Let $ S $ be the set of all $ \lambda \in R $ for which the system of linear equations $ 2 \mathrm{x}-\mathrm{y}+2 \mathrm{z}=2 $ $ \mathrm{x}-2 \mathrm{y}+\lambda \mathrm{z}=-4 $ $ \mathrm{x}+\lambda \mathrm{y}+\mathrm{z}=4 $ has no solution. Then the set $ \mathrm{S} $

Question

Let $ S $ be the set of all $ \lambda \in R $ for which the system of linear equations $ 2 \mathrm{x}-\mathrm{y}+2 \mathrm{z}=2 $ $ \mathrm{x}-2 \mathrm{y}+\lambda \mathrm{z}=-4 $ $ \mathrm{x}+\lambda \mathrm{y}+\mathrm{z}=4 $ has no solution. Then the set $ \mathrm{S} $, Contains more than two elements, Is an empty set, Is a singleton, Contains exactly two elements

MARKS App · Accepted Answer

For no. solution Δ = 0 and at least one of Δ 1 , Δ 2 , Δ 3 is non-zero. Δ = 2 - 1 2 1 - 2 λ 1 λ 1 = - λ - 1 2 λ + 1 Δ 1 = 2 - 1 2 - 4 - 2 λ 4 λ 1 = - 2 λ 2 + 6 λ - 4 Δ = 0 ⇒ λ = 1 , - 1 2 And for λ = 1 , - 1 2 , ∆ 1 ≠ 0 Hence, S = 1 , - 1 2

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