Let f(x)=cos 5 x+A cos 4 x+B cos 3 x+C cos 2 x+D cos x+E , and T=f(0)-f ((π)/5 )+f ((2 π)/5 )-f ((3 π)/5 )+…+f ((8 π)/5 )-f ((9 π)/5 ) . Then T

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Correct answer: (C) depends on B, D but independent of A, C, E. Clearly f(π+x)+f(π-x) ( every term contain cosine ) f ((π)/5 )=f ((9 π)/5 ), f ((2 π)/5 )=f ((8 π)/5 ), f ((3 π)/5 )=f ((7 π)/5 ) f ((4 π)/5 )=f ((6 π)/5 ) T=f(0)-2 [f ((π)/5 )+f ((3 π)/5 ) ]+2 [f ((2 π)/5 )+f ((4 π)/5 ) ]-f(π) f(0)-f(π)=2(1+B+D) f ((π)/5 )+f ((3 π)/5 )=f ((π)/5 )-f ((4 π)/5 )=2 (1+B cos (3 π)/5+D cos (π)/5 ) f ((2 π)/5 )+f ((4 π)/5 )=f ((2 π)/5 )-f ((3 π)/5 )=2 (1+B cos (6 π)/5+D cos (2 π)/5 ) T ⇒ contains only B, D terms

Let $f(x)=\cos 5 x+A \cos 4 x+B \cos 3 x+C \cos 2 x+D \cos x+E$, and $T=f(0)-f\left(\frac{\pi}{5}\right)+f\left(\frac{2 \pi}{5}\right)-f\left(\frac{3 \pi}{5}\right)+\ldots+f\left(\frac{8 \pi}{5}\right)-f\left(\frac{9 \pi}{5}\right) .$ Then $T$

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