MCQ(XII)

1. What type of a relation is R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} on the set A – {1, 2, 3, 4}
(a) Reflexive
(b) Transitive
(c) Symmetric
(d) None of these

2. Let E = {1, 2, 3, 4} and F = {1, 2} Then, the number of onto functions from E to F is
(a) 14
(b) 16
(c) 12
(d) 8

3. Let the functioin ‘f’ be defined by f (x) = 5x² + 2 ∀ x ∈ R, then ‘f’ is

(a) onto function
(b) one-one, onto function
(c) one-one, into function
(d) many-one into function

4. What type of relation is ‘less than’ in the set of real numbers?

(a) Only symmetric
(b) only transitive
(c) only reflexive
(d) equivalence

5. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as
aRb if a congruent to b ∀ a, b ∈ T. Then R is
(a) reflexive but-not transitive
(b) transitive but not symmetric
(c) equivalence
(d) None of these

6. The maximum number of equivalence relations on the set A = {1, 2, 3} are

(a) 1
(b) 2
(c) 3
(d) 5

7. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

(a) reflexive
(b) transitive
(c) symmetric
(d) None of these

8. Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then
R is
(a) reflexive but not symmetric
(b) reflexive-but not transitive.
(c) Symmetric and transitive
(d) neither symmetric, nor transitive

9. If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one
and onto mappings from A to B is
(a) 720
(b) 120
(c) 0
(d) None of these

10. If the set A contains 5 elements and the set B contains 6 elements, then the number of one-
one and onto mappings from A to B is
(a) 720
(b) 120
(c) 0
(d) None of these

11. Which of the following functions from Z into Z are bijective?

(a) f(x) = x³
(b) f(x) = x + 2
(c) f(x) = 2x + 1
(d) f{x) = x² + 1

12. Let P = {(x, y) | x² + y² = 1, x, y ∈ R]. Then, P is

(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric

13. Let R be an equivalence relation on a finite set A having n elements. Then, the number of
ordered pairs in R is
(a) Less than n
(b) Greater than or equal to n
(c) Less than or equal to n
(d) None of these
14. For real numbers x and y, we write xRy ⇔ x – y + √2 is an irrational number. Then, the
relational R is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) None of these

15. Which one of the following relations on R is an equivalence relation?

(a) aR1b ⇔ |a| = |b|
(b) aR2b ⇔ a ≥ b
(c) aR3b ⇔ a divides b
(d) aR4b ⇔ a < b

16. Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e.
n | m). Then, R is
(a) Reflexive and symmetric
(b) Transitive and symmetric
(c) Equivalence
(d) Reflexive, transitive but not symmetric

17. If x + 1/x = 2 then the principal value of sin-1 x is x

(a) π/4
(b) π/2
(c) π
(d) 3π/2

18. The value of cos-1(1/2) + 2sin-1(1/2) is equal to

(a) π/4
(b) π/6
(c) 2π/3
(d) 5π/6

19. cos-1(cos7π/6) is equal to

(a) 7π/6
(b) 5π/6
(c) π/3
(d) π/6

20. sin[π/3 – sin-1(-1/2)] is equal to

(a) 1/2
(b) 1/3
(c) 1/4
(d) 1
21. The domain of the following f(x) =√ sin−1 x   is
(a) [0, 1]
(b) [-1, 1]
(c) [-, 0]
(d) [0, 1]

22. The value of cos-1[cos(33π/5)] is

(a) 3π/5
(b) −3π/5
(c) π/10
(d) –−π/10

23. The domain of the function cos-1 (2x – 1) is

(a) [0, 1]
(b) [-1, 1]
(c) [-1, -1]
(d) [0, π]

24. If cos-1 α + cos-1 β + cos-1 γ = 3π, then α(β + γ) + β (γ + α) + γ(α + β) equals

(a) 0
(b) 1
(c) 6
(d) 12

25. If cos-1 x > sin-1 x, then

(a) 1/√2 < x ≤ 1
(b) 0 ≤ x < 1/√2
(c) -1 ≤ x < 1/√2 
(d) x > 0
26. A matrix A = [aij]m×n is said to be symmetric if
(a) aij = 0
(b) aij = aji
(c) aij = aij
(d) aij = 1

