion of (x³/4 − 2/x²)⁹ the 4ᵗʰ term from the end is
- (a) 48/x³
- (b) 84/x³
- (c) 64/x³
- (d) 72/x³
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Answer: (b)
64. A sphere after collision with a plane vertically downwards with velocity 20 cm per second returns upwards with velocity 4 cm/second then the value of the coefficient of restitution e is
- (a) 1/2
- (b) 1/3
- (c) 1/4
- (d) 1/5
Show answer
Answer: (d)
65. Consider the following statements I : d/dx sechx = sechx tanhx II : d/dx sinh⁻¹x = 1/√(x² + 1) which of the following is/are true ?
- (a) only I
- (b) only II
- (c) I and II both
- (d) neither I nor II
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Answer: (b)
66. The straight lines 3x − 4y + 4 = 0 and 6x − 8y + 13 = 0 are tangents to the same circle. The radius of the circle is
- (a) 1/2
- (b) 1/4
- (c) 3/2
- (d) 2
Show answer
Answer: (b)
67. The equation of the director sphere of the central conicoid x²/a² + y²/b² + z²/c² = 1 is
- (a) x² + y² + z² = a² + b² + c²
- (b) x² + y² + z² = 1/a² + 1/b² + 1/c²
- (c) ax² + by² + cz² = a² + b² + c²
- (d) x² + y² + z² = 1
Show answer
Answer: (a)
68. The probabilities of winning a race by three racers P, Q, R are 1/3, 1/4 and 1/5 respectively. The probability of none of them wins in the race is
- (a) 1/5
- (b) 13/60
- (c) 2/5
- (d) None of the above
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Answer: (c)
69. The area lying between line y = x, x axis and ordinates x = 0 and x = 2 is revolved about x axis. The centre of gravity of the solid thus generated is at the following point.
- (a) (1/2, 0)
- (b) (3/2, 0)
- (c) (3/4, 0)
- (d) (1/4, 0)
Show answer
Answer: (b)
70. If the plane x + 2y + 3z = p touches the conicoid x² − 2y² + 3z² = 2, then the value of p is
- (a) 0
- (b) 1
- (c) 4
- (d) 2
Show answer
Answer: (d)
71. If a, b, c are in arithmetic progression then the value of |x+1 x+2 x+a; x+2 x+3 x+b; x+3 x+4 x+c| is
- (a) 0
- (b) 1
- (c) 2
- (d) a b c
Show answer
Answer: (a)
72. If X = {1, 2, 3, 4} then the relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 2), (2, 3), (2, 1), (1, 2)} defined on X is
- (a) reflexive, symmetric and transitive
- (b) reflexive, symmetric but not transitive
- (c) symmetric, transitive but not reflexive
- (d) reflexive, transitive but not symmetric
Show answer
Answer: (b)
73. The solution of differential equation (dy/dx)² − 7 dy/dx + 12 = 0 is
- (a) (y − 4x + c) (y − 3x + c) = 0
- (b) (y + x + c) (y − x + c) = 0
- (c) (y + 4x + c) (y + 3x + c) = 0
- (d) (y + 2x + c) (y + 3x + c) = 0
Show answer
Answer: (a)
74. Equation of the curve passing through (1, 1) and satisfying the differential equation dy/dx + y/x = x² is
- (a) xy = x⁴ + 3
- (b) 4xy + x⁴ = 3
- (c) yeˣ = x⁴ + 3
- (d) 4xy = x⁴ + 3
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Answer: (d)
75. In the group (Z, +), the subgroup generated by 2 and 7 is
- (a) 9Z
- (b) 14Z
- (c) Z
- (d) 5Z
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Answer: (c)
76. If a⃗ and b⃗ are irrotational vectors then div(a⃗ × b⃗) is equal to
- (a) 1
- (b) 2
- (c) 3
- (d) 0
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Answer: (d)
77. The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and 2lm + 2nl − mn = 0 is
- (a) 45°
- (b) 90°
- (c) 120°
- (d) none of the above
Show answer
Answer: (c)
78. If the difference of the roots of the equation x² + px + 12 = 0 is one then the values of p are
- (a) ± 7
- (b) ± 2
- (c) ± 3
- (d) ± 1
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Answer: (a)
79. The dimension of the vector space of all 3 × 3 real symmetric matrices is
- (a) 3
- (b) 6
- (c) 3n
- (d) 9
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Answer: (b)
80. Sum of maximum and minimum values of 4(sin²θ + cos⁴θ) is
- (a) 3
- (b) 4
- (c) 5
- (d) 7
Show answer
Answer: (d)
81. The value of the determinant |1² 2² 3² 4²; 2² 3² 4² 5²; 3² 4² 5² 6²; 4² 5² 6² 7²| is
- (a) 60
- (b) 96
- (c) 120
- (d) None of the above
Show answer
Answer: (d)
82. If r̂ = r⃗/r, r = |r⃗|, then div r̂ is equal to
- (a) 0
- (b) −1
- (c) 1/r
- (d) 2/r
Show answer
Answer: (d)
83. If N is the set of natural numbers then the mapping f : N → N defined by f(x) = x + 1 if x is odd, x − 1 if x is even is
- (a) one-one and onto
- (b) many to one and onto
- (c) one-one and into
- (d) many to one and into
Show answer
Answer: (a)
84. Solution of the differential equation logₑ(dy/dx) = 2x − y, y(0) = 0 is
- (a) eʸ = 2e²ˣ + 1
- (b) 2eʸ = e²ˣ + 1
- (c) eʸ = 2e²ˣ + 3
- (d) 2eʸ = e²ˣ − 1
Show answer
Answer: (b)
85. If x = log(secθ + tanθ), then coshx is equal to
- (a) tan θ
- (b) cos θ
- (c) sin θ
- (d) sec θ
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Answer: (d)
