$A$ thin hollow sphere of mass $m$ is completely filled with a liquid of mass $m$. When the sphere rolls with a velocity $v$,the kinetic energy of the system is (neglect friction).

The angle between the lines whose direction cosines are $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$ and $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)$ is:

Four capacitors marked with their capacitances and breakdown voltages are connected as shown in the figure. Find the maximum $EMF$ of the source such that no capacitor breaks down. (in $kV$)

At $25 \, ^oC$,the equilibrium $SO_2Cl_2(g) \rightleftharpoons SO_2(g) + Cl_2(g)$ is established in a closed vessel. When $Cl_2$ is added to this equilibrium mixture,which of the following statements is true for the system?
$(1)$ The concentrations of $SO_2Cl_2$ and $SO_2$ change.
$(2)$ More $Cl_2$ is produced.
$(3)$ The concentration of $SO_2$ decreases and $SO_2Cl_2$ increases.

An inductor $(L = 0.03 \ H)$ and a resistor $(R = 0.15 \ k\Omega)$ are connected in series to a battery of $15 \ V$ $EMF$ in a circuit as shown below. The key $K_1$ has been kept closed for a long time. Then at $t = 0$,$K_1$ is opened and key $K_2$ is closed simultaneously. At $t = 1 \ ms$,the current in the circuit will be ....... $mA$ $(e^5 \cong 150)$.

The reagent used for converting acetylene to oxalic acid is