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Determinants

JEE Main 2024 (Online) 6th April Morning ShiftMCQMedium📈 Topic in 19% papers+4−1
Ques. 1JEE Main 2024, 6 April (Morning Shift)

For , R and a natural number n, let Ar = vmatrix r & 1 & n22 + \ 2r & 2 & n2 - \ 3r - 2 & 3 & n(3n - 1)2 vmatrix. Then 2A10 - A8 is

  1. 4 + 2
  2. 0
  3. 2n
  4. 2 + 4

Answer

(1) 4 + 2

Solution

Only the first column of Ar depends on r, and a determinant is linear in a column, so 2A10 - A8 is a single determinant with C1 = 2(10, 20, 28) - (8, 16, 22) = (12, 24, 34):

2A10 - A8 = vmatrix 12 & 1 & n22 + \ 24 & 2 & n2 - \ 34 & 3 & n(3n - 1)2 vmatrix

Applying R2 R2 - 2R1 and R3 R3 - 3R1 gives

2A10 - A8 = vmatrix 12 & 1 & n22 + \ 0 & 0 & -2 - \ -2 & 0 & - n2 - 3 vmatrix

Expanding along R2: 2A10 - A8 = -(-2 - ) vmatrix 12 & 1 \ -2 & 0 vmatrix = (2 + ) 2 = 4 + 2

The value is free of n, and 0, 2n, 2 + 4 do not match it, so (2), (3) and (4) are wrong.

Answer: (1)

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Ques. 2JEE Main 2024, 6 April (Morning Shift)

Let = 45; , , R. If x( , 1, 2) + y(1, , 2) + z(2, 3, ) = (0, 0, 0) for some x, y, z R, xyz 0, then 6 + 4 + is equal to .

Ques. 3JEE Main 2024, 6 April (Evening Shift)

If A is a square matrix of order 3 such that (A) = 3 and (adj(-4 adj(-3 adj(3 adj((2A)-1))))) = 2m3n, then m + 2n is equal to :

  1. 2
  2. 4
  3. 3
  4. 6
Ques. 4JEE Main 2024, 6 April (Evening Shift)

If the system of equations 2x + 7y + z = 3 3x + 2y + 5z = 4 x + y + 32z = -1 has infinitely many solutions, then ( - ) is equal to :

Ques. 5JEE Main 2024, 5 April (Morning Shift)

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |ATA(adj(2 A))-1(adj(4 B))(adj(AB))-1AAT| is equal to :

  1. 32
  2. 81
  3. 64
  4. 108
Ques. 6JEE Main 2024, 5 April (Morning Shift)

If the system of equations 11x + y + z = -5 2x + 3y + 5z = 3 8x - 19y - 39z = has infinitely many solutions, then 4 - is equal to :

  1. 51
  2. 45
  3. 47
  4. 49
Ques. 7JEE Main 2024, 5 April (Evening Shift)

The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz = n has infinitely many solutions, satisfy the equation :

  1. m2 + n2 - m - n = 46
  2. m2 + n2 + mn = 68
  3. m2 + n2 - mn = 39
  4. m2 + n2 + m + n = 64
Ques. 8JEE Main 2024, 5 April (Evening Shift)

Let 0 and A = bmatrix & & 3 \ & & \ - & & 2 bmatrix. If B = bmatrix 3 & -9 & 3 \ - & 7 & -2 \ -2 & 5 & -2 bmatrix is the matrix of cofactors of the elements of A, then (AB) is equal to :

  1. 64
  2. 343
  3. 125
  4. 216
Ques. 9JEE Main 2024, 4 April (Morning Shift)

Let (0, ) and A= bmatrix1 & 2 & \ 1 & 0 & 1 \ 0 & 1 & 2 bmatrix. If det ( adj (2A-AT ) adj (A-2AT ) )=28, then ( det(A))2 is equal to:

  1. 16
  2. 36
  3. 49
  4. 1
Ques. 10JEE Main 2024, 4 April (Morning Shift)

If the system of equations x+( 2 )y+( 2 )z=0 x+( )y+( )z=0 x+( )y-( )z=0 has a non-trivial solution, then (0, 2 ) is equal to :

  1. 5 24
  2. 11 24
  3. 7 24
  4. 3 4
Ques. 11JEE Main 2024, 4 April (Morning Shift)

Let A be a 3 3 matrix of non-negative real elements such that A bmatrix1\\1\\1 bmatrix=3 bmatrix1\\1\\1 bmatrix. Then the maximum value of (A) is .

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