Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations \[ \begin{aligned} & a x+2 y=\lambda \\ & 3 x-2 y=\mu \end{aligned} \] Which of the following statement(s) is(are) correct?

(B) If $a \neq-3$, then the system has a unique solution for all values of $\lambda$ and $\mu$; (C) If $\lambda+\mu=0$, then the system has infinitely many solutions for $a=-3$; (D) If $\lambda+\mu \neq 0$, then the system has no solution for $a=-3$

(A) If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$

Matrices — JEE Advanced · 2016 · 22 May 2016 · Paper 2 Question with Solution

Let

a, λ, μ \in R

. Consider the system of linear equations

a x + 2 y = λ 3 x - 2 y = μ

Which of the following statement(s) is(are) correct?

(A)If $a = - 3$ , then the system has infinitely many solutions for all values of $λ$ and $μ$
(B)If $a \neq = - 3$ , then the system has a unique solution for all values of $λ$ and $μ$ ✓ Correct
(C)If $λ + μ = 0$ , then the system has infinitely many solutions for $a = - 3$ ✓ Correct
(D)If $λ + μ \neq = 0$ , then the system has no solution for $a = - 3$ ✓ Correct

Answer: (B), (C), (D)

Solution

The system has coefficient matrix with determinant

Δ = - 2 a - 6

. For

a \neq = - 3

Δ \neq = 0

, so the system has a unique solution for all

λ, μ

, confirming (B). For

a = - 3

Δ = 0

. The system becomes

- 3 x + 2 y = λ

and

3 x - 2 y = μ

. Adding these:

0 = λ + μ

. If

λ + μ = 0

, the equations are consistent and have infinitely many solutions, confirming (C). If

λ + μ \neq = 0

, the system is inconsistent and has no solution, confirming (D). Option (A) is false because infinitely many solutions require

λ + μ = 0

, not all values.

JEE Advanced · 2016 · 22 May 2016 · Paper 2 · One or more correct options

JEE Advanced · 2016 · 22 May 2016 · Paper 2 · Matrices

Solution

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