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COMEDK2019MathematicsComplex Number

The value of _ K=1 ⁶ [ 2 K 7 -i 2 K 7 ] is

Options

  1. Ai
  2. B0
  3. C-i
  4. D-1

Correct answer

A. i

Step-by-step solution

We have, aligned & _ K=1 ⁶ ( 2 K 7 -i 2 K 7 ) &=-i _ K=1 ⁶ ( 2 K 7 +i 2 K 7 ) &=-i _ K=1 ⁶ e^ 2 K 7 &=-i [e^ 2 7 +e^ 4 7 +e^ 6 7 +e^ 8 7 +e^ 10 7 +e^ 12 7 ] &=-i [1+e^ 2 7 +e^ 4 7 +e^ 6 7 +e^ 8 7 +e^ 10 7 +e^ 12 7 -1 ] &=-i[0-1][ Sum of all nth roots of unit is zero] &=i &= [ array l [ array ] aligned

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