A) \[50\,\pi \]
B) \[51\,\pi \]
C) \[100\,\pi \]
D) \[315\,\pi \]
Correct Answer: B
Solution :
\[\because \] \[\cos (315\pi +x)={{(-1)}^{315}}.\cos x=-\cos x\] \[\therefore \] \[4{{\cos }^{3}}x-4{{\cos }^{2}}x-\cos (315\pi x+x)=1\] \[\Rightarrow \] \[4{{\cos }^{3}}x-4{{\cos }^{2}}x+\cos x-1=0\] \[\Rightarrow \] \[(4{{\cos }^{2}}x+1)(\cos x-1)=0\] \[\therefore \] \[cox=1,4{{\cos }^{2}}x+1\ne 0\] \[\Rightarrow \] \[\cos x=\cos 0{}^\circ \] \[\therefore \] \[x=2n\pi ,\] \[n\in I\] \[\therefore \]\[x=2\pi ,4\pi ,6\pi ,8\pi .....,100\pi \] \[(\because 0<x<315)\] \[\therefore \] Required arithmetic mean \[=\frac{2\pi +4\pi +6\pi +8\pi +...+100\pi }{50}\] \[=\frac{2\pi (1+2+3+4+...+50)}{50}\] \[=2\pi .\frac{\frac{50}{2}.51}{50}=51\pi \]
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