If $1+\frac{\cos \theta}{2}+\frac{\cos 2 \theta}{4}+\frac{\cos 3 \theta}{8}+\ldots \ldots=\frac{a-2 \cos \theta}{5+b \cos \theta}$ for some $a, b \in \mathbf{R}$, then $(a-b)^2=$

If $f(1)=3$, and $f(n+1)-f(n)=3\left(4^n-1\right)$, then $\forall n \in \mathbf{N}$, $f(n)=$

If $S_n$ is the sum of the first $n$ terms of the series $1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty$, then, when $n$ is even $S_n=$

If the roots of the equation, $8 x^3+6 p x^2+3 q x-27=0$ are in a geometric progression, then $q^2+9 p^2+6 p q+q / p=$