CAT (QA) PYQs

For some positive and distinct real numbers $x, y$ and $z$, if $\frac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\frac{1}{\sqrt{x}+\sqrt{z}}$ and $\frac{1}{\sqrt{x}+\sqrt{y}}$, then the relationship which will always hold true, is

Let both the series $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3 \dots$ be in arithmetic progression such that the common differences of both the series are prime numbers. If $a_5 = b_9$, $a_{19} = b_{19}$ and $b_2 = 0$, then $a_{11}$ equals

Let $a_1, a_2, ...$ be integers such that $a_1 - a_2 + a_3 - a_4 + ... + (-1)^{n - 1} a_n = n$, for all $n \ge 1$. Then $a_{51} + a_{52} + ... + a_{1023}$ equals