• # question_answer Let $f(\theta )=\frac{1}{{{\tan }^{9}}\theta }\,{{\{1+\tan \theta )}^{10}}+\,(2+\tan \theta )$$+...+{{(20+\tan \theta )}^{10}}\}-20\,\tan \theta$ A)  1900                           B)  2000             C)  2100                           D)  2200

$v'\frac{v}{\sqrt{2}}$ $E=\frac{1}{2}mv{{'}^{2}}+\frac{1}{2}mv{{'}^{2}}+\frac{1}{2}(2m){{v}^{2}}$             By binomial theorem, $=mv{{'}^{2}}+m{{v}^{2}}$   $\Rightarrow$ $m{{\left( \frac{v}{\sqrt{2}} \right)}^{2}}+m{{v}^{2}}=\frac{3}{2}m{{v}^{2}}$ ${{T}_{1}}-{{T}_{2}}=12a$         ${{T}_{2}}=3a$ ${{T}_{2}}$$Eq$ ${{T}_{1}}=15a$ Similarly, $\underset{^{\theta \to \frac{{{\pi }^{-}}}{2}}}{\mathop{\lim }}\,\,\frac{{{(1+\tan \,\theta )}^{10}}}{{{\tan }^{9}}\theta }=20+\,\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,\tan \theta$, $\underset{^{\theta \to \frac{{{\pi }^{-}}}{2}}}{\mathop{\lim }}\,\,\frac{{{(3+\tan \,\theta )}^{10}}}{{{\tan }^{9}}\theta }=30+\,\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,\tan \theta$ ?????????????????????????????????????????????????????? $\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,\frac{{{(20+\tan \theta )}^{10}}}{{{\tan }^{9}}\theta }=200+\,\,\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,\,\,\tan \theta$ On adding all these terms; we get $\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,f(\theta )=(10+20+...+200)+20\,\,\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,\,\,\tan \theta$ $-\underset{\theta \to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,20\,d\,\tan \,\theta$ $\Rightarrow$            $\underset{\theta -\frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,f(\theta )=10\,(1+2+3+...+20)$ $=\frac{10\times 20\times 21}{2}=2100$