Complex Numbers: exam review Exam review

Two complex numbers are given. Let \(z = 5 + 2i\) and \(w = 3 - i\). [6]

1. (a) Find \(3z - w\). [2]
2. (b) Find \(zw\), giving your answer in the form \(a + bi\). [2]
3. (c) Find \(\dfrac{z}{w}\), giving your answer in the form \(p + qi\) with \(p\) and \(q\) as fractions. [2]

The number \(i\) satisfies \(i^2 = -1\). [6]

1. (a) Simplify \(i^{43}\). [1]
2. (b) Write \(\sqrt{-49}\) in the form \(ki\), where \(k\) is a real number. [1]
3. (c) The complex number \(z = -4 + 3i\) is plotted on an Argand diagram. Write down the coordinates of the points that represent \(-z\) and \(z^*\). [2]
4. (d) Find \(|z|\) and \(zz^*\). [2]

Two equations in complex numbers are given. All the unknown letters \(a\), \(b\), \(x\) and \(y\) are real. [7]

1. (a) Find the real numbers \(a\) and \(b\) such that \((a + bi)(2 + i) = 4 + 7i\). [3]
2. (b) Solve the equation \(2z + iz^* = 5 + 4i\), giving \(z\) in the form \(x + iy\). [4]

A quadratic function \(f\) is defined for real \(x\). [6]

1. (a) Show that the discriminant of \(f(x) = x^2 - 6x + 25\) is negative, and state what this means for the graph of \(y = f(x)\). [2]
2. (b) Solve \(f(x) = 0\). [2]
3. (c) Write down the coordinates of the vertex of the graph of \(y = f(x)\). [1]
4. (d) The two roots are plotted on an Argand diagram. Find the distance between the two points. [1]

Complex numbers can be written in modulus-argument form. [6]

1. (a) Let \(z = -2 - 5i\). Write \(z\) in the form \(r\,\text{cis}\,\theta\), where \(0 \le \theta < 2\pi\). Give \(r\) and \(\theta\) to 3 significant figures. [3]
2. (b) Let \(w = 4\,\text{cis}\left(\frac{7 \pi}{6}\right)\). Write \(w\) in the form \(a + bi\), giving exact values. [2]
3. (c) Write \(z^*\) in the form \(r\,\text{cis}\,\theta\), where \(0 \le \theta < 2\pi\). [1]

Two complex numbers are given in modulus-argument form. Let \(z = 4\,\text{cis}\left(\frac{5 \pi}{6}\right)\) and \(w = 2\,\text{cis}\left(\frac{4 \pi}{3}\right)\). [6]

1. (a) Find \(zw\) in the form \(r\,\text{cis}\,\theta\), where \(0 \le \theta < 2\pi\). [2]
2. (b) Find \(\dfrac{z}{w}\) in the form \(re^{i\theta}\), where \(0 \le \theta < 2\pi\). [2]
3. (c) Find \(z^2\) in the form \(a + bi\), giving exact values. [2]

Complex numbers can be found from equations. In this question \(z^*\) is the complex conjugate of \(z\), and \(z\) and \(w\) are complex numbers. [6]

1. (a) Find the complex number \(z\) that satisfies \(z - 2i = 3z^* + 6\). [3]
2. (b) Solve the simultaneous equations \((1 + i)z + w = 5 + 2i\) and \(z - iw = 3\), giving \(z\) and \(w\) in the form \(a + bi\). [3]

The complex number \(z\) is plotted on an Argand diagram. [14]

1. (a) Let \(z = -3 + 3i\). Find the modulus and the argument of \(z\), where \(-\pi < \theta \le \pi\). Give the argument as a multiple of \(\pi\). [3]
2. (b) Use your answer to part (a) to find \(z^5\) in the form \(a + bi\). [3]
3. (c) Find the smallest positive integer \(n\) for which \(z^n\) is a real number. [2]
4. (d) Find the modulus and the argument of \(5 + 2e^{3i}\). Angles are in radians. [4]
5. (e) Let \(w = 2e^{\frac{\pi}{6}i}\). Write \(\dfrac{z}{w}\) in the form \(re^{i\theta}\), where \(-\pi < \theta \le \pi\). [2]

Two alternating voltages in a circuit, in volts, at time \(t\) seconds are \(V_1 = 8\cos(50t)\) and \(V_2 = 5\cos\left(50t + \dfrac{\pi}{3}\right)\). The total voltage is \(V_1 + V_2\). Use your calculator in radian mode. [16]

