UP PGT 2021 — Mathematics (Series A)

Mathematics

1. 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
   उत्तर देखें

   उत्तर: (a)

2. 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
   उत्तर देखें

   उत्तर: (c)

3. 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)]
   उत्तर देखें

   उत्तर: (a)

4. 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
   उत्तर देखें

   उत्तर: (b)

5. 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
   उत्तर देखें

   उत्तर: (b)

6. 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
   उत्तर देखें

   उत्तर: (a)

7. 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
   उत्तर देखें

   उत्तर: (b)

8. 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
   उत्तर देखें

   उत्तर: (a)

9. 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
   उत्तर देखें

   उत्तर: (d)

10. 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
    उत्तर देखें

    उत्तर: (d)

11. 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
    उत्तर देखें

    उत्तर: (c)

12. 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
    उत्तर देखें

    उत्तर: (d)

13. 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
    उत्तर देखें

    उत्तर: (c)

14. 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
    उत्तर देखें

    उत्तर: (d)

15. 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
    उत्तर देखें

    उत्तर: (a)

16. 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
    उत्तर देखें

    उत्तर: (a)

17. 17. ∫₀^∞ e^(−x²) dx is equal to

    • (a) (1/2)√π
    • (b) π/2
    • (c) π/√2
    • (d) π
    उत्तर देखें

    उत्तर: (a)

18. 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
    उत्तर देखें

    उत्तर: (b)

19. 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
    उत्तर देखें

    उत्तर: (c)

20. 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
    उत्तर देखें

    उत्तर: (b)

21. 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
    उत्तर देखें

    उत्तर: (d)

22. 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
    उत्तर देखें

    उत्तर: (c)

23. 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
    उत्तर देखें

    उत्तर: (b)

24. 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
    उत्तर देखें

    उत्तर: (a)

25. 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)
    उत्तर देखें

    उत्तर: (c)

26. 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
    उत्तर देखें

    उत्तर: (c)

27. 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
    उत्तर देखें

    उत्तर: (d)

28. 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
    उत्तर देखें

    उत्तर: (b)

29. 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
    उत्तर देखें

    उत्तर: (d)

30. 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
    उत्तर देखें

    उत्तर: (a)

31. 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
    उत्तर देखें

    उत्तर: (c)

32. 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
    उत्तर देखें

    उत्तर: (c)

33. 33. The radius of sphere x² + y² + z² + x + y + z = 4 is

    • (a) 3
    • (b) 4
    • (c) √19/4
    • (d) √19/2
    उत्तर देखें

    उत्तर: (d)

34. 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
    उत्तर देखें

    उत्तर: (d)

35. 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|
    उत्तर देखें

    उत्तर: (d)

36. 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)
    उत्तर देखें

    उत्तर: (b)

37. 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
    उत्तर देखें

    उत्तर: (a)

38. 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
    उत्तर देखें

    उत्तर: (b)

39. 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
    उत्तर देखें

    उत्तर: (d)

40. 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
    उत्तर देखें

    उत्तर: (c)

41. 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]
    उत्तर देखें

    उत्तर: (b)

42. 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
    उत्तर देखें

    उत्तर: (a)

43. 43. The value of 5² + 6² + 7² + .... + 20² is

    • (a) 2040
    • (b) 2540
    • (c) 2840
    • (d) 3840
    उत्तर देखें

    उत्तर: (c)

44. 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
    उत्तर देखें

    उत्तर: (a)

45. 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
    उत्तर देखें

    उत्तर: (b)

46. 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
    उत्तर देखें

    उत्तर: (a)

47. 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)
    उत्तर देखें

    उत्तर: (b)

48. 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
    उत्तर देखें

    उत्तर: (d)

49. 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
    उत्तर देखें

    उत्तर: (b)

50. 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)
    उत्तर देखें

    उत्तर: (c)

51. 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
    उत्तर देखें

    उत्तर: (b)

52. 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
    उत्तर देखें

    उत्तर: (d)

53. 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
    उत्तर देखें

    उत्तर: (d)

54. 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
    उत्तर देखें

    उत्तर: (b)

55. 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
    उत्तर देखें

    उत्तर: (d)

56. 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
    उत्तर देखें

    उत्तर: (b)

57. 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
    उत्तर देखें

    उत्तर: (b)

58. 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
    उत्तर देखें

    उत्तर: (c)

59. 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
    उत्तर देखें

    उत्तर: (d)

60. 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
    उत्तर देखें

    उत्तर: (b)

61. 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|
    उत्तर देखें

    उत्तर: (c)

62. 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
    उत्तर देखें

    उत्तर: (b)

63. 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
    उत्तर देखें

    उत्तर: (b)

64. 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
    उत्तर देखें

    उत्तर: (d)

65. 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
    उत्तर देखें

    उत्तर: (b)

66. 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
    उत्तर देखें

    उत्तर: (b)

67. 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
    उत्तर देखें

    उत्तर: (a)

68. 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
    उत्तर देखें

    उत्तर: (c)

69. 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)
    उत्तर देखें

    उत्तर: (b)

70. 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
    उत्तर देखें

    उत्तर: (d)

71. 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
    उत्तर देखें

    उत्तर: (a)

72. 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
    उत्तर देखें

    उत्तर: (b)

73. 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
    उत्तर देखें

    उत्तर: (a)

