CAT (QA) PYQs

If $5^{x}-3^{y}=13438$ and $5^{x-1}+3^{y+1}=9686$, then $x+y$ equals

Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of $2$. Then, the smallest possible value of $n + m$ is

The sum of all possible values of $x$ satisfying the equations $2^{4 x^{2}}-2^{2 x^{2}+x+16}+2^{2 x+30}=0$, is