JEE-Mains 2015 (7/30)

Consider $f (x)=(1-2\sqrt{x})^{50}+(1+2\sqrt {x})^{50}$

In the expansion of $f (x)$ , we only get terms of integral powers of $x$ . Terms with $\sqrt {x}$ will cancel out.

However, in both the summands $(1-2\sqrt{x})^{50}$ and $(1+2\sqrt{x})^{50}$ , the sum of coefficients of terms with integral powers is the same. (Why this is so as such a term will be in the form $( \pm 2\sqrt {x})^{2r}$ ${50} \choose {2r}$ for some integer $r$ which is independent of the sign $\pm$ .)

So the answer we want is half of the sum of coefficients of all terms of expansion of $f (x)$ . The sum of coefficients is when we substitute $x=1$ into $f (x)$ . Hence the answer is $\frac {1}{2}(3^{50}+1)$ .