Determinants

If A= bmatrix 2 & 1\ -1 & 2 bmatrix, B= bmatrix1 & 0\ 1 & 1 bmatrix, C=ABAT and X=ATC2A, then is equal to :

1. 243
2. 729
3. 27
4. 891

If the system of equations 2x+3y-z=5 x+ y+3z=-4 3x-y+ z=7 has infinitely many solutions, then 13 is equal to :

1. 1110
2. 1120
3. 1210
4. 1220

Let the system of equations x+2 y+3 z=5, 2 x+3 y+z=9, 4 x+3 y+ z= have infinite number of solutions. Then +2 is equal to :

1. 22
2. 17
3. 15
4. 28

If the system of linear equations x-2y+z=-4 2x+ y+3z=5 3x-y+ z=3 has infinitely many solutions, then 12 +13 is equal to

1. 60
2. 54
3. 64
4. 58

If f(x)= vmatrix x3 & 2x2+1 & 1+3x \ 3x2+2 & 2x & x3+6 \ x3-x & 4 & x2-2 vmatrix for all x R, then 2f(0)+f'(0) is equal to

1. 24
2. 18
3. 42
4. 48

Let A be a 3 3 real matrix such that A pmatrix1\\0\\1 pmatrix=2 pmatrix1\\0\\1 pmatrix, A pmatrix-1\\0\\1 pmatrix=4 pmatrix-1\\0\\1 pmatrix, A pmatrix0\\1\\0 pmatrix=2 pmatrix0\\1\\0 pmatrix. Then, the system (A-3I) pmatrixx\ \ pmatrix= pmatrix1\\2\\3 pmatrix has :

1. exactly two solutions
2. infinitely many solutions
3. unique solution
4. no solution

Let A be a 3 3 matrix and (A)=2. If n= ( 2024 times adj(adj( (adj A)))), then the remainder when n is divided by 9 is equal to .

If f(x)= vmatrix2 4x & 2 4x & 3+ 2 2x\ 3+2 4x & 2 4x & 2 2x\ 2 4x & 3+2 4x & 2 2x vmatrix, then 15f'(0) is equal to :

1. 2
2. 1
3. 0
4. 6

Consider the system of linear equations x+y+z=4 , x+2y+2 z=10 , x+3y+4 2z= 2+15 where , . Which one of the following statements is NOT correct ?

1. The system has unique solution if 12 and 1, 15
2. The system has infinite number of solutions if = 12 and =15
3. The system is consistent if 12
4. The system is inconsistent if = 12 and 1

Let R= pmatrixx&0&0\\0&y&0\\0&0&z pmatrix be a non-zero 3 3 matrix, where x =y ( + 2 3 )=z ( + 4 3 ) 0, (0,2 ). For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M. Then, among the statements: (I) Trace (R)=0 (II) If trace (adj(adj(R)))=0, then R has exactly one non-zero entry.

1. Only (I) is true
2. Only (II) is true
3. Both (I) and (II) are true
4. Neither (I) nor (II) is true