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Determinants · Question 19

JEE Main 2026 (Online) 22nd January Evening ShiftMCQMedium📈 Topic in 42% papers+4−1
1

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2)y + (n + 1)z = 8 (n - 1)y + z = 1 has a unique solution is k6, then the sum of k and all possible values of n is :

  1. 22
  2. 20
  3. 24
  4. 21

Answer

(1) 22

Solution

Unique solution 0

= vmatrix 1 & -n & 1 \ 1 & n-2 & n+1 \ 0 & n-1 & 1 vmatrix

= 1[(n - 2) - (n + 1)(n - 1)] + n[1 - 0] + 1[(n - 1) - 0]

= n - 2 - n2 + 1 + n + n - 1

= -n2 + 3n - 2 = -(n - 1)(n - 2)

0 for n 1, 2 n \3, 4, 5, 6\

Probability = 46 k = 4

Sum = 4 + (3 + 4 + 5 + 6) = 22

Answer: (1)

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2

If X = bmatrix x \ y \ z bmatrix is a solution of the system of equations AX = B, where adj A = bmatrix 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 bmatrix and B = bmatrix 4 \ 0 \ 2 bmatrix, then |x + y + z| is equal to :

  1. 3
  2. 2
  3. 32
  4. 1
JEE Main 2026 (Online) 22nd January Evening Shift
3

If the system of equations 3x + y + 4z = 3 2x + y - z = -3 x + 2y + z = 4 has no solution, then the value of is equal to:

  1. 13
  2. 4
  3. 19
  4. 23
JEE Main 2026 (Online) 21st January Evening Shift
4

Let A = bmatrix 2 & 2+p & 2+p+q \ 4 & 6+2p & 8+3p+2q \ 6 & 12+3p & 20+6p+3q bmatrix. If (adj(adj(3A))) = 2m 3n, m, n N, then m + n is equal to

  1. 22
  2. 20
  3. 24
  4. 26
JEE Main 2025 (Online) 8th April Evening Shift
5

Let A be a 3 3 matrix such that |adj(adj(adj A))| = 81. If S = \n Z : (|adj(adj A)| ) (n-1)22 = |A| (3n2 - 5n - 4 ) \, then n S |A (n2 + n ) | is equal to :

  1. 820
  2. 866
  3. 750
  4. 732
JEE Main 2025 (Online) 7th April Morning Shift
6

Let the system of equations : 2x + 3y + 5z = 9 7x + 3y - 2z = 8 12x + 3y - (4 + )z = 16 - have infinitely many solutions. Then the radius of the circle centred at ( , ) and touching the line 4x = 3y is :

  1. 75
  2. 215
  3. 7
  4. 175
JEE Main 2025 (Online) 7th April Morning Shift
7

The number of singular matrices of order 2, whose elements are from the set \2, 3, 6, 9\, is .

JEE Main 2025 (Online) 7th April Morning Shift
8

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
JEE Main 2025 (Online) 7th April Evening Shift
9

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
10

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
11

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

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