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Question

Δ1=∣ ∣ ∣y5z6(z3y3)x4z6(x3z3)x4y5(y3x3)y2z3(y6z6)xz3(z6x6)xy2(x6y6)y2z3(z3y3)xz3(x3z3)xy2(y3x3)∣ ∣ ∣ 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 Δ23
On solving
Δ2=∣ ∣ ∣xy2z3x4y5z6x7y8z9∣ ∣ ∣=(xy2z3)((z3y3)(z3x3)(y3x3))
Now
Δ1=∣ ∣ ∣y5z6(z3y3)x4z6(x3z3)x4y5(y3x3)y2z3(y6z6)xz3(z6x6)xy2(x6y6)y2z3(z3y3)xz3(x3z3)xy2(y3x3)∣ ∣ ∣=(y2z3(z3y3))(xz3(z3x3))(xy2(y3x3))∣ ∣ ∣y3z3x3z3x3y3(y3+z3)(z3+x3)(x3+y3)111∣ ∣ ∣
Applying C2C2+C1,C3C3C1
=x2y4z6(z3y3)(z3x3)(y3x3)∣ ∣ ∣y3z3z3(y3x3)y3(x3z3)(y3+z3)x3y3z3x3100∣ ∣ ∣=(xy2z3)2((z3y3)(z3x3)(y3x3))2=Δ23
Therefore, Δ=Δ23
Hence, option 'A' is correct.

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