Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Matrices

Question:

Let $a, λ, μ ∈ R$. Consider the system of linear equations

$ax + 2y = λ$

$3x-2y=μ$

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

Options:

If $a=-3$, then the system has infinitely many solutions for all values of λ and μ.

If $a≠ -3$, 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 $a=-3$. 

If $λ+μ ≠ 0$, then the system has no solution for $a = -3$.

Correct Answer:

If $a=-3$, then the system has infinitely many solutions for all values of λ and μ.

Explanation:

If $a=-3$, then the two equations in the given system represent parallel lines for $-λ≠u$ i.e. $λ+μ ≠ 0$ and so the system has no solution. So, option (4) is correct.

If $a=-3$ and $λ+μ=0$ i.e. $-λ=μ$, then the two lines given by the above system are coincident and hence the system has infinitely many solutions. So, option (3) is correct for all values of λ and μ.

If $a≠ -3$, then the two lines given in the system are not parallel i.e. they are intersecting and hence the system has a unique solution. So, option (2) is correct.

Clearly, option (1) is incorrect.