The number of electrons involved in the reduction of permanganate to manganese dioxide in acidic medium is _____________.

A solution containing $$2 \mathrm{~g}$$ of a non-volatile solute in $$20 \mathrm{~g}$$ of water boils at $$373.52 \mathrm{~K}$$. The molecular mass of the solute is ___________ $$\mathrm{g} ~\mathrm{mol}^{-1}$$. (Nearest integer)

Given, water boils at $$373 \mathrm{~K}, \mathrm{~K}_{\mathrm{b}}$$ for water $$=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$$

The energy of one mole of photons of radiation of frequency $$2 \times 10^{12} \mathrm{~Hz}$$ in $$\mathrm{J} ~\mathrm{mol}^{-1}$$ is ___________. (Nearest integer)

[Given : $$\mathrm{h}=6.626 \times 10^{-34} ~\mathrm{Js}$$

$$\mathrm{N}_{\mathrm{A}}=6.022 \times 10^{23} \mathrm{~mol}^{-1}$$]

Let the solution curve $$y=y(x)$$ of the differential equation

$$ \frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{6}\right)}}\right\} \text { pass through the origin. Then } y(1) \text { is equal to : } $$