Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :

The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 e^{-\sin x}=2$ is

If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is

The two roots of an equation ${x^3} - 9{x^2} + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots will be

A man standing on a railway platform noticed that a train took $21\, s$ to cross the platform (this means the time elapsed from the moment the engine enters the platform till the last compartment leaves the platform) which is $88\,m$ long, and that it took $9 s$ to pass him. Assuming that the train was moving with uniform speed, what is the length of the train in meters?

If $\alpha, \beta $ and $\gamma$ are the roots of equation ${x^3} - 3{x^2} + x + 5 = 0$ then $y = \sum {\alpha ^2} + \alpha \beta \gamma $ satisfies the equation