The correct option is A k=0 only
Δ=∣∣
∣∣11−22−311−54∣∣
∣∣
R2→R2−2R1,R3→R3−R1
Δ=∣∣
∣∣11−20−550−66∣∣
∣∣
=5×6∣∣
∣∣11−20−110−11∣∣
∣∣
Δ=0 (R2 and R3 are identical)
Hence, system is consistent only when Δ1=Δ2=Δ3=0
Consider Δ1=0
⇒∣∣
∣∣01−20−31k−54∣∣
∣∣=0
⇒−5k=0
⇒k=0
For k=0,
clearly, Δ2=Δ3=0
Therefore, system is consistent if k=0