[ ]
1 1 1
27. If A = 1 1 1   then A² is
1 1 1
(a) 27 A
(b) 2 A
(c) 3 A
(d) 1
28. Let A be a non-singular matrix of the order 2 × 2 then |A-1|=
(a) |A|
(b) 1/|A|
(c) 0
(d) 1

[ ] [ ]
29. If A = 
α 0
1 1
B = 
1 0
5 1
 where A² = B then the value of α is

(a) 1
(b) -1
(c) 4
(d) we cant calculate the value of α

[ ]
2 λ −3
30. If A =  0 2 5  then A-1 exists if
1 1 3
(a) λ = 2
(b) λ ≠ 2
(c) λ ≠ -2
(d) none of these

[ ]
31. If A = 
α 2
2 α
and |A³| = 25 then α is

(a) ±3
(b) ±2
(c) ±5
(d) 0
32. A² – A + I = 0 then the inverse of A
(a) A
(b) A + I
(c) I – A
(d) A – I

33. If A =[ 3 1
−1 2 ]   then A² – 5A – 7I is

(a) zero matrix

(b) a diagonal matrix
(c) identity matrix
(d) None of these

34 If A = [ ]
1 2
4 2
 then |2A| =

(a) 2|A|
(b) 4|A|
(c) 8|A|
(d) None of these.

[ ]
2 −1 4
35. The matrix 1 0 −5   is
−4 5 7
(a) a symmetric matix
(b) a skew-sybtmetric matrix
(c) a diagonal matrix
(d) None of these
36. If a matrix is both symmetric matrix and skew symmetric matrix then
(a) A is a diagonal matrix
(b) A is zero matrix
(c) A is scalar matrix
(d) None of these
37. If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the
order of matrix (5A – 2B) is
(a) m × 3
(b) 3 × 3
(c) m × n
(d) 3 × n
38. If matrix A = [aij]2×2 where aij = {1

{
f ( x )= 1 ,∧if i ≠ j  then A² is equal to
0 ,∧i= j
(a) I
(b) A
(c) O
(d) None of these
39. If A is matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then
order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n
40. A square matrix A = [aij]n×n is called a diagonal matrix if aij = 0 for
(a) i = j
(b) i < j
(c) i > j
(d) i ≠ j
41. If AB = A and BA = B, then
(a) B = 1
(b)A = I
(c) A² = A
(d) B² = I

42. If A =  [ 10 12][1012] then A  – 2  (A – I)

8 8

(a) I – A
(b) 2I – A
(c) I + A
(d) A – 2I
43. The area of a triangle with vertices (-3, 0) (3, 0) and (0, k) is 9 sq. units. The value of k will
be
(a) 9
(b) 3
(c) -9
(d) 6
44. If A and B are invertible matrices, then which of the following is not correct?
(a) adj A = |A|.A-1
(b) det (a)-1 = [det (a)]-1
(c) (AB)-1 = B-1A-1
(d) (A + B)-1 = B-1 + A-1
4−x 2
45. The function f(x) =  is
4 x−x 3
(a) discontinuous at only one point at x = 0
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of these
46. The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is
(a) R
(b) R - {1/2}
(c) (0, ∞)
(d) None of these
47.

