WBCS Mains 2019 Official Paper-VI

$A$ and $B$ are partners in a firm sharing profits and losses in the ratio of $5:3$. They admit $C$ as a partner for $\frac{1}{4}\text{th}$ share in the profits of the firm. Calculate the new profit sharing ratio of $A$, $B$ and $C$ if $C$ acquires his share as $\frac{1}{5}\text{th}$ from $A$ and $\frac{1}{20}\text{th}$ from $B$.

Evaluate the following expression:
$$\frac{3}{\sqrt{3}}-\frac{4}{\sqrt{7}+\sqrt{3}}+\frac{1}{\sqrt{12}+3}$$

What least number must be added to $8275$ to make the sum a perfect square?

What least number must be subtracted from $732$ to make the remainder a perfect square?

By what number must $192$ be multiplied, so that the product may be a perfect square?