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Question

The coefficient of x−5 in binomial expansion of ⎛⎝x+1x23−x13+1−x−1x−x12⎞⎠10, where x≠0,1 is :

A
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B
4
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C
4
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D
1
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Solution

The correct option is C 1⎛⎝x+1x23−x12+1−x−1x−x12⎞⎠10x+1x23−x13+1 =(x13)3+1x23−x13+1 =(x13+1)x23−x13+1 =(x13+1)(x23−x13+1)x23−x13+1 =x13+1x−1x−x12=(x12−1)(x12+1)x12(x12−1) =x12+1x12 =1+x−12∴ Expression =[x13+1−1−x−12]10 =[x13−x−12]10General term =10Cr(x13)r(−1)10−r(x−12)10−rFor coefficient of x−5,r3−12(10−r)=−5r3−5+r2=−5r3+r2=0⇒r=0Coefficient =10Cr(−1)10−r|r=0=10C0(−1)10=1Hence, option B is correct.

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