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Question:

Let a, λ, μ ∈ R. Consider the system of linear equations ax + 2y = λ
3x - 6y = μ
Which of the following statement(s) is(are) correct?

If a = -6, then the system has infinitely many solutions for all values of λ and μ

If a ≠ -6, then the system has a unique solution for all values of λ and μ

If λ + μ = 0, then the system has infinitely many solutions for a = -6

If λ + μ ≠ 0, then the system has no solution for a = -6

Solution:

ax + 2y = λ
3x - 6y = μ
(A) a = -6 gives λ = μ or λ + μ = 0 not for all λ, μ
(B) a ≠ -6 ⇒ Δ ≠ 0 where Δ = |a 2|
|3 -6| = -6a + 6 ≠ 0
∴ (B) is correct
(C) correct
(D) if λ + μ ≠ 0
3x + 2y = λ (1)
3x - 6y = μ (2)
Inconsistent ⇒ (D) correct