$\sinh^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ is equal to

The amplitude of a wave represented by the displacement equation $y = \frac{1}{\sqrt{a}}\sin \omega t \pm \frac{1}{\sqrt{b}}\cos \omega t$ will be

$A$ plane mirror is placed at the bottom of a tank containing a liquid of refractive index $\mu$. $P$ is a small object at a height $h$ above the mirror. An observer $O$,vertically above $P$ outside the liquid,sees $P$ and its image in the mirror. The apparent distance between these two will be

Which of the following quantities does not depend upon the orbital radius of the satellite?

The angle of elevation of a stationary cloud from a point $2500 \ m$ above a lake is $15^{\circ}$ and from the same point the angle of depression of its reflection in the lake is $45^{\circ}$. The height (in metres) of the cloud above the lake,given that $\cot 15^{\circ}=2+\sqrt{3}$,is

Two sides of a triangle are given by the roots of the equation $x^2-5x+6=0$ and the angle between the sides is $\frac{\pi}{3}$. Then,the perimeter of the triangle is