If $x, y$ are real numbers such that $3^{(x / y)+1}-3^{(x / y)-1}=24$ then the value of $(x+y) /(x-y)$ is

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 :

If $\alpha ,\beta$ are the roots of $x^2 -ax + b = 0$ and if $\alpha^n + \beta^n = V_n$, then -

The number of solutions of $\frac{{\log 5 + \log ({x^2} + 1)}}{{\log (x - 2)}} = 2$ is

Let $p$ and $q$ be two real numbers such that $p+q=$ 3 and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to

Let $\alpha $ and $\beta $ be the roots of the quadratic equation ${x^2}\,\sin \,\theta  - x\,\left( {\sin \,\theta \cos \,\,\theta  + 1} \right) + \cos \,\theta  = 0\,\left( {0 < \theta  < {{45}^o}} \right)$ , and $\alpha  < \beta $.  Then $\sum\limits_{n = 0}^\infty  {\left( {{\alpha ^n} + \frac{{{{\left( { - 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ is equal to