Question

Let $f ( x )$ be a continuous function such that the area bounded by the curve $y=f(x), x$ -axis and the lines $x =0$ and $x = a$ is $\frac{ a ^{2}}{2}+\frac{ a }{2} \sin a +\frac{\pi}{2} \cos a$, then $f \left(\frac{\pi}{2}\right)=$

• A. 1
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. None of these

Answer 1

Answer: (B)

We have, $\int\limits_{0}^{ a } f ( x ) dx =\frac{ a ^{2}}{2}+\frac{ a }{2} \sin a +\frac{\pi}{2} \cos a$
Differentiating w.r.t. a, we get
$f ( a )= a +\frac{1}{2}(\sin a + a \cos a )-\frac{\pi}{2} \sin a$
Put $a =\frac{\pi}{2} ; f \left(\frac{\pi}{2}\right)=\frac{\pi}{2}+\frac{1}{2}-\frac{\pi}{2}=\frac{1}{2}$