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Determinants

JEE Main 2025 (Online) 2nd April Morning ShiftMCQMedium📈 Topic in 38% papers+4−1
Ques. 1JEE Main 2025, 2 April (Morning Shift)

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

Answer

(4) 16

Solution

A = bmatrix 1 & a & 1 \ 2 & 1 & 0 \ a & 1 & 2 bmatrix - I = bmatrix 0 & a & 1 \ 2 & 0 & 0 \ a & 1 & 1 bmatrix

(A) = 0(0 - 0) - a(2 - 0) + 1(2 - 0) = 2 - 2a

2 - 2a = -4 a = 3

So we need ((a + 1) adj((a - 1)A)) = (4 adj(2A)).

For order 3, adj(kA) = k2 adjA, (kB) = k3 B and (adjA) = ( A)2, so

(4 22 adjA) = (16 adjA)

= 163 ( A)2 = 212 16 = 216 30

Hence m = 16, n = 0.

m + n = 16

Answer: (4)

Keep practising — Determinants

Ques. 2JEE Main 2025, 2 April (Morning Shift)

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
Ques. 3JEE Main 2025, 2 April (Evening Shift)

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
Ques. 4JEE Main 2025, 29 January (Morning Shift)

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
Ques. 5JEE Main 2025, 29 January (Morning Shift)

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
Ques. 6JEE Main 2025, 29 January (Evening Shift)

Let , ( ) be the values of m, for which the equations x + y + z = 1, x + 2y + 4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of n=110(n + n ) is equal to :

  1. 3410
  2. 560
  3. 3080
  4. 440
Ques. 7JEE Main 2025, 24 January (Morning Shift)

If the system of equations 2x-y+z=4 5x+ y+3z=12 100x-47y+ z=212 has infinitely many solutions, then -2 is equal to

  1. 56
  2. 59
  3. 57
  4. 55
Ques. 8JEE Main 2025, 24 January (Evening Shift)

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
Ques. 9JEE Main 2025, 24 January (Evening Shift)

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
Ques. 10JEE Main 2025, 23 January (Morning Shift)

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
Ques. 11JEE Main 2025, 23 January (Morning Shift)

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) )

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