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AP EAMCET202021 Sep 2020Evening ShiftMathematicsComplex NumberActual

For how many natural numbers ' (n ) ' such that (1 n 2021 ) is ( ( 1+i 1-i )^n=1 ) ?

Options

  1. A504
  2. B505
  3. C506
  4. D503

Correct answer

B. 505

Step-by-step solution

It is given that, ( aligned ( 1+i 1-i )^n=1 & ( 1+i^2+2 i 1-i^2 )^n=1 ( 2 i 2 )^n=1 & i^n=1 aligned ) If (n ) is a integral multiple of 4, then (i^n=1 ) ( 1 n 2021 ), so the possible values of (n ) are (4,8,12,16, ., 2020 ) and there are 505 values of (n ). Hence, option (b) is correct.

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