AP EAMCET202017 Sep 2020Morning ShiftMathematicsComplex NumberActual
If (1, a, a^2, , a^ n-1 ) are the (n )th roots of unity. Then ( _ i=1 ^ n-1 1 2-a^i ) is equal to
Options
- A((n-2) 2^n )
- B( (n-2) 2^ n-1 +1 2^n-1 )
- C( (n-2) 2^ n-1 2^n-1 )
- D( 1 (n-2) 2^n )
Correct answer
B. ( (n-2) 2^ n-1 +1 2^n-1 )
Step-by-step solution
( _ i=1 ^ n-1 1 2- ^i ) (1, , ^2, , ^ n-1 ) are the (n )th root of unity ( array ll & x^n-1=0 & x^n-1=(x-1)(x- ) (x- ^2 ) (x- ^ n-1 ) & (x^n-1 )= (x-1)+ (x- ) & + . . (x- ^ n-1 ) array ) Differentiating w.r.t ' (x ) ', we get ( aligned & n x^ n-1 x^n-1 = 1 (x-1) + 1 (x- ) + + 1 (x- ^ n-1 ) & At (x=2) & n 2^ n-1 2^n-1 =1+ 1 2- + 1 2- ^2 + . .+ 1 2- ^ n-1 & _ i=1 ^ n-1 1 2- ^i = ( n 2^ n-1 2^n-1 -1 )= (n-2) 2^ n-1 +1 2^n-1 aligned )