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

In an alpha particle scattering experiment distance of closest approach for the $$\alpha$$ particle is $$4.5 \times 10^{-14} \mathrm{~m}$$. If target nucleus has atomic number 80 , then maximum velocity of $$\alpha$$-particle is __________ $$\times 10^5 \mathrm{~m} / \mathrm{s}$$ approximately.

($$\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{SI}$$ unit, mass of $$\alpha$$ particle $$=6.72 \times 10^{-27} \mathrm{~kg}$$)

Radius of a certain orbit of hydrogen atom is 8.48 $$\mathop A\limits^o$$. If energy of electron in this orbit is $$E / x$$. then $$x=$$ ________ (Given $$\mathrm{a}_0=0.529$$ $$\mathop A\limits^o$$, $$E=$$ energy of electron in ground state).

The shortest wavelength of the spectral lines in the Lyman series of hydrogen spectrum is $$915\mathop A\limits^o$$. The longest wavelength of spectral lines in the Balmer series will be _______ $$\mathop A\limits^o$$.