1. (a) Show that \(V_1 + V_2 = \text{Re}\left(e^{50ti}\left(8 + 5e^{\frac{\pi}{3}i}\right)\right)\). [2]
2. (b) Calculate \(8 + 5e^{\frac{\pi}{3}i}\), giving your answer in the form \(x + iy\). [2]
3. (c) Find the modulus and the argument of your answer to part (b), where \(-\pi < \theta \le \pi\). Give both to 3 significant figures. [3]
4. (d) Hence write \(V_1 + V_2\) in the form \(R\cos(50t + \theta)\), and write down the greatest value of \(V_1 + V_2\). [2]
5. (e) Find the first time \(t > 0\) at which \(V_1 + V_2 = 0\). [3]
6. (f) Find the first time \(t > 0\) at which \(V_1 + V_2\) is greatest. [2]
7. (g) At \(t = 0.005\), find the percentage of \(V_1 + V_2\) that is due to \(V_2\). Give your answer to 3 significant figures. [2]

On a treasure map drawn on an Argand diagram, the origin O is the camp and the point A, the old oak, represents the complex number \(z = 4 + 2i\). [12]

1. (a) Find \(|z|\). [1]
2. (b) Point B represents the complex number \(b\), where \(b\) is \(z\) rotated through \(\dfrac{\pi}{3}\) anticlockwise about O. Show that \(b = z\,\text{cis}\left(\dfrac{\pi}{3}\right)\), and write \(b\) in the form \(x + iy\) with exact values. [3]
3. (c) Find \(|b - z|\), and hence state the type of triangle OAB. [2]
4. (d) The point C is the fourth vertex of the parallelogram OACB. Find the complex number that represents C. [2]
5. (e) Triangle OAB is transformed by multiplying each of its vertices by \(2i\). Describe fully the transformation. [2]
6. (f) Find the area of triangle OAB. [2]

Let \(z = 2 + ki\) and \(w = 3 - i\), where \(k\) is a real number. [10]

1. (a) Find \(\dfrac{z}{w}\) in the form \(x + iy\), where \(x\) and \(y\) are in terms of \(k\). [3]
2. (b) Find the value of \(k\) for which \(\dfrac{z}{w}\) is purely imaginary. [2]
3. (c) Find the value of \(k\) for which the real part and the imaginary part of \(\dfrac{z}{w}\) are equal. [2]
4. (d) Show that \((z^*)^2 = (z^2)^*\) for \(z = x + iy\), where \(x\) and \(y\) are real. [3]

A sequence of complex numbers \(u_1, u_2, u_3, \dots\) is geometric with first term \(u_1 = 6\) and common ratio \(r = \dfrac{\sqrt{3} + i}{4}\). [10]

1. (a) Write \(r\) in the form \(\rho e^{i\theta}\), where \(\rho > 0\) and \(-\pi < \theta \le \pi\). [2]
2. (b) Find \(u_3\) in the form \(a + bi\), giving exact values. [2]
3. (c) Show that the moduli \(|u_1|, |u_2|, |u_3|, \dots\) form a geometric sequence, and write down its common ratio. [2]
4. (d) Find the sum of the moduli of all the terms of the sequence. [2]
5. (e) Write \(u_2^*\) in the form \(Re^{i\theta}\) with \(-\pi < \theta \le \pi\), and hence find \(u_2 u_2^*\). [2]

In a plane drawn as an Argand diagram, a sensor is at the point \(S\) representing \(\alpha = 1 + 2i\) and a robot is at the point \(R\) representing \(z = 5 + 3i\). [14]

1. (a) The robot turns through \(\dfrac{\pi}{2}\) anticlockwise about \(S\). Find the complex number that represents its new position. [3]
2. (b) The robot's controller moves points using the mapping \(w = az + b\), where \(a = 1 + \sqrt{3}\,i\) and \(b = 3 - i\). Describe fully the geometric transformation that this mapping applies, in the order in which the steps happen. [3]
3. (c) Find the image of the point \(2 + i\) under this mapping. [2]
4. (d) The point \(Q\) is \(6\,\text{cis}\left(\dfrac{2\pi}{3}\right)\) and the point \(P\) is \(3\,\text{cis}\left(\dfrac{\pi}{6}\right)\). Find \(\left|\dfrac{Q}{P}\right|\) and \(\arg\dfrac{Q}{P}\), and describe what dividing \(Q\) by \(P\) does to the point \(Q\). [4]
5. (e) Find the area of triangle \(OPQ\), where \(O\) is the origin. [2]