$A$ light wave has a frequency of $4 \times 10^{14} ~Hz$ and a wavelength of $5 \times 10^{-7} ~m$ in a medium. The refractive index of the medium is:

Two substances $A$ and $B$ of equal mass $m$ are heated at a uniform rate of $6 \, cal \, s^{-1}$ under similar conditions. $A$ graph between temperature and time is shown in the figure. The ratio of heat absorbed $H_A/H_B$ by them for complete fusion is

If $\int \frac{5\tan x}{\tan x - 2} dx = x + a \ln |\sin x - 2\cos x| + k$,then $a$ is equal to

The polar equation of the line perpendicular to the line $\sin \theta - \cos \theta = \frac{1}{r}$ and passing through the point $\left(2, \frac{\pi}{6}\right)$ is

If $\cos A = \frac{3}{4}$,then $32 \sin \left(\frac{A}{2}\right) \sin \left(\frac{5A}{2}\right) = $

In a $\triangle ABC$,if $\frac{a}{b^2-c^2} + \frac{c}{b^2-a^2} = 0$,then $B$ is equal to