$A$ power of $100 \ W$ is produced by the fission of $1 \ kg$ of $^{235}U$. For approximately how long will the energy production continue?

$A$ star has $100 \%$ helium composition. It starts to convert three ${ }^4 He$ into one ${ }^{12} C$ via the triple alpha process as ${ }^4 He + { }^4 He + { }^4 He \rightarrow { }^{12} C + Q$. The mass of the star is $2.0 \times 10^{32} \ kg$ and it generates energy at the rate of $5.808 \times 10^{30} \ W$. The rate of converting these ${ }^4 He$ nuclei to ${ }^{12} C$ is $n \times 10^{42} \ s^{-1}$,where $n$ is. . . . . . . [Take,mass of ${ }^4 He = 4.0026 \ u$,mass of ${ }^{12} C = 12 \ u$,$1 \ u = 1.66 \times 10^{-27} \ kg$,$c = 3 \times 10^8 \ m/s$]

This question contains Statement-$1$ and Statement-$2$. Of the four choices given after the statements,choose the one that best describes the two statements.
Statement-$1$: Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion.
Statement-$2$: For heavy nuclei,binding energy per nucleon increases with increasing $Z$,while for light nuclei,it decreases with increasing $Z$.

Calculate the height of the potential barrier for a head-on collision of two deuterons. (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius $2.0 \; fm$.)

The thermonuclear reaction of hydrogen inside the stars takes place by a cycle of operations. The particular element which acts as a catalyst is

The binding energies of nuclei $X$ and $Y$ are $E_1$ and $E_2$ respectively. Two atoms of $X$ fuse to give one atom of $Y$ and an energy $Q$ is released. Then: