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Prove that |z1+z2|^2 = |z1|^2+|z2|^2 if ...

Prove that `|z_1+z_2|^2` = `|z_1|^2`+`|z_2|^2` if `z_1/z_2` is purely imaginary.

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. If f(z) = a0z^n+a1z^(n-1)+a2z^(n-2)+...+an where z is a complex numbe...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  16. Locate the region in the Argand plane for the complex number z satisfy...

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  17. Indicate the region represented by pi/6leargzlepi/4 .

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  18. Indicate the region in the Argand plane respresented by |z+1|^2+|z-1|^...

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  19. Prove that the product of any number of unimodular complex numbers is ...

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  20. Separate the real and imaginary parts of frac((cosalpha+isinalpha)(cos...

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