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Question

# The set of all values of λ for which the system of linear equations.x−2y−2z=λxx+2y+z=λy−x−y=λzhas a non-trivial solution.

A
contains more than two elements
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B
is a singleton
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C
is an empty set
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D
contains exactly two elements
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Solution

## The correct option is D is a singletonx−2y−2z=λx(1−λ)x−2y−2z=0.....(1)x+2y+z=λyx+(2−λ)y+z=0−x−y=λzx+y+λz=0Since the system of given linear equations has a non-trivial solution.Therefore,∣∣ ∣∣1−λ−2−212−λ111λ∣∣ ∣∣=0((1−λ)(λ(2−λ)−1)−2(λ−1)−2(1−(2−λ))=0(1−λ)3=0⇒λ=1Thus, the set of values of λ contains exactly one element.Hence the correct answer is (B).

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