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Question

If a, b, c and d are complex numbers, then the determinant
Δ=∣ ∣ ∣2a+b+c+dab+cda+b+c+d2(a+b)(c+d)ab(c+d)+cd(a+b)ab+cdab(c+d)+cd(a+b)2abcd∣ ∣ ∣
equals

A
0
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B
abcd
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C
a+b+c+d
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D
2abcd
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Solution

The correct option is A 0
Δ=∣ ∣ ∣2a+b+c+dab+cda+b+c+d2(a+b)(c+d)ab(c+d)+cd(a+b)ab+cdab(c+d)+cd(a+b)2abcd∣ ∣ ∣
=∣ ∣110c+da+b0cdab0∣ ∣∣ ∣110a+bc+d0abcd0∣ ∣ Row X Row multiplication
=0
Hence, option A.

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