86. Which of the following functions T : R² → R² is a linear transformation ?
- (a) T(x, y) = (x + 1, y)
- (b) T(x, y) = (x, y + 1)
- (c) T(x, y) = (x + y, 0)
- (d) T(x, y) = (x − 1, y)
Show answer
Answer: (c)
87. The radius of curvature of the curve s = a log tan(π/4 + ψ/2) at (s, ψ) is
- (a) a tanψ
- (b) a secψ
- (c) a sec²ψ
- (d) a secψ tanψ
Show answer
Answer: (b)
88. If f(2) = 4 and f′(2) = 1 then lim_x→2 (x f(2) − 2 f(x))/(x − 2) is equal to
- (a) 2
- (b) 0
- (c) 1
- (d) 4
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Answer: (a)
89. If A is a singular matrix, then A. adj(A) is
- (a) an identity matrix
- (b) a null matrix
- (c) a scalar matrix
- (d) none of the above
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Answer: (b)
90. A unit vector, which is normal to the surface x² − xy + z² = 1 at the point (1, 1, 1) is
- (a) (î − ĵ + 2k̂)/√6
- (b) (î + ĵ − 2k̂)/√6
- (c) (î + ĵ + 2k̂)/√6
- (d) (î − ĵ − 2k̂)/√6
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Answer: (a)
91. Statement A : Every isomorphic image of a cyclic group is cyclic. Statement B : Every cyclic group is abelian. Then
- (a) Both A and B are true
- (b) Both A and B are false
- (c) A is true only
- (d) B is true only
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Answer: (a)
92. The value of ∫₀¹ x⁷/(1 + x¹⁶) dx is equal to
- (a) π/4
- (b) 0
- (c) π/32
- (d) 1
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Answer: (c)
93. If normal at any point P of a common catenary meets the directrix at Q and ρ is the radius of curvature of the catenary at P then PQ is equal to
- (a) ρ
- (b) c sec ψ
- (c) c tan ψ
- (d) c² sec ψ
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Answer: (a)
94. The value of sin log(iⁱ) is
- (a) 0
- (b) 1
- (c) −1
- (d) 1/2
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Answer: (c)
95. A stone just clears a wall of height 75 meters situated at a distance 150 meter and goes in horizontal direction, then the angle of projection is
- (a) 30°
- (b) 60°
- (c) 45°
- (d) 75°
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Answer: (c)
96. The composite mapping fog(x) of the maps f : R → R, f(x) = sinx; g : R → R, g(x) = x², is
- (a) sin x + x²
- (b) (sin x)²
- (c) sin x²
- (d) x² sin x
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Answer: (c)
97. The area of the region bounded by the curve y = x³, x axis and the ordinates x = − 2 and x = 1 is
- (a) 1 square unit
- (b) 1/4 square unit
- (c) 3/4 square unit
- (d) 17/4 square unit
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Answer: (d)
98. Σ_(r=0)^n 3ʳ ⁿCᵣ is equal to
- (a) 2ⁿ
- (b) 3ⁿ
- (c) 4ⁿ
- (d) 1
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Answer: (c)
99. If w is cube root of unity, then 1 + w + w² + w³ + ..... + w⁵², w ≠ 1 is equal to
- (a) w
- (b) 1 − w
- (c) −w²
- (d) 1 + w²
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Answer: (c)
100. Two uniform solid spheres composed of the same material and having their radii 6 cm and 3 cm respectively are firmly united the distance of the centre of gravity of the whole body from the centre of the larger sphere is
- (a) 4 cms
- (b) 3 cms
- (c) 2 cms
- (d) 1 cm
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Answer: (d)
101. Let T be a linear transformation from R³ → R², defined by T(x, y, z) = (x + y, y − z) then the matrix T with respect to the ordered basis {(1, 1, 1), (1, −1, 0), (0, 1, 0)} and {(1,1), (1, 0)} is
- (a) [[0, −1, 1], [2, 1, 0]]
- (b) [[−2, 0, 1], [1, 1, −1]]
- (c) [[2, 1], [0, −1], [1, 1]]
- (d) [[0, 2], [−1, 1], [1, 0]]
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Answer: (a)
