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If $a, b, c$ are all different and the equations $a x + a^{2} y + (a^{3} + 1) z = 0,$ $b x + b^{2} y + (b^{3} + 1) z = 0,$ $c x + c^{2} y + (c^{3} + 1) z = 0$
have a non-zero solution, then

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abc+1=0

abc−1=0

a2+b2+c2=ab+bc+ca

a3+b3+c3+3abc=0

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detailed solution

Correct option is A

Given equations have non-zero solution⇒Δ=0⇒aa2a3+1bb2b3+1cc2c3+1=0 ⇒aa2a3bb2b3cc2c3+aa21bb21cc21=0 ⇒abc1aa21bb21cc2+1aa21bb21cc2=0 ⇒abc+1a-bb-cc-a=0 ⇒abc+1=0 ∵a, b, c are different

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If a,b,care all different and the equations ax+a2y+(a3+1)z=0,bx+b2y+(b3+1)z=0,cx+c2y+(c3+1)z=0 have a non-zero solution, then

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Ready to Test Your Skills?

free study material

If $a, b, c$ are all different and the equations $a x + a^{2} y + (a^{3} + 1) z = 0,$ $b x + b^{2} y + (b^{3} + 1) z = 0,$ $c x + c^{2} y + (c^{3} + 1) z = 0$
have a non-zero solution, then