question_answer

A system of binary stars of masses \[\log \left[ {{\tan }^{-1}}\left( x+\frac{1}{x} \right) \right]+C\]and mg are moving in circular orbits of radii \[\int_{{}}^{{}}{\frac{\cos x}{{{\sin }^{2}}x.(\sin x+\cos x)}dx}\]and \[\log \left| \frac{1+\tan x}{\tan x} \right|-\cot x+C\]respectively. If \[\log \left| \frac{\tan x}{1+\tan x} \right|+C\] and \[\log \left| \frac{\tan x}{1+\tan x} \right|-\tan x+C\] are the time periods of masses \[\log \left| \frac{\tan x}{1+\tan x} \right|+\cot x+C\] and \[\int_{{}}^{{}}{\frac{dx}{\sin x-\cos x+\sqrt{2}}}\] respectively, then

A) \[-\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+C\]                          

B) \[\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+C\]

C) \[\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+C\]            

D) \[-\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+C\]

Correct Answer: D

Solution :

\[2.75\text{ }g/c{{m}^{3}}.\] \[3.6\times {{10}^{18}}\]\[3.6\times {{10}^{16}}\] \[2.34\times {{10}^{23}}\]\[6.02\times {{10}^{24}}\]