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Determinants · Question 30

JEE Main 2025 (Online) 2nd April Morning ShiftMCQMedium📈 Topic in 42% papers+4−1
1

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

((a + 1) adj((a - 1)A)) = (4 adj(2A))

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

(4 22 adjA) = (16 adjA)

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

m = 16, n = 0

m + n = 16

Answer: (4)

JEE Main 2025 (Online) 2nd April Morning Shift

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2

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
JEE Main 2025 (Online) 2nd April Morning Shift
3

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
JEE Main 2025 (Online) 2nd April Evening Shift
4

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
JEE Main 2025 (Online) 29th January Morning Shift
5

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
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