Let $\mathrm{S}$ be the set of positive integral values of $a$ for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}<0, \forall \mathrm{x} \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is :

Let $\alpha$ and $\beta$ be the two disinct roots of the equation $x^3 + 3x^2 -1 = 0.$ The equation which has $(\alpha \beta )$ as its root is equal to

Let $S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }$ Then $n ( S )$ is equal to

The number of solutions of $\sin ^2 \mathrm{x}+\left(2+2 \mathrm{x}-\mathrm{x}^2\right) \sin \mathrm{x}-3(\mathrm{x}-1)^2=0$, where $-\pi \leq \mathrm{x} \leq \pi$, is....................

Let $\alpha$ and $\beta$ be the roots of the equation $5 x^{2}+6 x-2=0 .$ If $S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots$ then :

Number of positive integral values of $'K'$ for which the equation $k = \left| {x + \left| {2x - 1} \right|} \right| - \left| {x - \left| {2x - 1} \right|} \right|$ has exactly three real solutions, is