Determinants

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

Consider the system of linear equations x+y+z=5, x+2y+ 2 z=9, x+3y+ z= , where , R. Then, which of the following statements is NOT correct?

1. System is consistent if 1 and =13
2. System is inconsistent if =1 and 13
3. System has unique solution if 1 and 13
4. System has infinite number of solutions if =1 and =13

Let A= bmatrix1&0&0\\0& & \\0& & bmatrix and |2A|3=221 where , Z. Then a value of is

1. 9
2. 17
3. 3
4. 5

Let A= bmatrix2&1&2\\6&2&11\\3&3&2 bmatrix and P= bmatrix1&2&0\\5&0&2\\7&1&5 bmatrix. The sum of the prime factors of |P-1AP-2I | is equal to

1. 66
2. 27
3. 23
4. 26

Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax+by+c=0 be concurrent at the point P and Q( , ) be a point such that the system of equations x+y+z=6, 2x+5y+ z= and x+2y+3z=4, has infinitely many solutions. Then (PQ)2 is equal to .