17. Complex Numbers Problems Part 1

This is split into parts because MathJaX takes a lot of time to render if there is lots of Math on the same page.

Find the square root of following complex numbers:

1. $$7+8i$$

2. $$3+4i$$

3. $$a^2-b^2+2abi$$

4. $$7-25\sqrt{-2}$$

5. $$\sqrt[4]{-81}$$

6. Find the square root of

$\frac{x^2}{y^2}+\frac{y^2}{x^2}+\frac{1}{2i}\left(\frac{x}{y}+\frac{y}{x} \right) + \frac{31}{16}$
7. Find the square root of

$\frac{x^2}{y^2}+\frac{y^2}{x^2}-\frac{1}{i}\left(\frac{x}{y}-\frac{y}{x} \right) - \frac{9}{4}$
8. Find the square root of

$x^2+\frac{1}{x^2}+4i\left(x-\frac{1}{x}\right)-6$
9. Find $$\sqrt{2+3\sqrt{-5}}+\sqrt{2-3\sqrt{-5}}$$

10. Find $$\sqrt{i}\sqrt{-i}$$

Simplify following in the form of $$A+iB$$

11. $$i^{n+80}+i^{n+50}$$

12. $$\left(i^{17}+\frac{1}{i^{15}}\right)^3$$

13. $$\frac{(1+i)^2}{2+3i}$$

14. $$\left(\frac{1}{1+i} + \frac{1}{1-i}\right)\frac{7+8i}{7-8i}$$

15. $$\frac{(1+i)^{4n+7}}{(1-i)^{4n-1}}$$

16. $$\frac{1}{1-cos\theta + 2isin\theta}$$

17. $$\frac{(cosx+isinx)(cosy+isiny)}{(cotu+i)(i+tanv)}$$

18. Find the complex number $$z$$ such that $$z^2 + |z|=0$$

19. Show that for $$z\in C, |z|=0$$ if and only if $$z=0$$

20. If $$z_1$$ and $$z_2$$ are $$1-i$$ and $$2+7i$$ find $$Im\left(\frac{z_1z_2}{\overline{z_1}}\right)$$

Find $$x$$ and $$y$$ if

21. $$(x+5i)-(3-iy)=7+8i$$

22. $$\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$$

23. If $$|z|=1$$ then prove that $$\frac{z-1}{z+1}$$ is purely imaginary.

24. If $$|z-i|<1$$ then prove that $$|z+12-6i|<14$$

25. If $$|z+6|=|2z+3|$$ then prove that $$|z|=3$$

26. If $$\sqrt{a-ib}=x-iy$$ prove that $$\sqrt{a+ib}=x+iy$$

27. Show that the equation

$\frac{A^2}{x-a}+\frac{B^2}{x-b}+ ... + \frac{H^2}{x-h} = x+l$

where $$A, B, ..., H; a, b, ..., h \text{ and } l$$ are real; cannot have imaginary roots.

28. Show that a unimodular complex number, not purely real can always be expressed as $$\frac{c+i}{c-i}$$ for some real $$c$$.

29. If the expression

$\frac{sin\frac{x}{2}+cos\frac{x}{2}-itanx}{1+2isin\frac{x}{2}}$

is real, then find all the possible set of values for $$x$$

30. Find the conjugate, modulus and argument of $$\sqrt{3}+2i$$

31. Put $$\frac{a+ib}{x-iy}$$ in polar form.

For any two complex numbers $$z_1$$ and $$z_2$$ prove that

32. $$|z_1+z_2|^2+|z_2-z_2|^2 = 2(|z_1|^2 + |z_2|^2)$$

33. $$|z_1+z_2|^2=z_1^2+z_2^2+2Re(z_1\overline{z_2}) = z_1^2+z_2^2+2Re(\overline{z_1}z_2)$$

34. If $$z_1=1 \text{ and } |z_2|=1$$ then prove that

$|z_1+z_2|= \left|\frac{1}{z_1}+\frac{1}{z_2}\right|$
35. If $$|z-2|=2|z-1|$$ then show that $$|z|^2=\frac{4}{3}Re(z)$$

36. If $$\sqrt[3]{a+ib}=x+iy$$ then prove that $$\frac{a}{x} + \frac{b}{y} = 4(x^2 - y^2)$$

37. If $$x+iy=\sqrt{\frac{a+ib}{c+id}}$$ then prove that $$(x^2+y^2)^2= \frac{a^2+b^2}{c^2+d^2}$$

38. If $$\frac{3}{2+cos\theta+isin\theta}=a+ib$$ then prove that $$a^2+b^2=4a-3$$

39. If $$|2z-1|=|z-2|$$ then prove that $$|z|=1$$

40. If $$x$$ is real and $$\frac{1-ix}{1+ix}=m+in$$ then prove that $$m^2+n^2=1$$

41. If $$m+in=\frac{x+iy}{x-iy}, \text{ where } x,y,m,n$$ are real and $$x+iy\ne 0$$ and $$m+in\neq 0$$ then prove that $$m^2 + n^2 = 1$$

42. If $$\left(1+i\frac{x}{a}\right) \left(1+i\frac{x}{c}\right) \left(1+i\frac{x}{c}\right) ... = A+iB$$ then prove that $$\left(1+\frac{x^2}{a^2}\right) \left(1+\frac{x^2}{b^2}\right) \left(1+\frac{x^2}{c^2}\right) ... = A^2+B^2$$

43. Let $$z_1$$ and $$z_2$$ be complex numbers such that $$z_1\ne z_2$$ and $$|z_1|=|z_2|.$$ If $$z_1$$ has positive real part and $$z_2$$ has negative imaginary part then prove that $$\frac{z_1+z_2}{z_1-z_2}$$ is either zero or purely imaginary.

44. For complex numbers $$z_1=x_1+iy_1$$ and $$z_2=x_2+iy_2$$ the notation of $$z_1\cap z_2$$ if $$x_1\le x_2$$ and $$y_1\le y_2.$$ Show that for all complex numbers $$z$$ with $$1\cap z$$ we have $$\frac{1-z}{1+z}\cap 0.$$

45. If $$a>0, z|z|+az+1=0,$$ show that $$z$$ is a negative real number.

46. Find the range of real number $$\alpha$$ for which the equation $$z+\alpha|z-1|+2i=0; z=x+iy$$ has a solution. Also, find the solution.

47. For every real number $$a\ge 0,$$ find all the complex numbers satisfying the equation $$a|z|-4az+1+ia=0$$

48. Show that $$(x^2+y^2)^5=(x^5-10x^3y^2+5xy^4)^2+(5x^4y-10x^2y^3+y^5)^2$$

49. Express $$(x^2+a^2)(x^2+b^2)(x^2+c^2)$$ as sum of two squares.

50. If $$(1+x)^n=a_0+a_1x+a_2x^2+ ... +a_nx^n,$$ prove that $$2^n=(a_0-a_2+a_4- ...)^2 + (a_1-a_3+a_5- ...)^2$$