If $f(x)=(\cos x)(\cos 2 x) \ldots(\cos n x)$,then $f^{\prime}(x)+\sum_{r=1}^n(r \tan r x) f(x)$ is equal to

The coefficient of $x^3 y^4 z^5$ in the expansion of $(x y+y z+x z)^6$ is

Assertion $(A)$: Propagation of light through an optical fibre is due to total internal reflection taking place at the core-clad interface.
Reason $(R)$: Refractive index of the material of the core of the optical fibre is greater than that of air.

$A$ star initially has $10^{40}$ deuterons. It produces energy via the processes:
${}_1H^2 + {}_1H^2 \to {}_1H^3 + p$
${}_1H^2 + {}_1H^3 \to {}_2He^4 + n$
The masses of the nuclei are as follows: $M(H^2) = 2.014 \, amu$,$M(p) = 1.007 \, amu$,$M(n) = 1.008 \, amu$,$M(He^4) = 4.001 \, amu$. If the average power radiated by the star is $10^{16} \, W$,the deuteron supply of the star is exhausted in a time of the order of:

If the coefficient of $x$ in the expansion of $(x^2 + \frac{k}{x})^5$ is $270$,then $k =$

If $(a \times b)^2 + (a \cdot b)^2 = 144$ and $|a| = 4$,then $|b| = $