102. Let V(F) be a finite dimensional vector space over the field F and W be a subspace of V. If dim V = 5, dim W = 3 then dim W⁰ is
- (a) 2
- (b) 3
- (c) 1
- (d) 8
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Answer: (a)
103. If y = 4x − 5 is equation of the tangent to a curve y² = ax³ + b at (2, 3), then (a, b) is equal to
- (a) (2, 7)
- (b) (2, −7)
- (c) (−2, 7)
- (d) (−2, −7)
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Answer: (b)
104. Moment of inertia of a hollow sphere about a diameter whose mass is M and radius a, is
- (a) M.(2a²/3)
- (b) M.(2a²/5)
- (c) M.(a²/4)
- (d) M.(a²/3)
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Answer: (a)
105. If (G, ∗) is a group and x ∗ y = x + 2y − 3 ∀ x, y ∈ G, then inverse of x in the group is
- (a) (2x + 9)/4
- (b) (9 − 2x)/4
- (c) (x − 3)/4
- (d) (x + 2)/4
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Answer: (b)
106. If W (≠ 1) is a cube root of unity and (1 + W)⁷ = A + BW then the value of A² + B² is
- (a) 0
- (b) 1
- (c) 2
- (d) 4
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Answer: (c)
107. ∫₀^(π/4) (cos 2θ)^(3/2) cosθ dθ is equal to
- (a) π/√2
- (b) π/(4√2)
- (c) 3π/(16√2)
- (d) 3π/8
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Answer: (c)
108. If y = −1 when x = 0 then the solution of the differential equation (1 + e²ˣ) dy + (1 + y²) eˣ dx = 0 is
- (a) tan⁻¹y + tan⁻¹eˣ = 0
- (b) tan⁻¹xy + tan⁻¹eˣ = 0
- (c) tan⁻¹y + tan⁻¹(xeˣ) = 0
- (d) none of the above
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Answer: (a)
109. If f(x) = ax² + 2bx + 1, a and b are positive real numbers and b² < a, then which of the following is correct ?
- (a) f(x) = 0 ∀ x ∈ R
- (b) f(x) > 0 ∀ x ∈ R
- (c) f(x) < 0 ∀ x ∈ R
- (d) None of the above
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Answer: (b)
110. If H and K are subgroups of a group G such that O(H) = 3 and O(K) = 5, then what will be O(H∩K) ?
- (a) 1
- (b) 3
- (c) 5
- (d) 15
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Answer: (a)
111. The equation of second degree x² + 2√2 xy + 2y² + 4x + 4√2 y + 1 = 0 represents a pair of straight lines, the distance between them is
- (a) 4
- (b) 4/√3
- (c) 2
- (d) 2√3
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Answer: (c)
112. If xʸ = yˣ, then (x/y)(dy/dx) is equal to
- (a) (x logy + y)/(y logx + x)
- (b) (x logy − y)/(y logx − x)
- (c) (y logx + x)/(x logy + y)
- (d) (y logx − x)/(x logy − y)
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Answer: (b)
113. A particle moves along the curve x = t³ − 2, y = t² + t, z = 2t + 1. The component of its acceleration at t = 1 in the direction î − ĵ + k̂ is
- (a) 4
- (b) 4√3
- (c) 4/√3
- (d) 2
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Answer: (c)
114. If V = (x² + y² + z²)^(−1/2), then x ∂V/∂x + y ∂V/∂y + z ∂V/∂z is equal to
- (a) V
- (b) (1/2)V
- (c) −V
- (d) 0
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Answer: (c)
115. If f(x) = |x − 1| + |x| then f′(1) is equal to
- (a) 0
- (b) 1
- (c) −1
- (d) does not exist
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Answer: (d)
116. The solution of the partial differential equation (mz − ny) ∂z/∂x + (nx − lz) ∂z/∂y = ly − mx is
- (a) f(x² + xz, y² + yz) = 0
- (b) f(z² + xy, y² + xz) = 0
- (c) f(x² + y², lx + my) = 0
- (d) f(x² + y² + z², lx + my + nz) = 0
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Answer: (d)
117. If the equation hxy + gx + fy = c, h ≠ 0 represents a pair of straight lines, then
- (a) fc + gh = 0
- (b) fh + cg = 0
- (c) gf + ch = 0
- (d) gc + f² = 0
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Answer: (c)
