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Determinants · Question 26

JEE Main 2025 (Online) 7th April Evening ShiftMCQHard📈 Topic in 42% papers+4−1
1

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

Answer

(3) 3

Solution

For infinitely many solutions, = 0:

vmatrix 1 & 5 & -1 \ 4 & 3 & -3 \ 24 & 1 & vmatrix = (3 + 3) - 5(4 + 72) - (4 - 72) = 0

-17 - 289 = 0 = -17

The third equation must be a combination a(I) + b(II):

a + 4b = 24, 5a + 3b = 1 b = 7, a = -4 (z-coefficient: -a - 3b = -17 ✓)

= a(1) + b(7) = -4 + 49 = 45

Solve the first two equations with z = t:

x + 5y = 1 + t, 4x + 3y = 7 + 3t

y = t - 317, x = 12t + 3217

For integer y, put t = 17k + 3 (k Z):

x = 12k + 4, y = k, z = 17k + 3 (all integers)

x + y + z = 30k + 7

7 30k + 7 77 0 k 73 k = 0, 1, 2

number of integer solutions = 3

Answer: (3)

JEE Main 2025 (Online) 7th April Evening Shift

Keep practising — Determinants

2

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
JEE Main 2025 (Online) 3rd April Morning Shift
3

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
JEE Main 2025 (Online) 3rd April Morning Shift
4

Let I be the identity matrix of order 3 3 and for the matrix A = bmatrix & 2 & 3 \ 4 & 5 & 6 \ 7 & -1 & 2 bmatrix, |A| = -1. Let B be the inverse of the matrix adj (A adj (A2 ) ). Then |( B + I)| is equal to

JEE Main 2025 (Online) 3rd April Evening Shift
5

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
JEE Main 2025 (Online) 2nd April Morning Shift
6

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
7

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
8

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
9

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
JEE Main 2025 (Online) 29th January Morning Shift

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