61. If S is a
non-empty set of real numbers, then

(A) Inf S = Sup S

(B) Inf S = -Sup (-S)

(C) Inf S = Sup (-S)

(D) Inf S = -Sup S

Answer: B

62. Every infinite
set has

(A) an uncountable subset

(B) a countable subset

(C) both countable and uncountable subsets

(D) none of the above

Answer: B

63. A real valued
function f has discontinuity of the second kind at x=a if

(A) f (a+) exist only

(B) f (a-) exist only

(C) Neither f(a+) nor f(a-) exist

(D) Both f(a+) and f(a-) exist

Answer: C

64. For the
sequence {x

_{n}}, where x_{n}= (-1)^n n , the ullim x_{n}is
(A) 1

(B) 0

(C) +∞

(D) -∞

Answer: D

65. Every open set
of real numbers is the union of

(A) countable collection of disjoint closed intervals

(B) uncountable collection of disjoint closed intervals

(C) countable collection of disjoint open intervals

(D) uncountable collection of disjoint open intervals

Answer: C

66. A set E is
nowhere dense if

(A) closure of E contains non-empty open sets

(B) closure of E contains no non-empty open sets

(C) closure of E contains empty open set

(D) none of the above

Answer: B

67. If f

_{1}and f_{2}are two real-valued bounded functions defined on [a,b] then for every partition P on [a,b]
(A) U (P, f

_{1}+f_{2}) = U (P, f_{1}) + U (P, f_{2})
(B) U (P, f

_{1}+f_{2})<= U (P, f_{1}) + U (P, f_{2})
(C) U (P, f

_{1}+f_{2})>= U (P, f_{1}) + U (P, f_{2})
(D) None of the above

Answer: B

68. If f : [a,b]
-> R is continuous and monotonic function then

(A) f is Riemann integrable on [a,b]

(B) f is not Riemann integrable on [a,b]

(C) f is Riemann integrable on R

(D) None of the above

Answer: A

69. Which of the
following is true?

(A) The set [0,1] is not countable

(B) If E

_{1}and E_{2}are Lebesgue measurable, then E_{1}U E_{2}is Lebesgue measurable
(C) The family M of Lebesgue measurable sets is an algebra of sets

(D) All of the above

Answer: D

70. Given int

_{0^1}(sin {1/(x)})/(sqrt(x))dx , then
(A) Integral is divergent

(B) Integral is absolutely convergent

(C) Integral is not absolutely convergent

(D) None of the above

Answer: B

71. If f satisfies
the conditions of Lagrange's mean value theorem and if f’(x) = 0 ∀ x in [a,b], then which of the following is true?

(A) f is constant on [a,b]

(B) f is strictly increasing in [a,b]

(C) f is strictly decreasing in [a,b]

(D) None of the above

Answer: A

72. lim

_{(z->0)}(bar z)/(z) is
(A) 0

(B) 1

(C) (1)/(2)

(D) Does not exist

Answer: D

73. The radius of
convergence of the power series ∑

^{∞}_{n=0}(2n!)/((n!)^2) (2-3i)^n is
(A) 1

(B) 0

(C) (1)/(2)

(D) (1)/(4)

Answer: D

74. A function is
said to be harmonic if

(A) (∂

^{2}u)/( ∂x^{2}) + (∂^{2}v)/( ∂x^{2}) = 0
(B) (∂

^{2}u)/( ∂x^{2}) + (∂^{2}u)/( ∂y^{2}) = 0
(C) (∂u)/( ∂x) + (∂u)/( ∂y) = 0

(D) (∂v)/( ∂x) + (∂v)/( ∂y) = 0

Answer: B

75. The value of int_c
log z dz where c is the unit circle is

(A) Pi i

(B) 2Pi i

(C) 4Pi i

(D) 0

Answer: B

76. The image of
the unit circle |z| = 1 under the transformation w=2z+z

^{2}is
(A) Circle

(B) Straight line

(C) Parabola

(D) Cardioid

Answer: D

77. If X is any
set, T is a collection of all subsets of X then (X, T) is

(A) Discrete topology

(B) Indiscrete topology

(C) Trivial topology

(D) None of the above

Answer: A

78. Let X and Y
are topological spaces. The function f is a homeomorphism if

(A) f : X -> Y is a bijective function

(B) f` is continuous

(C) f^{-1} : Y ->X is continuous

(D) All of the above

Answer: D

79. Every compact
subset of a Hausdorff space is

(A) Closed set

(B) Open set

(C) Null set

(D) None of the above

Answer: A

80. The order and
degree of the differential equation (d)/(dx) ((d

^{2}y)/(dx^{2}))^4 =0 is
(A) 1, 4

(B) 2, 4

(C) 3, 1

(D) 3, 4

Answer: C

## 0 comments:

## Post a Comment