The electric current through a wire varies with time as $$I=I_0+\beta t$$, where $$I_0=20 \mathrm{~A}$$ and $$\beta=3 \mathrm{~A} / \mathrm{s}$$. The amount of electric charge crossed through a section of the wire in $$20 \mathrm{~s}$$ is :

The potential energy function (in $$J$$ ) of a particle in a region of space is given as $$U=\left(2 x^2+3 y^3+2 z\right)$$. Here $$x, y$$ and $$z$$ are in meter. The magnitude of $$x$$-component of force (in $$N$$ ) acting on the particle at point $$P(1,2,3) \mathrm{m}$$ is :

The explosive in a Hydrogen bomb is a mixture of $${ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3$$ and $${ }_3 \mathrm{Li}^6$$ in some condensed form. The chain reaction is given by

$$\begin{aligned} & { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \\ & { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1 \end{aligned}$$

During the explosion the energy released is approximately

[Given ; $$\mathrm{M}(\mathrm{Li})=6.01690 \mathrm{~amu}, \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \mathrm{~amu}, \mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388$$ $$\mathrm{amu}$$, and $$1 \mathrm{~amu}=931.5 \mathrm{~MeV}]$$

In a double slit experiment shown in figure, when light of wavelength $$400 \mathrm{~nm}$$ is used, dark fringe is observed at $$P$$. If $$\mathrm{D}=0.2 \mathrm{~m}$$, the minimum distance between the slits $$S_1$$ and $$S_2$$ is _________ $$\mathrm{mm}$$.