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KVPY2011MathematicsFunctions

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

Options

  1. Adepends on A, B, C, D, E
  2. Bdepends on A, C, E but independent of B and D
  3. Cdepends on B, D but independent of A, C, E
  4. Dis independent of A, B, C, D, E

Correct answer

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

Step-by-step solution

Clearly f ( + x )+ f ( - x ) ( every term contain cosine )f ( 5 )=f ( 9 5 ), f ( 2 5 )=f ( 8 5 ), f ( 3 5 )=f ( 7 5 )f ( 4 5 )=f ( 6 5 ) T = f (0)-2 [f ( 5 )+f ( 3 5 ) ]+2 [f ( 2 5 )+f ( 4 5 ) ]-f( ) f (0)- f ( )=2(1+ B + D )f ( 5 )+f ( 3 5 )=f ( 5 )-f ( 4 5 )=2 (1+B 3 5 +D 5 )f ( 2 5 )+f ( 4 5 )=f ( 2 5 )-f ( 3 5 )=2 (1+B 6 5 +D 2 5 ) T contains only B , D terms

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