KVPY Sample Paper KVPY Stream-SX Model Paper-21

\[\int\limits_{\pi /4}^{\pi /2}{\sqrt{2+\sqrt{2+2\cos 4x}}}\,\,dx\] is equal to:

A) \[\sqrt{2}\]

B) \[\sqrt{2}\,(\sqrt{2}-1)\]

C) 2

D) none of these

Correct Answer: A

Solution :

\[l=\int\limits_{\pi /4}^{\pi /2}{\sqrt{2+\sqrt{2+2\cos 4x}}}\,\,dx=\int\limits_{\pi /4}^{\pi /2}{\sqrt{2-2\cos 2x}\,\,dx}\]

\[=2\int\limits_{\pi /4}^{\pi /2}{\sin x\,\,dx}\] \[=2\,(-\cos x)_{\pi /4}^{\pi /2}=\sqrt{2}\]