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Question

If ∣ ∣ ∣(1+x)(1+x)2(1+x)3(1+x)4(1+x)5(1+x)6(1+x)7(1+x)8(1+x)9∣ ∣ ∣ =a0+a1x+a2x2+. , then, a1 is equal to

A
1
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B
2
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C
0
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D
3
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Solution

The correct option is D 0
∣ ∣ ∣(1+x)(1+x)2(1+x)3(1+x)4(1+x)5(1+x)6(1+x)7(1+x)8(1+x)9∣ ∣ ∣=a0+a1x+a2x2+.....
Put x=0
∣ ∣111111111∣ ∣=a0
a0=0
So,∣ ∣ ∣(1+x)(1+x)2(1+x)3(1+x)4(1+x)5(1+x)6(1+x)7(1+x)8(1+x)9∣ ∣ ∣=a1x+a2x2+.....
Differentiating w.r.t. x,
∣ ∣ ∣12(1+x)3(1+x)2(1+x)4(1+x)5(1+x)6(1+x)7(1+x)8(1+x)9∣ ∣ ∣+∣ ∣ ∣(1+x)(1+x)2(1+x)34(1+x)35(1+x)46(1+x)5(1+x)7(1+x)8(1+x)9∣ ∣ ∣+
∣ ∣ ∣(1+x)(1+x)2(1+x)3(1+x)4(1+x)5(1+x)67(1+x)68(1+x)79(1+x)8∣ ∣ ∣=a1+2a2x+....
Put x=0
∣ ∣123111111∣ ∣+∣ ∣111456111∣ ∣+∣ ∣111111789∣ ∣=a1
a1=0

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