The system of linear equations $\mathrm{x}+\lambda \mathrm{y}-\mathrm{z}=0$ $\lambda \mathrm{x}-\mathrm{y}-\mathrm{z}=0$ $\mathrm{x}+\mathrm{y}-\lambda \mathrm{z}=0$ has a non-trivial solution for :

(C) exactly three values of $\lambda$.

(A) exactly one value of $\lambda$.

(B) exactly two values of $\lambda$.

(D) infinitely many values of $\lambda$.

JEE Main · 2016 · 03 Apr 2016 · Offline · Determinants

The system of linear equations

x + λ y - z = 0

λ x - y - z = 0

x + y - λ z = 0

has a non-trivial solution for :

Answer: (C)

A homogeneous system has a non-trivial solution only when its coefficient determinant is zero. Expanding

1 λ 1 λ - 1 1 - 1 - 1 - λ = 0

gives

1 (λ + 1) - λ (- λ^{2} + 1) - 1 (λ + 1) = 0

, which tidies to

λ^{3} - λ = 0

, i.e.

λ (λ - 1) (λ + 1) = 0

. That gives

λ = 0, 1, - 1

, exactly three values.

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option