Consider the system of linear equation $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^{2}z=\mu^{2}+15$, where $\lambda, \mu \in \mathrm{R}$. Which one of the following statements is NOT correct?

(B) The system is inconsistent if $\lambda=\frac{1}{2}$ and $\mu \neq 1$

(A) The system has unique solution if $\lambda \neq \frac{1}{2}$ and $\mu \neq 1,15$

(C) The system has infinite number of solutions if $\lambda=\frac{1}{2}$ and $\mu=15$

(D) The system is consistent if $\lambda \neq \frac{1}{2}$

Matrices — JEE Main · 2024 · 30 Jan 2024 · Shift 1 Question with Solution

Consider the system of linear equation

x + y + z = 4 μ

x + 2 y + 2 λ z = 10 μ

x + 3 y + 4 λ^{2} z = μ^{2} + 15

, where

λ, μ \in R

. Which one of the following statements is NOT correct?

(A)The system has unique solution if $λ \neq = \frac{1}{2}$ and $μ \neq = 1, 15$
(B)The system is inconsistent if $λ = \frac{1}{2}$ and $μ \neq = 1$ ✓ Correct
(C)The system has infinite number of solutions if $λ = \frac{1}{2}$ and $μ = 15$
(D)The system is consistent if $λ \neq = \frac{1}{2}$

Answer: (B)

Δ = 111123 1 2 λ 4 λ^{2} = (2 λ - 1)^{2}

, zero only at

λ = \frac{1}{2}

— confirming a unique solution whenever

λ \neq = \frac{1}{2}

(so statements 1 and 4 are correct).

At

λ = \frac{1}{2}

Δ_{x} = Δ_{z} = (μ - 1) (μ - 15)

; the system has infinitely many solutions when this also vanishes, i.e.

μ = 1

μ = 15

(statement 3 is correct), and is inconsistent for every other

μ

.

Statement 2 claims inconsistency whenever

λ = \frac{1}{2}

and

μ \neq = 1

— but at

μ = 15

(which satisfies

μ \neq = 1

) the system actually has infinitely many solutions, not none. So statement 2 is the one that is NOT correct.

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Single correct option