Let f(x) = \cos5x + A\cos4x + B\cos3x + C\cos2x + D\cos x + E, and T=f(0)-f(\pi/5)+f({2\pi}/5)-f({3\pi}/5)+...+f({8\pi}/5)-f({9\pi}/5)Then T

Correct Answer is : depends on B, D but independent of A, C, E

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

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

KVPY SB SX Exam 2011 Question Paper

Question : 9 of 120

Marks: +1, -0

Let

f(x) = \cos5x + A\cos4x + B\cos3x

+ C\cos2x + D\cos x + E,

and

T=f(0)-f(π/5)+f({2π}/5)-f({3π}/5)+...+f({8π}/5)-f({9π}/5)

Then T

Solution:

Clearly f(π + x) + f(π – x) (every term contain cosine)

f(π/5)=f({9π}/5),f({2π}/5)=f({8π}/5)

f({3π}/5)=f({7π}/5)

f({4π}/5)=f({6π}/5)

T=f(0)-2[f(π/5) +f({3π}/5)]

+2[f({2π}/5)+f({4π}/5)]

-f(π)

f(0) - f(π) = 2(1 + B + D)

f({ π}/5)+f({3π}/5)

=f({ π}/5)-f({4π}/5)

=2(1+B\cos {3π}/5+D\cosπ/5)

f({ 2π}/5)+f({4π}/5)

=f({ 2π}/5)-f({3π}/5)

=2(1+B\cos {6π}/5+D\cos{2π}/5)

T ⇒ contains only B, D terms

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