Let S denote the set of all real values of $\lambda$ such that the system of equations $\lambda \mathrm{x}+\mathrm{y}+\mathrm{z}=1$ $\mathrm{x}+\lambda \mathrm{y}+\mathrm{z}=1$ $\mathrm{x}+\mathrm{y}+\lambda \mathrm{z}=1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$ is equal to

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Matrices

Let S denote the set of all real values of

λ

such that the system of equations

λ x + y + z = 1

x + λ y + z = 1

x + y + λ z = 1

is inconsistent, then

\sum_{λ \in S} (∣ λ ∣^{2} + ∣ λ ∣)

is equal to

Answer: (D)

The coefficient determinant is

λ 11 1 λ 1 11 λ = (λ + 2) (λ - 1)^{2}

, which vanishes at

λ = 1

and

λ = - 2

. At

λ = 1

all three equations become

x + y + z = 1

, giving infinitely many solutions, so the system is consistent. At

λ = - 2

adding the three equations yields

0 = 3

, so the system is inconsistent. Hence

S = {- 2}

, and

∣ λ ∣^{2} + ∣ λ ∣ = 4 + 2

, so the required sum is

6

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Single correct option