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