{
π
mx +1 ,if ∧x ≤
2
If f(x) = f ( x )= is continuous at x = π/2, then
π
sinnx+1 ,if x >
2
(a) m = 1, n = 0
(b) m = nπ/2 + 1
(c) n = mπ/2
(d) m = n = π2
48. If y = log(1−x2/1+x2), then dydx is equal to
(a) 4x3/1−x4
(b) −4x/1−x4
(c) 1/4−x4
(d) −4x3/1−x4
48. Let f(x) = |sin x| Then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
(c) f is everywhere continuous but no differentiable at x = (2n + 1) π/2 n ∈ Z
(d) None of these
49. If y = √ sinx + y  then dy/dx is equal to
(a) cosx/2y−1
(b) cosx/1−2y
(c) sinx/1−xy
(d) sinx/2y−1
50. The derivative of cos-1 (2x² – 1) w.r.t cos-1 x is
(a) 2
(b) −1/2√ 1−x 2
(c) 2/x
(d) 1 – x²
51. The sides of an equilateral triangle are increasing at the rate of 2cm/sec. The rate at which the
are increases, when side is 10 cm is
(a) 10 cm²/s
(b) √3 cm²/s
(c) 10√3 cm²/s
(d) 10/3 cm²/s
52. A ladder, 5 meter long, standing oh a horizontal floor, leans against a vertical wall. If the top
of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between
the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is
(a) 1/10 radian/sec
(b) 1/20 radian/sec
(c) 20 radiah/sec
(d) 10 radiah/sec
53. The interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is
(a) [-1, ∞]
(b) [-2, -1]
(c) [-∞, -2]
(d) [-1, 1]
54. y = x (x – 3)² decreases for the values of x given by
(a) 1 < x < 3
(b) x < 0
(c) x > 0
(d) 0 < x <3/2
55. The function f(x) = 4 sin³ x – 6 sin²x + 12 sin x + 100 is strictly
(a) increasing in (π, 3π/2)
(b) decreasing in (π/2, π)
(c) decreasing in [−π/2,π/2]
(d) decreasing in [0, π/2]
56. Which of the following functions is decreasing on(0, π/2)?
(a) sin 2x
(b) tan x
(c) cos x
(d) cos 3x
57. If x is real, the minimum value of x² – 8x + 17 is
(a) -1
(b) 0
(c) 1
(d) 2
58. The function f(x) = 2x³ – 3x² – 12x + 4 has
(a) two points of local maximum
(b) two points of local minimum
(c) one maxima and one minima
(d) no maxima or minima
2x
59. The function f(x) = log (1 + x) –   is increasing on
2+ x
(a) (-1, ∞)
(b) (-∞, 0)
(b) (-∞, ∞)
(d) None of these
60. The function f(x) = cot-1 x + x increases in the interval
(a) (1, ∞)
(b) (-1, ∞)
(c) (0, ∞)
(d) (-∞, ∞)
61. The function f(x) = x/logx increases on the interval
(a) (0, ∞)
(b) (0, e)
(c) (e, ∞)
(d) None of these
62. The value of b for which the function f (x) = sin x – bx + c is decreasing for x ∈ R is given by
(a) b < 1
(b) b ≥ 1
(c) b > 1
(d) b ≤ 1
63. If f (x) = x³ – 6x² + 9x + 3 be a decreasing function, then x lies in
(a) (-∞, -1) ∩ (3, ∞)
(b) (1, 3)
(c) (3, ∞)
(d) None of these
64. The length of the longest interval, in which the function 3 sin x – 4 sin³ x is increasing is
(a) π/3
(b) π/2
(c) 3π/2
(d) π
65. Maximum slope of the curve y = -x³ + 3x² + 9x – 27 is
(a) 0
(b) 12
(c) 16
(d) 32
66. f(x) = xx has a stationary point at
(a) x = e
(b) x = 1/e
(c) x = 1
(d) x = √e
67. ∫dx/1+cosx =
(a) tan x/2 + k
(b) 1/2 tan x/2 + k
(c) 2 tan x/2 + k
(d) tan² x/2 + k
68. ∫cos2xdx/(sinx+cosx)2 =
(a) –1/(sinx+cosx) + c
(b) log | sin x + cos x | + c
(c) log | sin x – cos x | + c
(d) 1/(sinx+cosx)2
2

