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Determinants

JEE Main 2024 (Online) 9th April Evening ShiftNumericalMedium📈 Topic in 59% papers+4
Ques. 1JEE Main 2024, 9 April (Evening Shift)

Consider the matrices : A = bmatrix 2 & -5 \ 3 & m bmatrix, B = bmatrix 20 \ m bmatrix and X = bmatrix x \ y bmatrix. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 ab |A| dm is equal to .

Answer

450

Solution

AX = B gives 2x - 5y = 20 and 3x + my = m, with |A| = 2m + 15.

By Cramer's rule: x = 20m + 5m2m + 15 = 25m2m + 15 and y = 2m - 602m + 15

x < 0 gives - 152 < m < 0

y < 0 gives - 152 < m < 30

So both hold for m (- 152, 0 ) = (a, b)

8 -15/20(2m + 15) dm = 8 [m2 + 15m ]-15/20

= 8 [0 - ( 2254 - 2252 ) ] = 8 2254 = 450

Answer: 450

Keep practising — Determinants

Ques. 2JEE Main 2024, 8 April (Evening Shift)

If a, b, c and vmatrix & b & c \ a & & c \ a & b & vmatrix = 0, then a - a + b - b + - c is equal to :

  1. 2
  2. 3
  3. 1
  4. 0
Ques. 3JEE Main 2024, 8 April (Evening Shift)

If the system of equations x + 4y - z = , 7x + 9y + z = -3, 5x + y + 2z = -1 has infinitely many solutions, then (2 + 3 ) is equal to :

  1. -2
  2. 2
  3. 3
  4. -3
Ques. 4JEE 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
Ques. 5JEE 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. 6JEE 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. 7JEE 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. 8JEE 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. 9JEE 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. 10JEE 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. 11JEE 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

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