Determinants

If the system of linear equations : x+y+z=6, x+2y+5z=10, 2x+3y+ z= has infinitely many solutions, then the value of + equals :

1. 12
2. 16
3. 22
4. 28

Let A= bmatrix & 1 & 2\ 2 & 3 & 0\ 0 & 4 & 5 bmatrix and B= bmatrix1 & 0 & 0\ 0 & -5 & 0\ 0 & 4 & -2 bmatrix+adj(A). If (B)=66, then (adj(A)) equals :

1. 289
2. 361
3. 441
4. 529

Let A= bmatrix-1 & 1 & -1\ 1 & 0 & 1\ 0 & 0 & 1 bmatrix satisfy A2+ (adj (adj(A) ) )+ (adj(A) (adj (adj(A) ) ) )= bmatrix2 & -2 & 2\ -2 & 0 & -1\ 0 & 0 & -1 bmatrix for some , R. Then ( - )2 is equal to .

The sum of all possible values of [0, 2 ], for which the system of equations : x 3 -8y-12z=0 x 2 +3y+3z=0 x+y+3z=0 has a non-trivial solution, is equal to :

2. 2
3. 3
4. 4

Let A be a 3 3 matrix such that AT bmatrix1\ 0\ 1 bmatrix= bmatrix5\ 2\ 2 bmatrix, AT bmatrix0\ 0\ 1 bmatrix= bmatrix3\ 1\ 1 bmatrix, A bmatrix1\ 0\ 1 bmatrix= bmatrix3\ 4\ 4 bmatrix and A bmatrix0\ 0\ 1 bmatrix= bmatrix1\ 3\ 1 bmatrix. If (A)=1, then (adj (A2+A ) ) is equal to:

1. 16
2. 25
3. 49
4. 64

Consider the system of linear equations in x, y, z : x+2y+tz=0 6x+y+5tz=0 3x+t2y+f(t)z=0 where f : R R is a differentiable function. If this system has infinitely many solutions for all t R, then f

1. is a constant function
2. is strictly increasing on R
3. is strictly decreasing on R
4. has two critical points

If f : N Z is defined by f(n)= vmatrixn & -1 & -5\ -2n2 & 3(2k+1) & 2k+1\ -3n3 & 3k(2k+1) & 3k(k+2)+1 vmatrix, k N, and n=1k f(n)=98, then k is equal to :

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

Let A= bmatrix1 & 1 & 2\ -2 & 0 & 1\ 1 & 3 & 5 bmatrix. Then the sum of all elements of the matrix adj (adj (2(adj A)-1 ) ) is equal to:

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

If the system of equations : x+y+z=5 x+2y+3z=9 x+3y+ z= has infinitely many solutions, then the value of + is :

1. 16
2. 18
3. 19
4. 21

Let A= bmatrix1 & 2 & 7\ 4 & -2 & 8\ 3 & 8 & -7 bmatrix and (A- I)=0, where is a real number. If the largest possible value of is p, then the circle (x-p)2+(y-2p)2=320, intersects the co-ordinate axes at

1. 1 point
2. 2 points
3. 3 points
4. 4 points