Need your help on GR9768 24,28,34,47,55
Posted: Wed Apr 09, 2008 11:23 pm
Hey guys, here are some problems in GR9768 practise book. Please help. Thank you!
24. which of the following sets of vectors is a basis for the subspace of Euclidean 4-subspace consisting all vectors that are orthogonal to both (0,1,1,1) and (1,1,1,)
(A) {(0,-1,-1,0)}
(B) {(1,0,0,0),(0,0,0,1)}
(C) {(-2,-1,1,-2),(0,1,-1,0)}
D,E are obvious wrong.
Ans: C.
28. V1 and V2 are 6-dimensional subspaces of a 10-dimensional vector space V. What is the smallest possible dimension of V1 intersect V2?
(A) 0 (B)1 (C)2 (D)4 (E)6
Ans: C (but why is cant be 1? )
34. f is a differentiable function for which both limf(x) and limf’(x) as x->infinity exist and finite. Which must be true?
(A) limf’(x)(x->infinity)=0
(B) limf’’(x)(x->infinity)=0
(C) limf(x) as x->infinity = limf’(x) as x->infinity
D,E are obvious wrong.
Ans:A
47. x, y are uniformly distributed, independent random variables on [0,1], what is the probability that distance between x and y is less than ½?
(A) 1/4 (B)1/3 (C)1/2 (D)2/3 (E)3/4
Ans:E
55. f is twice-differentiable function on set of real numbers, and f(0), f’(0) and f’’(0) are negative. Suppose f’’ has all the following three properties:
I. It is increasing on [0,infinity)
II. It has a unique zero in [0,infinity)
III. It is unbounded on [0,infinity)
Then which of them does f necessarily have?
Ans: II and III only
24. which of the following sets of vectors is a basis for the subspace of Euclidean 4-subspace consisting all vectors that are orthogonal to both (0,1,1,1) and (1,1,1,)
(A) {(0,-1,-1,0)}
(B) {(1,0,0,0),(0,0,0,1)}
(C) {(-2,-1,1,-2),(0,1,-1,0)}
D,E are obvious wrong.
Ans: C.
28. V1 and V2 are 6-dimensional subspaces of a 10-dimensional vector space V. What is the smallest possible dimension of V1 intersect V2?
(A) 0 (B)1 (C)2 (D)4 (E)6
Ans: C (but why is cant be 1? )
34. f is a differentiable function for which both limf(x) and limf’(x) as x->infinity exist and finite. Which must be true?
(A) limf’(x)(x->infinity)=0
(B) limf’’(x)(x->infinity)=0
(C) limf(x) as x->infinity = limf’(x) as x->infinity
D,E are obvious wrong.
Ans:A
47. x, y are uniformly distributed, independent random variables on [0,1], what is the probability that distance between x and y is less than ½?
(A) 1/4 (B)1/3 (C)1/2 (D)2/3 (E)3/4
Ans:E
55. f is twice-differentiable function on set of real numbers, and f(0), f’(0) and f’’(0) are negative. Suppose f’’ has all the following three properties:
I. It is increasing on [0,infinity)
II. It has a unique zero in [0,infinity)
III. It is unbounded on [0,infinity)
Then which of them does f necessarily have?
Ans: II and III only