The set of all values of $\lambda$ for which the system of linear equations : $$ \begin{aligned} & 2 \mathrm{x}_{1}-2 \mathrm{x}_{2}+\mathrm{x}_{3}=\lambda \mathrm{x}_{1} \\ & 2 \mathrm{x}_{1}-3 \mathrm{x}_{2}+2 \mathrm{x}_{3}=\lambda \mathrm{x}_{2} \\ & -\mathrm{x}_{1}+2 \mathrm{x}_{2} \quad=\lambda \mathrm{x}_{3} \end{aligned} $$ has a non-trivial solution,

(C) contains more than two elements.

JEE Main · 2015 · 04 Apr 2015 · Offline · Matrices

The set of all values of

λ

for which the system of linear equations :

2 x_{1} - 2 x_{2} + x_{3} = λ x_{1} 2 x_{1} - 3 x_{2} + 2 x_{3} = λ x_{2} - x_{1} + 2 x_{2} = λ x_{3}

has a non-trivial solution,

Answer: (B)

This is an eigenvalue problem:

A x = λ x

has a nontrivial solution only where

det (A - λ I) = 0

. Building that determinant from the coefficients and expanding, it factors as

(1 - λ) (λ^{2} + 2 λ - 3) = 0

, i.e.

(1 - λ) (λ + 3) (λ - 1) = 0

. The roots are

λ = 1

(repeated) and

λ = - 3

, so the set of admissible values is

{1, - 3}

-- exactly two distinct elements.

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option