118. For the function f(x) = |x − 5|, which of the following is not correct ?
- (a) The function f(x) is continuous at x = 5
- (b) The function f(x) is not continuous at x = − 5
- (c) The function f(x) is differentiable at x = 0
- (d) The function f(x) is differentiable at x = − 5
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Answer: (b)
119. The order and degree of the differential equation k d²y/dx² = [1 + (dy/dx)²]^(3/2) are
- (a) order 2 degree 3
- (b) order 2 degree 2
- (c) order 3 degree 2
- (d) none of the above
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Answer: (b)
120. If the normal to curve y = f(x) at the point (a, b) makes an angle 3π/4 with the positive x axis then f′(a) is equal to
- (a) 1
- (b) −1
- (c) a/b
- (d) b/a
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Answer: (a)
121. If u = sin⁻¹((x² + y²)/(x + y)), then x ∂u/∂x + y ∂u/∂y is equal to
- (a) cos 2u
- (b) tan u
- (c) tan 2u
- (d) cot u
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Answer: (b)
122. If W = {(x, y, z) ∈ R³ : x + y − z = 0} is a subspace of the vector space R³, then dim W is
- (a) 0
- (b) 1
- (c) 2
- (d) 3
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Answer: (c)
123. If y = cos(3 cos⁻¹x), then d³y/dx³ is equal to
- (a) 0
- (b) 3
- (c) 16
- (d) 24
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Answer: (d)
124. A non commutative group has at least
- (a) 2 elements
- (b) 3 elements
- (c) 5 elements
- (d) 6 elements
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Answer: (d)
125. The equation of the tangents drawn from the point (−2, −1) to the hyperbola 2x² − 3y² = 6 are
- (a) 3x + y + 5 = 0, x − y + 1 = 0
- (b) 3x + y + 5 = 0, x + y + 1 = 0
- (c) 3x − y + 5 = 0, x + y + 1 = 0
- (d) 3x − y + 5 = 0, x − y + 1 = 0
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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: (a)
Q 2: (c)
Q 3: (a)
Q 4: (b)
Q 5: (b)
Q 6: (a)
Q 7: (b)
Q 8: (a)
Q 9: (d)
Q 10: (d)
Q 11: (c)
Q 12: (d)
Q 13: (c)
Q 14: (d)
Q 15: (a)
Q 16: (a)
Q 17: (a)
Q 18: (b)
Q 19: (c)
Q 20: (b)
Q 21: (d)
Q 22: (c)
Q 23: (b)
Q 24: (a)
Q 25: (c)
Q 26: (c)
Q 27: (d)
Q 28: (b)
Q 29: (d)
Q 30: (a)
Q 31: (c)
Q 32: (c)
Q 33: (d)
Q 34: (d)
Q 35: (d)
Q 36: (b)
Q 37: (a)
Q 38: (b)
Q 39: (d)
Q 40: (c)
Q 41: (b)
Q 42: (a)
Q 43: (c)
Q 44: (a)
Q 45: (b)
Q 46: (a)
Q 47: (b)
Q 48: (d)
Q 49: (b)
Q 50: (c)
Q 51: (b)
Q 52: (d)
Q 53: (d)
Q 54: (b)
Q 55: (d)
Q 56: (b)
Q 57: (b)
Q 58: (c)
Q 59: (d)
Q 60: (b)
Q 61: (c)
Q 62: (b)
Q 63: (b)
Q 64: (d)
Q 65: (b)
Q 66: (b)
Q 67: (a)
Q 68: (c)
Q 69: (b)
Q 70: (d)
Q 71: (a)
Q 72: (b)
Q 73: (a)
Q 74: (d)
Q 75: (c)
Q 76: (d)
Q 77: (c)
Q 78: (a)
Q 79: (b)
Q 80: (d)
Q 81: (d)
Q 82: (d)
Q 83: (a)
Q 84: (b)
Q 85: (d)
Q 86: (c)
Q 87: (b)
Q 88: (a)
Q 89: (b)
Q 90: (a)
Q 91: (a)
Q 92: (c)
Q 93: (a)
Q 94: (c)
Q 95: (c)
Q 96: (c)
Q 97: (d)
Q 98: (c)
Q 99: (c)
Q 100: (d)
Q 101: (a)
Q 102: (a)
Q 103: (b)
Q 104: (a)
Q 105: (b)
Q 106: (c)
Q 107: (c)
Q 108: (a)
Q 109: (b)
Q 110: (a)
Q 111: (c)
Q 112: (b)
Q 113: (c)
Q 114: (c)
Q 115: (d)
Q 116: (d)
Q 117: (c)
Q 118: (b)
Q 119: (b)
Q 120: (a)
Q 121: (b)
Q 122: (c)
Q 123: (d)
Q 124: (d)
Q 125: (d)