Determinants

For some a,b, let f(x)= vmatrixa+ xx&1&b\ &1+ xx&b\ &1&b+ xx vmatrix, x 0, x 0f(x)= + a+ b. Then ( + + )2 is equal to :

1. 25
2. 16
3. 9
4. 36

If the system of equations x+2y-3z=2 2x+ y+5z=5 14x+3y+ z=33 has infinitely many solutions, then + is equal to :

1. 13
2. 10
3. 12
4. 11

If the system of equations ( -1)x+( -4)y+ z=5 x+( -1)y+( -4)z=7 ( +1)x+( +2)y-( +2)z=9 has infinitely many solutions, then 2+ is equal to

1. 20
2. 10
3. 6
4. 12

If A, B, and (adj (A-1 )+adj (B-1 ) ) are non-singular matrices of same order, then the inverse of A (adj (A-1 )+adj (B-1 ) )-1B, is equal to

1. AB-1|A|+ BA-1|B|
2. adj (B-1 )+adj (A-1 )
3. AB-1+A-1B
4. 1|AB| (adj(B)+adj(A) )

The system of equations x+y+z=6, x+2y+5z=9, x+5y+ z= , has no solution if

1. =17, =18
2. =17, 18
3. =15, 17
4. 17, 18

Let A be a square matrix of order 3 such that (A)=-2 and (3 adj (-6 adj(3A) ) )=2m+n 3mn, m>n. Then 4m+2n is equal to .

If the system of linear equations : x+y+2z=6 2x+3y+az=a+1 -x-3y+bz=2b where a,b , has infinitely many solutions, then 7a+3b is equal to :

1. 12
2. 9
3. 22
4. 16

For a 3 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 3 matrix such that |A|= 12 and trace (A)=3. If B=adj(adj(2A)), then the value of |B|+ trace (B) equals :

1. 56
2. 132
3. 174
4. 280

Let , R. If the system of equations 3x + 5y + z = 3 7x + 11y - 9z = 2 97x + 155y - 189z = has infinitely many solutions, then + 2 is equal to :

1. 24
2. 25
3. 27
4. 22

Let A be a non-singular matrix of order 3. If (3 adj(2 adj(( A)A))) = 3-13 2-10 and (3 adj(2A)) = 2m 3n, then |3m + 2n| is equal to .