Differentiation Question 338
Question: If $ r={{[2\varphi +{{\cos }^{2}}(2\varphi +\pi /4)]}^{1/2}} $ then what is the value of the derivative of $ dr/d\varphi $ at $ \varphi =\pi /4 $
[Orissa JEE 2005]
Options:
A) $ 2{{( \frac{1}{\pi +1} )}^{1/2}} $
B) $ 2{{( \frac{2}{\pi +1} )}^{-1/2}} $
C) $ 2{{( \frac{1}{\pi +1} )}^{-1/2}} $
D) $ 2{{( \frac{2}{\pi +1} )}^{1/2}} $
Show Answer
Answer:
Correct Answer: D
Solution:
$ r={{[ 2\varphi +{{\cos }^{2}}( 2\varphi +\frac{\pi }{4} ) ]}^{1/2}} $
$ \Rightarrow \frac{dr}{d\varphi }=\frac{1}{2}{{[ 2\varphi +{{\cos }^{2}}( 2\varphi +\frac{\pi }{4} ) ]}^{-1/2}} $
$ [ 2-2\times 2\sin ( 2\varphi +\frac{\pi }{4} )\times \cos ( 2\varphi +\frac{\pi }{4} ) ] $
$ {{( \frac{dr}{d\varphi } )} _{r=\frac{\pi }{4}}}=\frac{1}{2}{{[ \frac{\pi }{2}+{{\cos }^{2}}\frac{3\pi }{4} ]}^{-1/2}}\times 2[ ( 1-\sin ( \pi +\frac{\pi }{2} ) ) ] $
$ {{( \frac{dr}{d\varphi } )} _{r=\frac{\pi }{4}}}=\frac{1}{2}{{( \frac{\pi }{2}+\frac{1}{2} )}^{-1/2}}\times 2(1+1)=2\times {{( \frac{2}{\pi +1} )}^{1/2}} $ .
BETA
