$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

The major product of the following reaction is

$A$ rectangular glass slab $ABCD,$ of refractive index $n_1,$ is immersed in water of refractive index $n_2$ $(n_1 > n_2)$. $A$ ray of light is incident at the surface $AB$ of the slab as shown. The maximum value of the angle of incidence $\alpha_{\max},$ such that the ray comes out only from the other surface $CD,$ is given by:

$\cos ^2\left(\frac{\pi}{6}+\theta\right)-\sin ^2\left(\frac{\pi}{6}-\theta\right)$ is equal to

With an increase in atomic number,the melting point of alkali metals .......

Considering only the principal values of inverse functions,the set $A = \{x \geq 0, \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$