74. 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
    उत्तर देखें

    उत्तर: (d)

75. 75. In the group (Z, +), the subgroup generated by 2 and 7 is

    • (a) 9Z
    • (b) 14Z
    • (c) Z
    • (d) 5Z
    उत्तर देखें

    उत्तर: (c)

76. 76. If a⃗ and b⃗ are irrotational vectors then div(a⃗ × b⃗) is equal to

    • (a) 1
    • (b) 2
    • (c) 3
    • (d) 0
    उत्तर देखें

    उत्तर: (d)

77. 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
    उत्तर देखें

    उत्तर: (c)

78. 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
    उत्तर देखें

    उत्तर: (a)

79. 79. The dimension of the vector space of all 3 × 3 real symmetric matrices is

    • (a) 3
    • (b) 6
    • (c) 3n
    • (d) 9
    उत्तर देखें

    उत्तर: (b)

80. 80. Sum of maximum and minimum values of 4(sin²θ + cos⁴θ) is

    • (a) 3
    • (b) 4
    • (c) 5
    • (d) 7
    उत्तर देखें

    उत्तर: (d)

81. 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
    उत्तर देखें

    उत्तर: (d)

82. 82. If r̂ = r⃗/r, r = |r⃗|, then div r̂ is equal to

    • (a) 0
    • (b) −1
    • (c) 1/r
    • (d) 2/r
    उत्तर देखें

    उत्तर: (d)

83. 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
    उत्तर देखें

    उत्तर: (a)

84. 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
    उत्तर देखें

    उत्तर: (b)

85. 85. If x = log(secθ + tanθ), then coshx is equal to

    • (a) tan θ
    • (b) cos θ
    • (c) sin θ
    • (d) sec θ
    उत्तर देखें

    उत्तर: (d)

86. 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)
    उत्तर देखें

    उत्तर: (c)

87. 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
    उत्तर देखें

    उत्तर: (b)

88. 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
    उत्तर देखें

    उत्तर: (a)

89. 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
    उत्तर देखें

    उत्तर: (b)

90. 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
    उत्तर देखें

    उत्तर: (a)

91. 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
    उत्तर देखें

    उत्तर: (a)

92. 92. The value of ∫₀¹ x⁷/(1 + x¹⁶) dx is equal to

    • (a) π/4
    • (b) 0
    • (c) π/32
    • (d) 1
    उत्तर देखें

    उत्तर: (c)

93. 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
    उत्तर देखें

    उत्तर: (a)

94. 94. The value of sin log(iⁱ) is

    • (a) 0
    • (b) 1
    • (c) −1
    • (d) 1/2
    उत्तर देखें

    उत्तर: (c)

95. 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°
    उत्तर देखें

    उत्तर: (c)

96. 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
    उत्तर देखें

    उत्तर: (c)

97. 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
    उत्तर देखें

    उत्तर: (d)

98. 98. Σ_(r=0)^n 3ʳ ⁿCᵣ is equal to

    • (a) 2ⁿ
    • (b) 3ⁿ
    • (c) 4ⁿ
    • (d) 1
    उत्तर देखें

    उत्तर: (c)

99. 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
    उत्तर देखें

    उत्तर: (c)

100. 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
     उत्तर देखें

     उत्तर: (d)

101. 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]]
     उत्तर देखें

     उत्तर: (a)

102. 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
     उत्तर देखें

     उत्तर: (a)

103. 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)
     उत्तर देखें

     उत्तर: (b)

104. 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)
     उत्तर देखें

     उत्तर: (a)

105. 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
     उत्तर देखें

     उत्तर: (b)

106. 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
     उत्तर देखें

     उत्तर: (c)

107. 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
     उत्तर देखें

     उत्तर: (c)

108. 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
     उत्तर देखें

     उत्तर: (a)

109. 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
     उत्तर देखें

     उत्तर: (b)

110. 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
     उत्तर देखें

     उत्तर: (a)

111. 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
     उत्तर देखें

     उत्तर: (c)

112. 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)
     उत्तर देखें

     उत्तर: (b)

113. 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
     उत्तर देखें

     उत्तर: (c)

114. 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
     उत्तर देखें

     उत्तर: (c)

115. 115. If f(x) = |x − 1| + |x| then f′(1) is equal to

     • (a) 0
     • (b) 1
     • (c) −1
     • (d) does not exist
     उत्तर देखें

     उत्तर: (d)

116. 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
     उत्तर देखें

     उत्तर: (d)

117. 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
     उत्तर देखें

     उत्तर: (c)

118. 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
     उत्तर देखें

     उत्तर: (b)

119. 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
     उत्तर देखें

     उत्तर: (b)

120. 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
     उत्तर देखें

     उत्तर: (a)

121. 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
     उत्तर देखें

     उत्तर: (b)

122. 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
     उत्तर देखें

     उत्तर: (c)

123. 123. If y = cos(3 cos⁻¹x), then d³y/dx³ is equal to

     • (a) 0
     • (b) 3
     • (c) 16
     • (d) 24
     उत्तर देखें

     उत्तर: (d)

124. 124. A non commutative group has at least

     • (a) 2 elements
     • (b) 3 elements
     • (c) 5 elements
     • (d) 6 elements
     उत्तर देखें

     उत्तर: (d)

125. 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
     उत्तर देखें

     उत्तर: (d)