CAT (QA) PYQs

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

Let $a_{1}, a_{2}, \ldots, a_{52}$ be positive integers such that $a_{1}<a_{2}<\ldots<a_{52}$. Suppose, their arithmetic mean is one less than the arithmetic mean of $a_{2}, a_{3}$, $\ldots, a_{52}$. If $a_{52}=100$, then the largest possible value of $a_{1}$ 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