KVPY Sample Paper KVPY Stream-SX Model Paper-29

If \[\int\limits_{0}^{x}{f(t)}dt={{x}^{2}}+\int\limits_{x}^{1}{{{t}^{2}}f(t)}dt,\] then \[f'\left( \frac{1}{2} \right)\] is:

A) \[\frac{24}{25}\]

B) \[\frac{18}{25}\]

C) \[\frac{4}{5}\]

D) \[\frac{6}{25}\]

Correct Answer: A

Solution :

Differentiability we get \[f(x)=2x-{{x}^{2}}f(x)\]

\[f(x)=\frac{2x}{1+{{x}^{2}}}\Rightarrow f''(x)=2\frac{(1-{{x}^{2}})}{{{(1+{{x}^{2}})}^{2}}}\]

\[f'\left( \frac{1}{2} \right)=\frac{24}{25}\]