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Let f(x)-cos5x+Acos4x+Bcos3x+Ccos2x+Dco...

Let `f(x)-cos5x+Acos4x+Bcos3x+Ccos2x+Dcosx E and T-f(0)-f(pi/5)+f((2pi)/5)-f((3pi)/5)+.........+f((8pi)/5)-f((9pi)/5)` then T

A

depends on A, B, C, D, E

B

depends on A, C, E but independent of B and D

C

depends on B, D but independent of A, C, E

D

is independent of A, B, C, D, E

Text Solution

Verified by Experts

The correct Answer is:
C
Clearly `f(pi+x)+f(pi-x)` (every term contain cosine)
`f(pi/5)=f((9pi)/5), f((2pi)/5)=f((8pi)/5), f((3pi)/5)=f((7 pi)/5)`
`f((4pi)/5)=f((6pi)/5)`
`T=f(0)-2[f(pi/5)+f((3pi)/5)]+2[f((2pi)/5)+f((4pi)/5)]-f(pi)`
`f(0)-f(pi)=2(1+B+D)`
`f(pi/5)+f((3pi)/5)=f(pi/5)-f((4pi)/5)=2 (1+B" cos"(3pi)/5+D " cos"pi/5)`
`f((2pi)/5)+f((4pi)/5)=f((2pi)/5)-f((3pi)/5)=2(1+B" cos"(6pi)/5+D" cos" (2pi)/5)`
`T rArr` contains only B, D terms
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