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Show that (x + omega y + omega^(2)z) (x ...

Show that `(x + omega y + omega^(2)z) (x + omega^(2)y + omega z) = x^(2) + y^(2) + z^(2)- yz - zx - xy`

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If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |[x/(1+omega), y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[(z)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

Let S = { z: x = x+ iy, y ge 0,|z-z_(0)| le 1} , where |z_(0)|= |z_(0) - omega|= |z_(0) - omega^(2)|, omega and omega^(2) are non-real cube roots of unity. Then

If the real numbers x, y, z are such that x^2 + 4y^2 + 16z^2 = 48 and xy + 4yz + 2zx = 24 . what is the value of x^(2) +y^(2) z^(2) =?

ICSE-COMPLEX NUMBERS-Exercise (F )
  1. If omega is a cube root of unity, then omega^(3)= ……

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  2. If 1, omega, omega^(2) are three cube roots of unity, prove that (1...

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  3. If 1, omega, omega^(2) are three cube roots of unity, prove that (1...

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  4. If 1, omega, omega^(2) are three cube roots of unity, prove that (1...

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  5. If 1, omega, omega^(2) are three cube roots of unity, prove that (1...

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  6. If 1, omega, omega^(2) are three cube roots of unity, prove that (1...

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  7. If 1, omega, omega^(2) are three cube roots of unity, prove that (...

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  8. If 1, omega, omega^(2) are three cube roots of unity, prove that (3...

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  9. If 1, omega, omega^(2) are three cube roots of unity, prove that om...

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  10. Prove that ((-1 + isqrt3)/(2))^(n) + ((-1-isqrt3)/(2))^(n) is equal to...

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  11. If 1, omega, omega^(2) are the cube roots of unity, prove that omega^(...

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  12. Prove the following (1- omega + omega^(2)) (1 + omega- omega^(2)) (...

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  13. Prove the following (1+ omega) (1+ omega^(2)) (1 + omega^(4)) (1 + ...

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  14. Prove the following (1- omega + omega^(2)) (1- omega^(2) + omega^(4...

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  15. Prove the following (a + b omega + c omega^(2))/(b + c omega + a ome...

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  16. Prove the following (a + b omega + c omega^(2))/(c + a omega + b ome...

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  17. If omega is a cube root of unity and n is a positive integer which is ...

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  18. Show that (x + omega y + omega^(2)z) (x + omega^(2)y + omega z) = x^(2...

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  19. Show that x^(3) +y^(3) = (x + y) (omega x + omega^(2)y) (omega^(2)x +...

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  20. If 1, omega, omega^(2) are cube roots of unity, prove that 1, omega, o...

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