Statement -1 : The term independent of x in the expansion of ${(x^{2} + \frac{1}{x^{2}} + 2)}^{25} i s^{50} C_{25} .$
Statement-2 : In a binomial expansion middle term is independent of x.

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Statement -1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement -1

Statement -1 is true, Statement-2 is true; Statement-2 is not correct explanation for Statement -1

Statement -1 is false, Statement -2 is true

Statement -1 is true, Statement -2 is false

answer is C.

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Detailed Solution

We have, ${(x^{2} + \frac{1}{x^{2}} + 2)}^{25} = {(x + \frac{1}{x})}^{50} .$

$∴ T_{r + 1} =^{50} C_{r} . x^{50 - r} . {(\frac{1}{x})}^{r} =^{50} C_{r} x^{50 - 2 r}$

For independent of x, we put $50 - 2 r = 0 \Rightarrow r = 25$

$∴ T_{25 + 1} =^{50} C_{25}$

But in binomial expansion of ${(x + a)}^{n},$ middle terms is independent of x, iff $x . a = 1.$

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