The maximum value of $xy$ subject to $x+y=7$ is

In the given figure,a ring of mass $m$ is kept on a horizontal surface,while a body of equal mass $m$ is attached through a string wound on the ring. When the system is released,the ring rolls without slipping. Consider the following statements and choose the correct option.
$(i)$ Acceleration of the centre of mass of the ring is $g/3$.
$(ii)$ Acceleration of the hanging particle is $2g/3$.
$(iii)$ Frictional force (on the ring) acts along the forward direction.
$(iv)$ Frictional force (on the ring) acts along the backward direction.

The major product in the following reaction is:

If $x=e^{y+e^{y+e^{y+\ldots}}}$,then $\frac{d y}{d x}=$

The radius of the smallest electron orbit in a hydrogen-like ion is $(0.51 \times 10^{-10} / 4) \text{ m}$. This ion is:

If $x = \tan 15^{\circ}$,$y = \operatorname{cosec} 75^{\circ}$ and $z = 4 \sin 18^{\circ}$,then :