CAT (QA) PYQs

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

A function $f$ maps the set of natural numbers to whole numbers, such that $f (xy) = f (x) f (y) + f (x) + f (y)$ for all $x, y$ and $f (p) = 1$ for every prime number $p$. Then, the value of $f (160000)$ is

For all real values of x, the range of the function f(x) = $\frac{x^2+2x+4}{2x^2+4x+9}$ is