# 16. Complex Numbers Problems Part 4¶

1. If $$O$$ be the center of the circle circumscribing the equilateral $$\triangle ABC$$ and its radius be unity and $$A$$ lies on the $$x$$-axis. Find the complex numbers represented by $$B$$ and $$C.$$

2. $$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at a point $$M$$ and satisfy $$BD = 2AC.$$ If the points $$D$$ and $$M$$ represent the complex numbers $$1 + i$$ and $$2 - i$$ respectively, then find the complex number represented by $$A$$.

3. If $$z_1, z_2, z_3$$ and $$z_4$$ are the vertices of a square taken in anticlockwise order, prove that $$z_3 = -iz_1 + (1 + i)z_2$$ and $$z_4 = (1 - i)z_1 + iz_2.$$

4. Let $$z_1, z_2$$ and $$z_3$$ are the vertices of an equilateral triangle in the circle $$|z| = 2.$$ If $$z_1 = 1 + i\sqrt{3},$$ then find $$z_2$$ and $$z_3.$$

5. If $$a$$ and $$b$$ are real numbers between 0 and 1 such that the points $$z_1 = a + i, z_2 = 1 + bi,$$ and $$z_3 = 0$$ form an equilateral triangle then find $$a$$ and $$b.$$

6. Let $$ABCD$$ be a square described in the anticlockwise sense in the Argand plane. If $$A$$ represents $$(3 + 5i)$$ and the center of the square represents $$\left(\frac{7}{2} + \frac{5}{2}i\right).$$ Find the numbers represented by $$B, C$$ and $$D$$.

7. Find the vertices of a regular polygon of $$n$$ sides, if its center is located at origin and one of its vertices is $$z_1.$$

8. Prove that the points $$a(cos\alpha + i sin\alpha), b(cos\beta + i sin\beta)$$ and $$c(cos\gamma + i sin\gamma)$$ in the Argand plane are collinear if $$bcsin(\beta - \gamma) + casin(\gamma - \alpha) + absin(\alpha -\beta) = 0.$$

9. $$A$$ represents the number $$6i, B$$ the number $$3$$ and $$P$$ the complex number $$z$$. If $$P$$ moves such that $$PA : PB = 2 : 1,$$ show that $$z\overline{z} = (4 + 2i)z + (4 - 2i)\overline{z}.$$ Also, show that the locus of P is a circle, find its radius and center.

10. Show that if the points $$z_1, z_2, z_3$$ and $$z_4$$ taken in order are concyclic then the expression $$\frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 -z_1)}$$ is purely real.

11. Let $$z_1, z_2, z_3$$ and $$z_4$$ be the vertices of a quadrilateral. Prove that quadrilateral is cyclic if $$z_1z_2 + z_3z_4 = 0$$ and $$z_1 + z_2 = 0.$$

12. Show that the triangles whose vertices are $$z_1, z_2, z_3$$ and $$z_1', z_2', z_3'$$ are similar if

$\begin{split}\begin{vmatrix}z_1 & z_1' & 1\\z_2 & z_2' & 1\\z_3 & z_3' & 1\end{vmatrix} = 0\end{split}$
13. If $$a, b, c$$ and $$u, v, \omega$$ are the complex numbers representing two triangles such that $$c = (1 - r)a + rb$$ and $$\omega = (i - r)u + rv,$$ where $$r$$ is a complex number, prove that the two triangles are similar.

14. Find the equation of perpendicular bisector of the line segment joining points $$z_1$$ and $$z_2.$$

15. Find the equation of the circle having the line segment joining $$z_1$$ and $$z_2$$ as diameter.

16. If $$\left|\frac{z - z_1}{z - z_2}\right| = c, c \ne~0,$$ then show that the locus of $$z$$ is a circle.

17. If $$|z| = 1,$$ find the locus of the point $$\frac{2}{z}.$$

18. If for any two complex numbers $$z_1$$ and $$z_2, |z_1 + z_2| = |z_1| + |z_2|$$ prove that $$arg(z_1) - arg(z_2) = 2n\pi.$$

19. Find the complex number $$z$$ the least in absolute value which satisfies the condition $$|z - 2 + 2i| = 1.$$

20. Find the point in the first quadrant, on the curve $$|z - 5i| = 3$$ whose argument is minimum.

21. Find the set of points of the coordinate plane which satisfy the inequality

$log_\frac{1}{2}\left(\frac{|z - 1| + 4}{3|z - 1| - 2}\right) > 1$
22. Find the set of all points on the $$xy$$-plane whose coordinates satisfy the following conditions: the number $$z^2 + z + 1$$ is real and positive.

23. Find the real values of the parameter $$a$$ for which at least one complex number $$z$$ satisfies the equality $$|z - az| = a + 4$$ and the inequality $$|z - 2| < 1.$$

24. Find the real values of parameter $$t$$ for which at least one complex number $$z$$ satisfies the equality $$|z + \sqrt{2}| = t^2 - 3t +2$$ and the inequality $$|z + t\sqrt{2}| < t^2.$$

25. Among the complex numbers which satisfy the condition $$|z - 15i| \le 15,$$ find the number having least positive argument.

26. Find $$z$$ for which $$|z - 5i| \le 1$$ and $$arg(z)$$ is minimum.

27. Find the real value of $$a$$ for which there is at least one complex number satisfying $$|z + 4i| = \sqrt{a^2 - 12a + 28}$$ and $$|z - 4\sqrt{3}| < a.$$

28. Find the set of points belonging to the coordinate plane $$xy,$$ for which the real part of complex number $$(1 + i)z^2$$ is positive.

