CAT (QA) PYQs

For any positive integer $n$, let $f(n)=n(n+1)$ if $n$ is even, and $f(n)=n+3$ if $n$ is odd. If $m$ is a positive integer such that $8 f(m+1)-f(m)=2$, then $m$ equals

The number of the real roots of the equation $2\cos (x(x + 1)) = 2^x + 2^{-x}$ is

Consider a function $f$ satisfying $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) \mathrm{f}(\mathrm{y})$ where $x , y$ are positive integers, and $f(1)=2$. If $f(\mathrm{a}+1)+f(\mathrm{a}+2)+\ldots \ldots+f(\mathrm{a}+\mathrm{n})=$ $16\left(2^{n}-1\right)$ then a is equal to