JEE Main 2021 · 24 February · morning · Single correct · +4 / −1

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

  1. (a) k ≠ 3, m ∈ R
  2. (b) k = 3, m ≠ 4/5
  3. (c) k = 3, m = 4/5
  4. (d) k ≠ 3, m ≠ 4/5
Answer and solution

Answer (b) k = 3, m ≠ 4/5

Solution

Maths sir · 4 steps

First make the determinant zero to fix k, then check whether the right-hand sides break the system.

  1. Coefficient determinant. A system with Δ ≠ 0 has a unique solution, so it can only be inconsistent when Δ = 0. Expand along the first row.

    Δ = |3, -2, -k; 2, -4, -2; 1, 2, -1| = 3(8) + 2(0) - k(8) = 8(3 - k)

  2. Fix k. Δ = 0 only when k = 3. For any k ≠ 3 the system is consistent, whatever m is.
  3. Compare the equations. With k = 3, adding the second and third left sides gives (2x - 4y - 2z) + (x + 2y - z) = 3x - 2y - 3z, the first left side. So the right sides must match too: 6 + 5m = 10, that is m = 4/5.
  4. So. If m = 4/5 the first equation is just the sum of the other two, giving infinitely many solutions. If m ≠ 4/5 the equations contradict each other, so the system is inconsistent for k = 3, m ≠ 4/5: option (b).

The idea to keep. Inconsistency needs Δ = 0 first. Then look for the combination of equations whose left sides match: if the right sides do not match the same way, there is no solution.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Determinants

Asked about 190 times in JEE Main since 2020. This question is from 2021.