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

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

Let , R be such that the system of linear equations x+2y+z=5 2x+y+ z=5 8x+4y+ z=18 has no solution. Then is equal to :

1. -4
2. 4
3. 8
4. -8

If the system of equations x+5y+6z=4, 2x+3y+4z=7, x+6y+az=b has infinitely many solutions, then the point (a, b) lies on the line

1. y-x=3
2. x-y=3
3. x+y=11
4. x+y=12

Let P = [pij] and Q = [qij] be two square matrices of order 3 such that qij = 2(i+j-1)pij and (Q) = 210. Then the value of ( adj( adj P)) is:

1. 81
2. 16
3. 124
4. 32

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

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 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

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

Let A be a matrix of order 3 3 and |A| = 5. If |2 adj(3A adj(2A))| = 2 3 5 , , , N, then + + is equal to

1. 26
2. 27
3. 25
4. 28

Let a R and A be a matrix of order 3 3 such that (A) = -4 and A + I = bmatrix 1 & a & 1 \ 2 & 1 & 0 \ a & 1 & 2 bmatrix, where I is the identity matrix of order 3 3. If ((a + 1) adj((a - 1)A)) is 2m 3n, m, n \0, 1, 2, , 20\, then m + n is equal to :

1. 14
2. 17
3. 15
4. 16

If the system of linear equations 3x + y + z = 3 2x + y - z = -3 x + 2y + z = 4 has infinitely many solutions, then the value of 22 - 9 is :

1. 31
2. 37
3. 43
4. 49

If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( 2 + 2) is equal to :

1. 30
2. 26
3. 22
4. 18

Let A = [aij] = bmatrix 5 128 & 4 5 \ 5 8 & 4 25 bmatrix. If Aij is the cofactor of aij, Cij = k=12 aikAjk, 1 i, j 2, and C = [Cij], then 8|C| is equal to :

1. 288
2. 262
3. 222
4. 242

Let M and m respectively be the maximum and the minimum values of f(x) = vmatrix 1 + 2 x & 2 x & 4 4x \ 2 x & 1 + 2 x & 4 4x \ 2 x & 2 x & 1 + 4 4x vmatrix, x R Then M4 - m4 is equal to :

1. 1280
2. 1040
3. 1215
4. 1295