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1 + omega + omega^(2) + ….. + omega^(300...

`1 + omega + omega^(2) + ….. + omega^(300)`

A

0

B

`-omega^(2)`

C

`-omega`

D

`1`

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Verified by Experts

The correct Answer is:
D
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(1 + omega) (1 + omega^(2)) (1 + omega^(3)) (1 + omega^(4)) (1 + omega^(5)) (1 + omega^(6)) …. (1 + omega^(3n)) =

If omega is a cube root of unity , then |{:( 1 , omega , 2 omega^(2)) , (2 , 2omega^(2) , 4 omega^(3)) , (3 , 3 omega^(3) , 6 omega^(4)):}| =

omega + 2omega^(2) + 3 omega^(3) + ….. 90 omega^(90) =

If 1 , omega , omega^(2) are the cube roots of unity , then (x + y + z) (z + y omega + zomega^(2)) ( x + y omega^(2) + z omega) =

( 2 - omega) ( 2 - omega^(2)) ( 2 - omega^(10)) ( 2 - omega^(11)) = 49 .

(1 - omega + omega^(2)) ( 1 - omega^(2) + omega^(4)) ( 1- omega^(4) + omega^(8)) to 2n factors =

(1 - omega) (1 - omega^(2)) (1 - omega^(4)) (1 - omega^(5)) (1 - omega^(7)) (1 - omega^(8)) =

If 1 , omega , omega^(2) are the cube roots of unity prove that (1 - omega + omega^(2))^(6) + ( 1 - omega ^(2) + omega)^(6) = 128 = (1 - omega + omega^(2))^(7) + ( 1 + omega - omega^(2))^(7)

If 1 , omega + omega^(2) are the cube roots of unity prove that (i) (1 - omega + omega^(2))^(6) + (1 - omega^(2) + omega)^(6) = 128 = ( 1 - omega + omega^(2))^(7) + ( 1 + omega - omega^(2))^(7) (ii) ( a+ b) ( aomega+b omega^(2))( a omega^(2) + b omega) = a^(3) + b^(3) (iii) x^(2) + 4x + 7 = 0 " where " x = omega - omega^(2) - 2 .

AAKASH SERIES-COMPLEX NUMBERS-EXERCISE - I
  1. If amp (z-i) =pi//3 then the locus of z is

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  2. The locus of Z satisfying |z| + |z - 1| = 3 is

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  3. If the amplitude of z - 2 - 3i " is " pi//4 , then the locus of z = x ...

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  4. The complex number (1 + 2i)/(1 - i) lies in the

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  5. If three complex number are in A.P then they lie on

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  6. In the Argand plane, the points represented by the complex numbers 2-6...

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  7. The condition for the points z(1) , z(2) , z(3) , z(4) taken in order ...

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  8. The equation of the line joining the points represented by 2- 3i and -...

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  9. The number of complex number z such |z-1|=|z+1|=|z-i| equals

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  10. Statement - I : If z(1) and z(2) are two nonzero complex numbers such ...

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  11. Statement - I : If a and b are positive real numbers then sqrt(-a) xx ...

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  12. Statement - I : If z = barz then z is purely imaginary Statement- II...

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  13. In Argand plane , the quadrant in which (1 + 2i)/(1 -i) lies is

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  14. The descending order of the moduli of z(1) = (3 - 4i) (4 + 3i) , z(...

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  15. Match the following:

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  16. (3 - omega) ( 3- omega^(2)) ( 3- omega^(4)) (3 - omega^(8)) =

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  17. i+ i^(2) + i^(3) + i^(4) + …. + i^(100)

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  18. 1 + omega + omega^(2) + ….. + omega^(300)

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  19. i+ 2i^(2) + 3i^(3) + 4i^(4) + …. + 100i^(100) =

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  20. omega + 2omega^(2) + 3 omega^(3) + ….. 90 omega^(90) =

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