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Question

If ∣ ∣ ∣a2b2c2(a+1)2(b+1)2(c+1)2(a1)2(b1)2(c1)2∣ ∣ ∣=k(ab)(bc)(ca), then find the value of k

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Solution

Δ=∣ ∣ ∣a2b2c2(a+1)2(b+1)2(c+1)2(a1)2(b1)2(c1)2∣ ∣ ∣
=∣ ∣ ∣a2b2c2a2+2a+1b2+2b+1c2+2c+1a22a+1b22b+1c22c+1∣ ∣ ∣
=∣ ∣ ∣a2b2c22a+12b+12c+14a4b4c∣ ∣ ∣ R2R2R1, and R3R3R2
=∣ ∣ ∣a2b2c24a4b4c2a+12b+12c+1∣ ∣ ∣=4∣ ∣ ∣a2b2c2abc2a+12b+12c+1∣ ∣ ∣
=4∣ ∣ ∣a2b2c2abc111∣ ∣ ∣ R3R22R2
=4∣ ∣ ∣111abca2b2c2∣ ∣ ∣=4(ab)(bc)(ca)

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