Saturday, 4 November 2017

HSST Mathematics Solved Paper 36/2016/OL - Part 3

41.       If the identity element ‘e in S’ exists in a semigroup (S, `*` ), then it is a
(A) Group
(B) Groupoid
(C) Monoid
(D) None of the above
Answer: C
42.       The number of generators of (Z24, +) is
(A) 2
(B) 6
(C) 8
(D) 10
Answer: C
43.       A Sylow 3-subgroup of a group of order 12 has order
(A) 2
(B) 3
(C) 1
(D) 12
Answer: B
44.       Consider Z5 and Z20 as rings modulo 5 and 20 respectively. Then the number of homomorphism φ :Z5 -> Z20 is
(A) 1
(B) 4
(C) 5
(D) 2
Answer: D
45.       Let Q be the field of rational numbers and Z2 is a field modulo 2. Then the polynomial f(x) = x3 -9x2 + 9x + 3 is
(A) irreducible over Q but reducible over Z2
(B) irreducible over both Q and Z2
(C) reducible over Q but irreducible over Z2
(D) reducible over both Q and Z2
Answer: A
46.       Let A =[[3,1,-1],[2,2,-1],[2,2,0]]. The characteristic polynomial of A is
(A) x3+5x2+8x+4
(B) x2+5x
(C) x3-5x2+8x-4
(D) x3+8x+4
Answer: C
47.       The eigen values of the matrix [[4,-2],[-2,1]] are
(A) 1, 4
(B) -1, 2
(C) 0, 5
(D) Cannot be determined
Answer: C
48.       Let V be a finite dimensional vector space, ‘I’ be the identity transformation on V , then the null space of ‘I’ is
(A) {0}
(B) phi
(C) V
(D) None of the above
Answer: A
49.       If V is a vector space with dim V=n , then the dimension of the hyperspace of V is
(A) n
(B) n-1
(C) n+1
(D) 0
Answer: B
50.    Let V be a vector space of all 2 × 2 matrices over R . Let T be the linear mapping T:V-> V such that T(A) = AB-BA where B = [[2,1],[0,3]] . Then the nullity of T is
(A) 1
(B) 2
(C) 3
(D) 4
Answer: A

51.    Banach space is a
(A) Complete normed vector space
(B) Normed vector space
(C) Complete vector space
(D) None of the above
Answer: A
52.    Which of the following is true?
(A) All normed spaces are inner product spaces
(B) All inner product spaces are normed spaces
(C) All inner product spaces are Banach spaces
(D) All inner product spaces are Hilbert spaces
Answer: B
53.    Banach space is a Hilbert space if
(A) Pythagorean theorem holds
(B) Projection theorem holds
(C) Parallelogram law holds
(D) None of the above
Answer: C
54.    If T is a bounded linear operator on a Hilbert space H, which of the following is not true?
(A) T is normal if T is self-adjoint
(B) T is normal if T is unitary
(C) T is self-adjoint if T is normal
(D) None of the above
Answer: C
55.    The equation of the normal at the point (a sec Theta, b tanTheta) on the hyperbola (x2)/(a2)- (y2)/(b2) = 1 is
(A) (x)/(a) sec Theta - (y)/(b) tan Theta = 1
(B) (x)/(a) sec Theta + (y)/(b) tan Theta = 1
(C) (ax)/(sec Theta) - (by)/(tan Theta) = a2 + b2
(D) (ax)/(sec Theta) + (by)/(tan Theta) = a2 + b2
Answer: D
56.    lim_(x->∞) (log x)/(xn) is
(A) ∞
(B) -∞
(C) 1
(D) 0
Answer: D
57.    (x * y) + (x' + y') is equal to
(A) x * y
(B) x' + y'
(C) 0
(D) 1
Answer: D
58.    Let ‘a’ be any element in a Boolean algebra B . If a+x=1 and ax=0 , then
(A) x=1
(B) x=0
(C) x=a
(D) x=a'
Answer: D
59.    Which of the following is reflexive?
(A) l2
(B) l1
(C) L1 [a,b]
(D) l∞
Answer: A
60.    If 1 < p < ∞ and q is conjugate of p , then
(A) l^{p'} = l^q
(B) l^{p'} = l^p
(C) l^{p'} < l^q
(D) l^{p'}> l^q
Answer: A

Pages   2   3   4   5 

0 comments:

Post a Comment