29. Solve the equation $$2z = |z| + 2i$$ in complex numbers.

30. Show that all the roots of the equation $$a_1z^3 + a_2z^2 + a_3z + a_4 = 3,$$ where $$|a_i| \le 1, i= 1, 2, 3, 4$$ lie outside the circle with center as origin and radius $$\frac{2}{3}.$$

31. Given that $$\sum_{i = 1}^nb_i = 0$$ and $$\sum_{i = 1}^nb_iz_i = 0$$ where $$b_i$$’s are non-zero real numbers, no three of $$z_i$$’s form a straight line. Prove that $$z_i$$’s are concyclic if $$b_1b_2|z_2 - z_2|^2 = b_3b_4|z_3 - z_4|^2.$$

32. A cubic equation $$f(x) = 0$$ has one real root $$\alpha$$ and two complex roots $$\beta + i\gamma.$$ Points $$A, B$$ and $$C$$ represent these roots respectively. Show that the roots of the derived equation $$f'(x) = 0$$ are complex if $$A$$ falls inside one of the two equilateral triangles described on base $$BC$$.

33. Prove that reflection of $$\overline{a}z + a\overline{z} = 0$$ in the real axis is $$\overline{a}\overline{z} + az = 0.$$

34. If $$\alpha, \beta, \gamma, \delta$$ are four complex numbers such that $$\frac{\gamma}{\delta}$$ is real and $$\alpha\delta - \beta\gamma \ne 0$$ then prove that $$z = \frac{\alpha + \beta t}{\gamma + \delta t}, t \in \mathbf{R}$$ represents a straight line.

35. If $$\omega, \omega^2$$ are cube roots of unity, then prove that

1. $$(3 + 3\omega + 5\omega^2)^2 - (2 + 6\omega + 2\omega^2) = 0$$

2. $$(2 - \omega)(2 - \omega^2)(2 -\omega^{10})(2 - \omega^{11}) = 49$$

3. $$(1 - \omega)(1 - \omega^2)(1 - \omega^4)(1 - \omega^5) = 9$$

4. $$(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = 32$$

5. $$1 + \omega^n + \omega^{2n} = 3$$ where $$n > 0, n \in \mathbf{I}$$ and is a multiple of 3.

6. $$1 + \omega^n + \omega^{2n} = 0$$ where $$n > 0, n \in \mathbf{I}$$ and is not a multiple of 3.

36. Resolve into linear factors $$a^2 + b^2 + c^2 -ab -bc -ca$$

37. Prove that

$\sqrt{-1 - \sqrt{-1 - \sqrt{-1 - ... to \infty}}} = \omega \text{ or } \omega^2$
38. If $$x = a + b, y = a\omega + b\omega^2, z = a\omega^2 + b\omega,$$ prove that $$x^3 + y^3 + z^3 = 3(a^3 + b^3)$$ and $$xyz = a^3 + b^3.$$

39. Resolve into linear factors:

1. $$a^2 -ab + b^2$$

2. $$a^2 + ab + b^2$$

3. $$a^3 + b^3$$

4. $$a^3 - b^3$$

5. $$a^3 + b^3 +c^3 -3abc$$

40. Show that $$x^{3p} + x^{3q+1} + x^{3r+2},$$ where $$p, q, r$$ are positive integers is divisible by $$x^2 + x + 1.$$

41. Show that $$x^{3p} + x^{3q+1} + x^{3r+2} + x^{4s+3},$$ where $$p, q, r, s$$ are positive integers is divisible by $$x^3 + x^2 + x + 1.$$

42. If $$p = a + b + c, q = a + b\omega + c\omega^2, r = a + b\omega^2 + c\omega,$$ where $$\omega$$ is a cube root of unity, prove that $$p^3 + q^3 + r^3 - 3pqr = 27abc.$$

43. If $$\omega$$ be an imaginary root of unity, prove that $$(a + b\omega + c\omega^2)^3 + (a + b\omega^2 + c\omega)^3 = (2a - b -c)(2b -a -c)(2c -a -b).$$

44. If $$ax + cy + bz = X, cx + by + az = Y, bx + ay + cz = Z$$ show that

1. $$(a^2 + b^2 + c^2 - ab - bc - ca)(x^2 + y^2 + z^2 - xy - yz -zx) = X^2 + Y^2 + Z^2 - XY -YZ -ZY$$

2. $$(a^3 + b^3 - c^3 - 3abc)(x^3 + y^3 + z^3 -3xyz) = X^3 + Y^3 + Z^3 - 3XYZ$$

45. Prove that

$\left(\frac{cos\theta + isin\theta}{sin\theta + icos\theta}\right)^4 = cos8\theta + isin8\theta$
46. If $$z^2 -2zcos\theta +1 =0,$$ show that $$z^2 + z^{-2} = 2cos2\theta$$

47. If $$x_r = cos\frac{\pi}{2^r} + isin\frac{\pi}{2^r},$$ prove that $$x_1x_2x_3 ... to \infty = -1.$$

48. Prove that $$(1 + i)^n + (1 - i)^n = 2^{\frac{n}{2} + 1}cos\frac{n\pi}{4}.$$

49. Show that the value of

$\sum_{k = 1}^6\left(sin\frac{2\pi k}{7} -icos\frac{2\pi k}{7}\right)$

is $$i.$$

50. Show that

$e^{2mi cot^{-1}p}\left(\frac{pi + 1}{pi - 1}\right)^m = 1$