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Prove that for nonzero complex numbers `|z_1+z_2|` `|frac(z_1)(|z_1|)+frac(z_2)(|z_2|)|le2(|z_1|+|z_2|)`.

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. Prove that |1-barz1z2|^2-|z1-z2|^2=(1-|z1|^2)(1-|z2|^2).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  16. Prove that (frac(1+costheta+isintheta)(1+costheta-isintheta))^n=cos nt...

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  17. If alpha=pi/12 ,prove that frac((cosalpha+isinalpha)(cos2alpha+isin2al...

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  18. Prove that (frac(cosx+isinx)(sinx+icosx))^5=cos(10x-frac(5pi)(2))+isin...

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  19. Find the value of : root(4)(1).

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  20. Find the value of : 1^(1//5).

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