5. Let M = (a), i, j = {1, 2, 3}, be the 3 3 matrix such that a = 1 if j + 1 is divisible by i, otherwise aij = 0. Then which of the following statements is (are) true? (A) M is invertible (B) There exists a nonzero column matrix a such that Ma a3 -a -a3 kas (C) The set (X = R: MX = 0} = {0}, where 0 = 0 (D) The matrix (M-2/) is invertible, where / is the 3 3 ide! S matrix 6. f: Let f : (0, 1) R be the function defined as f(x) = [4x](x-1) () 4) (X-2), where [] denotes the greatest integer less than or equal to x. Then which of the following is (are) true? (A) The function f is discontinuous exactly at one point in (0,1) (B) There is exactly one point in (0, 1) at which the function f is continuous but NOT differentiable (C) The function f is NOT differentiable at more than three points in (0, 1) (D) The minimum value of the function f is 1 512 7. Let S be the set of all twice differentiable functions f from R to R such that d'f dx(x)>0 for all x = (-1,1). For f = S, let X; be the number of points x = (-1,1) for which f (x) = x. Then which of the following statements is (are) true? (A) There exists a function fe S such that X = 0 (B) For every function f = S, we have X 2 (C) There exists a function f = S, such that X = 2 (D) There does NOT exist any function fin S such that X = 1 8. For XR, let tan 1(x) =) Then the minimum value of the function 2'2 f: RR defined by f(x) = ! x tan-1x(t-cost) e dt is 1+12023 0 9. For x = R, let y(x) be a solution of the differential equation (x-5)dy-2xy = -2x(x-5) such that y(2) = 7. Then the maximum value of the function y(x) is 10. Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X. Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of 38p is equal to