When a capacitor is connected in series to an $LR$ circuit,the alternating current flowing in the circuit

If $\cos (\alpha+\beta)=\frac{4}{5}$,$\sin (\alpha-\beta)=\frac{5}{13}$ and $\alpha, \beta$ are between $0$ and $\frac{\pi}{4}$,then $\tan 2\alpha$ is equal to

The wavelengths of light in two liquids '$x$' and '$y$' are $3500 \ \mathring A$ and $7000 \ \mathring A$ respectively. The critical angle of '$x$' with respect to '$y$' will be ...... $^o$.

Two circular coils can be arranged in any of the three situations shown in the figure. Their mutual inductance will be:

$A$ very small hole in an electric furnace is used for heating metals. The hole nearly acts as a black body. The area of the hole is $200 ~mm^2$. To keep a metal at $727^{\circ} C$,the heat energy flowing through this hole per second,in joules,is $\left(\sigma = 5.67 \times 10^{-8} ~W m^{-2} K^{-4}\right)$.

$A$ student records $\Delta Q, \Delta U$ and $\Delta W$ for a thermodynamic cycle $A \to B \to C \to A$. Certain entries are missing. Find the correct entry from the following options.

Process	$AB$	$BC$	$CA$
$\Delta W$	$40 \ J$	$ $	$30 \ J$
$\Delta U$	$ $	$50 \ J$	$ $
$\Delta Q$	$150 \ J$	$10 \ J$	$ $