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Question

Assertion :The number of terms in the expansion (x+(1/x)+1)n are 2n+1. Reason: The number of terms in the expansion (nr=1ar)n is n+r1Cr1.

A
Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
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B
Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
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C
Assertion is true but Reason is false,
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D
Assertion is false but Reason is true.
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Solution

The correct option is B Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
(x+(1x)+1)n=(x2+x+1)nxn=a0+a1x+a2x2+...+a2nx2nxn=a0xn+a1xn1+a2xn2+...+an1x+an+an+1x+an+2x2+...+a2nxn

Hence, (x+(1x)+1)n has 2n+1 terms.
Now, (nr=1ar)n has n+r1Cr terms, where a0,a1,a2...ar are indepnendent variables.


So, the Assertion and reason are both True but the reason is true only if a0,a1,a2...ar are independent variables but x and 1/x are not independent variables. They vary with each other.


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