Determinants · Question 13

Let f(x) = 7x10 + 9x8(1 + x2 + 2x9)2 dx, x > 0, x 0 f(x) = 0 and f(1) = 14. If A = bmatrix 0 & 0 & 1 \ 14 & f'(1) & 1 \ 2 & 4 & 1 bmatrix and B = adj(adj A) be such that |B| = 81, then 2 is equal to

1. 2
2. 4
3. 3
4. 1

Among the statements : I: If vmatrix 1 & & \ & 1 & \ & & 1 vmatrix = vmatrix 0 & & \ & 0 & \ & & 0 vmatrix, then 2 + 2 + 2 = 32, and II: If vmatrix x2 + x & x + 1 & x - 2 \ 2x2 + 3x - 1 & 3x & 3x - 3 \ x2 + 2x + 3 & 2x - 1 & 2x - 1 vmatrix = px + q, then p2 = 196q2.

1. both are true
2. both are false
3. only I is true
4. only II is true

Let |A| = 6, where A is a 3 3 matrix. If |adj (3 adj (A2 adj(2A) ) ) | = 2m 3n, m, n N, then m + n is equal to .

The system of linear equations x + y + z = 6 2x + 5y + az = 36 x + 2y + 3z = b has :

1. unique solution for a = 8 and b = 16
2. infinitely many solutions for a = 8 and b = 14
3. infinitely many solutions for a = 8 and b = 16
4. unique solution for a = 8 and b = 14

Let A be a 3 3 matrix such that A + AT = O. If A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix 3 \ 3 \ 2 bmatrix, A2 bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -3 \ 19 \ -24 bmatrix and (adj(2 adj(A + I))) = (2) (3) (11) , , , are non-negative integers, then + + is equal to

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2)y + (n + 1)z = 8 (n - 1)y + z = 1 has a unique solution is k6, then the sum of k and all possible values of n is :

1. 22
2. 20
3. 24
4. 21

If X = bmatrix x \ y \ z bmatrix is a solution of the system of equations AX = B, where adj A = bmatrix 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 bmatrix and B = bmatrix 4 \ 0 \ 2 bmatrix, then |x + y + z| is equal to :

1. 3
2. 2
3. 32
4. 1

If the system of equations 3x + y + 4z = 3 2x + y - z = -3 x + 2y + z = 4 has no solution, then the value of is equal to:

1. 13
2. 4
3. 19
4. 23

Let A = bmatrix 2 & 2+p & 2+p+q \ 4 & 6+2p & 8+3p+2q \ 6 & 12+3p & 20+6p+3q bmatrix. If (adj(adj(3A))) = 2m 3n, m, n N, then m + n is equal to

1. 22
2. 20
3. 24
4. 26

Let A be a 3 3 matrix such that |adj(adj(adj A))| = 81. If S = \n Z : (|adj(adj A)| ) (n-1)22 = |A| (3n2 - 5n - 4 ) \, then n S |A (n2 + n ) | is equal to :

1. 820
2. 866
3. 750
4. 732