KVPY Sample Paper KVPY Stream-SX Model Paper-30

If is a twice derivable function such that\[f'(2010)=f(2010)=\frac{1}{2010}\] and \[\int\limits_{0}^{2010}{f(x)dx=\frac{3}{2}}\], then \[\int\limits_{0}^{2010}{{{x}^{2}}}f''(x)dx=\]

A) 2009

B) 2010

C) 2011

D) none of these

Correct Answer: C

Solution :

\[\int\limits_{0}^{a}{{{x}^{2}}f''(x)dx={{a}^{2}}f'(a)-}\]\[\int\limits_{0}^{a}{2x\,f'(x)dx={{a}^{2}}f'(a)-2[a\,f(a)-\int\limits_{0}^{a}{f(x)dx]}}\]

\[{{a}^{2}}f'(a)-2af(a)+2\int\limits_{0}^{a}{f(x)}dx\]

Put \[a=2010\]

\[\int\limits_{0}^{2010}{{{x}^{2}}f''(x)}dx=2010-2+3=2011\]