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Question

The system of equations
x+y2z=0
2x3y+z=0
x5y+4z=k is consistent for

A
k=0 only
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B
all non-zero real values of k
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C
all real values of k
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D
two real values of k
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Solution

The correct option is A k=0 only
Δ=∣ ∣112231154∣ ∣

R2R22R1,R3R3R1
Δ=∣ ∣112055066∣ ∣

=5×6∣ ∣112011011∣ ∣

Δ=0 (R2 and R3 are identical)
Hence, system is consistent only when Δ1=Δ2=Δ3=0
Consider Δ1=0
∣ ∣012031k54∣ ∣=0

5k=0
k=0
For k=0,
clearly, Δ2=Δ3=0
Therefore, system is consistent if k=0

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