Sunday, 5 November 2017

HSST Mathematics Solved Paper 36/2016/OL - Part 4

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 {xn}, where xn= (-1)^n n , the ullim xn 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 f1 and f2 are two real-valued bounded functions defined on [a,b] then for every partition P on [a,b]
(A) U (P, f1+f2) = U (P, f1) + U (P, f2)
(B) U (P, f1+f2)<= U (P, f1) + U (P, f2)
(C) U (P, f1+f2)>= U (P, f1) + U (P, f2)
(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 E1 and E2 are Lebesgue measurable, then E1 U E2 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 int0^1 (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) (∂2u)/( ∂x2) + (∂2v)/( ∂x2) = 0
(B) (∂2u)/( ∂x2) + (∂2u)/( ∂y2) = 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+z2 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) ((d2y)/(dx2))^4 =0 is
(A) 1, 4
(B) 2, 4
(C) 3, 1
(D) 3, 4
Answer: C

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