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If x = a + b, y= aw + bw^2 and z = aw^2 ...

If `x = a + b, y= aw + bw^2 `and `z = aw^2 + bw`, where `w` is an imaginary cube. root of unity, prove that `x^2 + y^2 + z^2 = 6ab.`

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. (i) If alpha,beta be the imaginary cube root of unity, then show that ...

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  2. If x=a+b,y=aomega+bomega^2 nd z=omega^2+bomega, prove that x^3+y^3+z^3...

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  3. If x = a + b, y= aw + bw^2 and z = aw^2 + bw, where w is an imaginary ...

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  4. Prove that ((i+sqrt3)/2)^100+((i-sqrt3)/2)^100+1=0.

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  5. What is the value of ((i+sqrt3)/(-i+sqrt3))^(200)+((i-sqrt3)/(i+sqrt3)...

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  6. Prove that sqrt(-1-sqrt(-1-sqrt1-… prop=w or w^2 .

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  7. Resolve into the linear factor: a^3+b^3

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  8. Resolve into the linear factor: a^3+b^3+c^3-3abc

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  9. value of the series a(0)-a(2) + a(4)-a(6) +….. is

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  10. If (1+x)^n=a0+a1x+a2x^2+...+anx^n then find : a1-a3+a5-a7+… .

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  11. If (1+x)^n=a0+a1x+a2x^2+...+anx^n then find : a0+a3+a6+a9+… .

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  12. If (1+x+x^2)^n=a0+a1x+a2x^2++a(2n)x(2n), find the value of a0+a6++ ,n ...

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  13. If 1,a1,a2,a3,…,a(n-1) are the nth roots of unity then prove that : (1...

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  14. If 1,a1,a2,a3,…,a(n-1) are the nth roots of unity then prove that : 1+...

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  15. If alpha is an n^(th) roots of unity, then 1+2alpha+3alpha^(2)+……..+n...

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  16. If alpha (!=1) is a nth root of unity then S = 1 + 3alpha+ 5alpha^2 + ...

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  17. sum(p=1)^pi(3p+2)[sum(q=1)^10sin(2qpi)/11-icos(2qpi)/11]^p

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  18. Show that the polynomial x^(4p)+x^(4q+1)+x^(4r+2)+x^(4s+3) is divisibl...

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  19. If alpha ia ROOT OF x^2+x+1= 0 then show that alpha is also a root of...

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  20. If p,q,r are three consecutive integers ge3, prove that x^p+x^q+x^r is...

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