CAT (QA) PYQs

If $12^{12x}\times4^{24x+12}\times5^{2y}=8^{4z}\times20^{12x}\times243^{3x-6}$, where $x$, $y$ and $z$ are natural numbers, then $x+y+z$ equals

How many distinct positive integer-valued solutions exist to the equation $\left(x^{2}-7 x+11\right)^{\left(x^{2}-13 x+42\right)}=1 ?$

If $9^{2 x-1}-81^{x-1}=1944$, then $x$ is