CAT (QA) PYQs

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

Suppose for all integers x, there are two function f and g such that $f(x) + f(x-1) - 1 = 0$ and $g(x) = x^2$. If $f(x^2 - x) = 5$, then the value of the sum $f(g(5)) + g(f(5))$ is

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