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 \(2^n=(a_0-a_2+a_4- ...)^2 + (a_1-a_3+a_5- ...)^2\)