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  4. The arithmetic mean of roots of the equation 4 cos 3x-4 cos 2x- cos (315π +x)=1 is

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Q. The arithmetic mean of roots of the equation $ 4{{\cos }^{3}}x-4{{\cos }^{2}}x-\cos (315\pi +x)=1 $ is

JamiaJamia 2013

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