A vessel contains $$14 \mathrm{~g}$$ of nitrogen gas at a temperature of $$27^{\circ} \mathrm{C}$$. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be :

Take $$\mathrm{R}=8.32 \mathrm{~J} \mathrm{~mol}^{-1} \,\mathrm{k}^{-1}$$.

A slab of dielectric constant $$\mathrm{K}$$ has the same cross-sectional area as the plates of a parallel plate capacitor and thickness $$\frac{3}{4} \mathrm{~d}$$, where $$\mathrm{d}$$ is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be :

(Given $$\mathrm{C}_{0}$$ = capacitance of capacitor with air as medium between plates.)

A uniform electric field $$\mathrm{E}=(8 \mathrm{~m} / \mathrm{e}) \,\mathrm{V} / \mathrm{m}$$ is created between two parallel plates of length $$1 \mathrm{~m}$$ as shown in figure, (where $$\mathrm{m}=$$ mass of electron and e = charge of electron). An electron enters the field symmetrically between the plates with a speed of $$2 \mathrm{~m} / \mathrm{s}$$. The angle of the deviation $$(\theta)$$ of the path of the electron as it comes out of the field will be _________.

Given below are two statements :

Statement I : A uniform wire of resistance $$80 \,\Omega$$ is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be $$5 \,\Omega$$.

Statement II: Two resistances 2R and 3R are connected in parallel in a electric circuit. The value of thermal energy developed in 3R and 2R will be in the ratio $$3: 2$$.

In the light of the above statements, choose the most appropriate answer from the option given below