Binomial Theorem Questions (605)

The $50^{\text{th}}$ root of a number $x$ is 12 and $50^{\text{th}}$ root of another number $y$ is 18. Then the remainder obtained on dividing $(x+y)$ by 25 is ___.

Suppose $2-p,\, p,\, 2-\alpha,\, \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6\alpha+2p$ equals [Note: In the official NTA paper no option was correct.]

Let n be an odd natural number greater than 1. Then, find the number of zeros at the end of the sum \(99^n + 1\).

If the fourth term in the binomial expansion of \(\left(x^{\sqrt {\left (\frac{1}{1+\log _{10} x}\right)}}+x^{\frac{1}{12}}\right)^{6}\) is equal to 200, and x > 1, then the value of x is

The sum of all rational terms in the expansion of $\left(2^{1/3}+5^{1/5}\right)^{15}$ is equal to: