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

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

The minimum number of elements that must be added to the relation $$ \mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}$$ on the set $$\{a, b, c\}$$ so that it becomes symmetric and transitive is :

Let the system of linear equations

$$x+y+kz=2$$

$$2x+3y-z=1$$

$$3x+4y+2z=k$$

have infinitely many solutions. Then the system

$$(k+1)x+(2k-1)y=7$$

$$(2k+1)x+(k+5)y=10$$

has :