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If (a1+ib1)(a2+ib2)...(an+ibn) = x+iy , ...

If `(a_1+ib_1)(a_2+ib_2)...(a_n+ib_n)` = x+iy , prove that : `tan^-1frac(b_1)(a_1)+tan^-1frac(b_2)(a_2)+...+tan^-1frac(b_n)(a_n)=tan^-1fracyx` .

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. Show that a real value of x will satisfy hte equation (1-i x)//(1+i x)...

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  2. If (a1+ib1)(a2+ib2)...(an+ibn) = x+iy , prove that : (a1^2+b1^2)(a2^2+...

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  3. If (a1+ib1)(a2+ib2)...(an+ibn) = x+iy , prove that : tan^-1frac(b1)(a1...

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  4. If the complex numbers z1, z2,.......zn lie on te unit circle |z| = 1 ...

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  5. Show that if iz^3+z^2-z+i=0, then |z|=1

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  6. If f(z) = a0z^n+a1z^(n-1)+a2z^(n-2)+...+an where z is a complex numbe...

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  7. If |z1+z2| = |z1-z2|, prove that ampz1 - ampz2 = pi/2 .

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  8. Prove that |z1+z2|^2 = |z1|^2+|z2|^2 if z1/z2 is purely imaginary.

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  9. Prove that |z1+z2|^2+|z1-z2|^2 =2|z1|^2+2|z2|^2.

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  10. Prove that |alpha+sqrt(alpha^2-beta^2)|+|alpha-sqrt(alpha^2-beta^2)|= ...

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  11. Prove that |1-barz1z2|^2-|z1-z2|^2=(1-|z1|^2)(1-|z2|^2).

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  12. If |z-1| <3, prove that |iz+3-5i| < 8.

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  13. Prove that for nonzero complex numbers |z1+z2| |frac(z1)(|z1|)+frac(z2...

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  14. Prove that following inequalities: (i) |(z)/(|z|) -1| le |arg z| (...

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  15. Find the value of w^4+w^6+w^8, if w is a complex cube root of unity.

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  16. Find the range of real number alpha for which the equation z+alpha|z-1...

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  17. If a, b are real, prove that the equation z^2+az+b=0 will not have any...

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  18. Find the maximum and minimum values of |z| satisfying |z+(1)/(z)|=2

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  19. If |z|ge3, then determine the least value of |z+(1)/(z)|.

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  20. If z is a complex number such that |z+1/z|=1 show that Re(z)=0 when |z...

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