$\frac{\sin 5 \theta}{\sin \theta}$ is equal to

In the following reaction,$X$ and $Y$ are respectively:
$CH_3COOH + NH_3$ $\longrightarrow X \stackrel{\Delta}{}$ ${\longrightarrow} Y + H_2O$

$A$ photon collides with a stationary hydrogen atom in the ground state inelastically. The energy of the colliding photon is $10.2 \, eV$. After a time interval of the order of a microsecond,another photon collides with the same hydrogen atom inelastically with an energy of $15 \, eV$. What will be observed by the detector?

The value of $\cos ^{-1} \left[ \cot \left\{ \sin ^{-1} \sqrt{\frac{2 - \sqrt{3}}{4}} + \cos ^{-1} \frac{\sqrt{12}}{4} + \sec ^{-1} \sqrt{2} \right\} \right]$ is

An object $2.4 \, m$ in front of a lens forms a sharp image on a film $12 \, cm$ behind the lens. $A$ glass plate $1 \, cm$ thick,of refractive index $1.50$,is interposed between the lens and the film with its plane faces parallel to the film. At what distance (from the lens) should the object be shifted to be in sharp focus on the film? (in $m$)

$(\tan ^{-1} x)^2+(\cot ^{-1} x)^2=\frac{5 \pi^2}{8} \Rightarrow x=$