1
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Question

# Δ1=∣∣ ∣ ∣∣y5z6(z3−y3)x4z6(x3−z3)x4y5(y3−x3)y2z3(y6−z6)xz3(z6−x6)xy2(x6−y6)y2z3(z3−y3)xz3(x3−z3)xy2(y3−x3)∣∣ ∣ ∣∣ and Δ2=∣∣ ∣ ∣∣xy2z3x4y5z6x7y8z9∣∣ ∣ ∣∣. then Δ1 is equal to

A
Δ23
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B
Δ22
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C
Δ24
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D
none of these
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Solution

## The correct option is A Δ23On solving Δ2=∣∣ ∣ ∣∣xy2z3x4y5z6x7y8z9∣∣ ∣ ∣∣=(xy2z3)((z3−y3)(z3−x3)(y3−x3))Now Δ1=∣∣ ∣ ∣∣y5z6(z3−y3)x4z6(x3−z3)x4y5(y3−x3)y2z3(y6−z6)xz3(z6−x6)xy2(x6−y6)y2z3(z3−y3)xz3(x3−z3)xy2(y3−x3)∣∣ ∣ ∣∣=(y2z3(z3−y3))(xz3(z3−x3))(xy2(y3−x3))∣∣ ∣ ∣∣y3z3−x3z3x3y3−(y3+z3)(z3+x3)−(x3+y3)1−11∣∣ ∣ ∣∣Applying C2→C2+C1,C3→C3−C1=x2y4z6(z3−y3)(z3−x3)(y3−x3)∣∣ ∣ ∣∣y3z3z3(y3−x3)y3(x3−z3)−(y3+z3)x3−y3z3−x3100∣∣ ∣ ∣∣=(xy2z3)2((z3−y3)(z3−x3)(y3−x3))2=Δ23Therefore, Δ=Δ23Hence, option 'A' is correct.

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