69. ∫ ¿ x∨¿ ¿dx =

−2

(a) 0
(b) 2
(c) 1
(d) 4
70. ∫(x4+1)/(x2+1) dx is equal to
(a) x3/3 + x + tan-1 x + c
(b) x3/3 – x + tan x + c
(c) x3/3 + x + 2tan-1 x + c
(d) x3/3 – x + 2tan-1 x + c
d
71. ∫f(x)dx is equal to
dx
(a) f'(x)
(b) f(x)
(c) f'(x’)
(d) f(x) + c
π
2
72. What is the value of∫ (sinx – cosx )   dx?
0 (1+ sinxcosx)

(a) 1
(b) π/2
(c) 0
(d) –π/2
1

73. What is the value of ∫ sin ³ x cos ² x  dx?

−1

(a) 0
(b) 1
(c) 1/2
(d) 2
1

74. ∫ x (1 – x )99 dx  is equal to

0

(a) 1/10010
(b) 1/10100
(c) 1/1010
(d) 11/10100
75. ∫log10 xdx =
(a) loge 10.x loge (x/e) + c
(b) log10 e.x loge (x/e) + c
(c) (x – 1) loge x + c
(d) 1/x + c
dx
76. Value of ∫
√2 x−x 2
(a) sin-1 (x – 1) + c
(b) sin-1 (1 + x) + c
(c) sin-1 (1 + x²) + c
(d) –√ 2 x− x2 + c
77. Area of the region bounded by the curve y = cos x between x = 0 and x = π is
(a) 2 sq. units
(b) 4 sq, units
(c) 3 sq.units
(d) 1 sq. units
78. The area of the region bounded by the curve x = 2y + 3 and the lines y = 1 and y = -1 is
(a) 4 sq. units
(b) 32 sq. units
(c) 6 sq. units
(d) 8 sq, units
79. The area of the region bounded by the line y = | x – 2 |, x = 1, x = 3 and x-axis is
(a) 4 sq. units
(b) 2 sq, units
(c) 3 sq. units
(d) 1 sq. unit
80. The area bounded by the line y = 2x – 2, y = -x and x-axis is given by
(a) 92 sq. units
(b) 436 sq. units
(c) 356 sq. units
(d) None pf these
81. The radius of a circle is increasing at the rate of 0.4 cm/ s. The rate of increasing of its
circumference is
(a) 0.4 π cm/s
(b) 0.8 π cm/s
(c) 0.8 cm/s
(d) None of these
82. The degree of the differential equation
2
d y dy dy
( 2 )² + ( )² = x sin   is
dx dx dx
(a) 1
(b) 2
(c) 3
(d) not defined
83. The degree of differential equation
dy 3/2 d 2 y
[1 + ( )²]  =  2  is
dx dx
(a) 4
(b) 32
(c) 2
(d) not defined
84. The order and degree of the differential equation
d2 y
2  + (dy/dx)  + x  = 0 respectvely, are
1/4 1/3

dx
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3
dy
85. Integrating factor of the differential equation cos x  + y sin x = 1 is
dx
(a) cos x
(b) tan x
(c) sec x
(d) sin x
86. Solution of the differential equation tan y sec² x dx + tan x sec² y dy + 0 is .
(a) tan x + tan y = k
(b) tan x – tan y = k
(c) tanx/tany = k
(d) tan x.tan y = k
dy y +1
87. The number of solutions of   =   when y(1) = 2 is
dx x−1
(a) none
(b) one
(c) two
(d) infinite
88. Which of the following is a second order differential equation?
(a) (y’)² + x = y²
(b) y’y” + y = sin x
(c) y” + (y”)² + y = 0
(d) y’ = y²
89. tan-1 x + tan-1 y = c is the general solution of the differential equation
dy
(a)   = (1+y2)/(1+x2)
dx
dy
(b)  = (1+x2)/(1+y2)
dx
(c) (1 + x²)dy + (1 + y²)dx = 0
(d) (1 +x²)dx+(1 + y²)dy = 0
dy
90. Integrating factor of the differential equation  d + y tan x – sec x = 0 is
dx
(a) cos x
(b) sec x
(c) ecos x
(d) esec x