CAT (QA) PYQs

The number of real-valued solutions of the equation $2^x + 2^{-x} = 2 - (x - 2)^2$ is

If $a$ and $b$ are non-negative real numbers such that $a + 2b = 6$, then the average of the maximum and minimum possible values of $(a + b)$ is

For natural numbers $x$, $y$, and $z$, if $xy + yz = 19$ and $yz + xz = 51$, then the minimum possible value of $xyz$ is