Determinants · Question 17

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

Let the system of equations : 2x + 3y + 5z = 9 7x + 3y - 2z = 8 12x + 3y - (4 + )z = 16 - have infinitely many solutions. Then the radius of the circle centred at ( , ) and touching the line 4x = 3y is :

1. 75
2. 215
3. 7
4. 175

The number of singular matrices of order 2, whose elements are from the set \2, 3, 6, 9\, is .

Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + z = , R, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 x + y + z 77, is :

1. 4
2. 5
3. 3
4. 6

If y(x) = vmatrix x & x & x+ x+1 \ 27 & 28 & 27 \ 1 & 1 & 1 vmatrix, x R, then d2ydx2 + y is equal to

1. 28
2. 27
3. -1
4. 1