A) \[2\,{{\left( \frac{1}{\pi +1} \right)}^{1/2}}\]
B) \[2\,{{\left( \frac{2}{\pi +1} \right)}^{-1/2}}\]
C) \[2\,{{\left( \frac{1}{\pi +1} \right)}^{-1/2}}\]
D) \[2\,{{\left( \frac{2}{\pi +1} \right)}^{1/2}}\]
Correct Answer: D
Solution :
\[r={{\left[ \,2\varphi +{{\cos }^{2}}\left( 2\varphi +\frac{\pi }{4} \right) \right]}^{1/2}}\] \[\Rightarrow \,\,\frac{dr}{d\varphi }=\frac{1}{2}{{\left[ 2\varphi +{{\cos }^{2}}\left( 2\varphi +\frac{\pi }{4} \right) \right]}^{-1/2}}\] \[\left[ 2-2\times 2\sin \,\left( 2\varphi +\frac{\pi }{4} \right)\times \cos \left( 2\varphi +\frac{\pi }{4} \right) \right]\] \[{{\left( \frac{dr}{d\varphi } \right)}_{r=\frac{\pi }{4}}}=\frac{1}{2}{{\left[ \frac{\pi }{2}+{{\cos }^{2}}\frac{3\pi }{4} \right]}^{-1/2}}\times 2\left[ \left( 1-\sin \left( \pi +\frac{\pi }{2} \right) \right) \right]\] \[{{\left( \frac{dr}{d\varphi } \right)}_{r=\frac{\pi }{4}}}=\frac{1}{2}{{\left( \frac{\pi }{2}+\frac{1}{2} \right)}^{-1/2}}\times 2\,(1+1)=2\times {{\left( \frac{2}{\pi +1} \right)}^{1/2}}\].
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