Complex Numbers: exam review Exam review One question at a time

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

1. (a) Find \(3z - w\). [2]

   • M1 for collecting the real parts and the imaginary parts
   • A1 for \(12 + 7i\)

2. (b) Find \(zw\), giving your answer in the form \(a + bi\). [2]

   • M1 for expanding the brackets and using \(i^2 = -1\)
   • A1 for \(17 + i\)

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

   • M1 for multiplying top and bottom by the conjugate \(3 + i\), giving denominator 10
   • A1 for \(\frac{13}{10} + \frac{11}{10}i\)

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

1. (a) Simplify \(i^{43}\). [1]

   • A1 for \(-i\) (the powers of \(i\) repeat every 4, so \(i^{43} = i^{3}\))

2. (b) Write \(\sqrt{-49}\) in the form \(ki\), where \(k\) is a real number. [1]

   • A1 for \(7i\)

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]

   • A1 for \(-z\) at (4, -3)
   • A1 for \(z^*\) at (-4, -3)

4. (d) Find \(|z|\) and \(zz^*\). [2]

   • A1 for \(|z| = 5\)
   • A1 for \(zz^* = 25\)

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]

   • M1 for expanding the product and using \(i^2 = -1\)
   • A1 for the two equations \(- b + 2 a = 4\) and \(a + 2 b = 7\) (real parts, imaginary parts)
   • A1 for \(a = 3\) and \(b = 2\)

2. (b) Solve the equation \(2z + iz^* = 5 + 4i\), giving \(z\) in the form \(x + iy\). [4]

   • M1 for substituting \(z = x + iy\) and \(z^* = x - iy\)
   • A1 for \(iz^* = y + ix\), so \(2z + iz^* = (y + 2 x) + (x + 2 y)i\)
   • M1 for equating real parts and imaginary parts: \(y + 2 x = 5\), \(x + 2 y = 4\)
   • A1 for \(z = 2 + i\)

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]

   • M1 for \(b^2 - 4ac = (-6)^2 - 4(1)(25)\)
   • A1 for \(-64 < 0\), so there are no real roots and the graph does not cross the \(x\)-axis

2. (b) Solve \(f(x) = 0\). [2]

   • M1 for using the calculator in complex mode (or the quadratic formula with \(\sqrt{-1} = i\))
   • A1 for \(3 + 4i\) and \(3 - 4i\)

3. (c) Write down the coordinates of the vertex of the graph of \(y = f(x)\). [1]

   • A1 for \((3, 16)\)

4. (d) The two roots are plotted on an Argand diagram. Find the distance between the two points. [1]

   • A1 for \(8\) (the points are \((3, 4)\) and \((3, -4)\))

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]

   • M1 for a sketch or a check that \(z\) is in the third quadrant, so \(\theta = \pi + \alpha\) with \(\alpha = \tan^{-1}\dfrac{5}{2}\)
   • A1 for \(r = \sqrt{29} = 5.39\)
   • A1 for \(\theta = 4.33\)

2. (b) Let \(w = 4\,\text{cis}\left(\frac{7 \pi}{6}\right)\). Write \(w\) in the form \(a + bi\), giving exact values. [2]

   • M1 for \(a = 4\cos\left(\frac{7 \pi}{6}\right)\) and \(b = 4\sin\left(\frac{7 \pi}{6}\right)\)
   • A1 for \(a = - 2 \sqrt{3}\) and \(b = -2\)

3. (c) Write \(z^*\) in the form \(r\,\text{cis}\,\theta\), where \(0 \le \theta < 2\pi\). [1]

   • A1 for \(r = 5.39\), \(\theta = 2\pi - 4.33 = 1.95\)

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]

   • M1 for multiplying the moduli and adding the arguments: \(8\) and \(\frac{13 \pi}{6}\) (this is not less than \(2\pi\), so subtract \(2\pi\))
   • A1 for \(8\,\text{cis}\left(\frac{\pi}{6}\right)\)

2. (b) Find \(\dfrac{z}{w}\) in the form \(re^{i\theta}\), where \(0 \le \theta < 2\pi\). [2]

   • M1 for dividing the moduli and subtracting the arguments: \(2\) and \(- \frac{\pi}{2}\) (this is negative, so add \(2\pi\))
   • A1 for \(2e^{\frac{3 \pi}{2}i}\)

3. (c) Find \(z^2\) in the form \(a + bi\), giving exact values. [2]

   • M1 for \(z^2 = 16\,\text{cis}\left(\frac{5 \pi}{3}\right)\) (square the modulus, double the argument)
   • A1 for \(8 - 8 \sqrt{3}i\)

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]

   • M1 for writing \(z = x + iy\), so \(z^* = x - iy\), and equating real and imaginary parts: \(x = 3x + 6\) and \(y - 2 = -3y\)
   • A1 for \(x = -3\)
   • A1 for \(y = \dfrac{1}{2}\)

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]

   • M1 for eliminating one unknown: from the first equation \(w = 5 + 2i - (1 + i)z\), substituted into the second to give \(iz = 1 + 5i\)
   • A1 for \(z = 5 - i\)
   • A1 for \(w = -1 - 2i\)

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]

   • M1 for a sketch to find the quadrant, and \(|z| = \sqrt{18}\)
   • A1 for \(|z| = 3 \sqrt{2}\)
   • A1 for \(\arg z = \frac{3 \pi}{4}\)

2. (b) Use your answer to part (a) to find \(z^5\) in the form \(a + bi\). [3]

   • M1 for \(z^5 = \left(3 \sqrt{2}\right)^5 \text{cis}\left(5 \times \frac{3 \pi}{4}\right)\)
   • A1 for \(972 \sqrt{2}\,\text{cis}\left(\frac{15 \pi}{4}\right)\)
   • A1 for \(972 - 972i\) (the argument \(\frac{15 \pi}{4}\) is the same direction as \(- \frac{\pi}{4}\))

3. (c) Find the smallest positive integer \(n\) for which \(z^n\) is a real number. [2]

   • M1 for \(n\arg z\) being a multiple of \(\pi\), that is \(n \times \frac{3 \pi}{4} = k\pi\)
   • A1 for \(n = 4\) (check: \(z^{4} = -324\))

4. (d) Find the modulus and the argument of \(5 + 2e^{3i}\). Angles are in radians. [4]

   • M1 for \(e^{3i} = \cos 3 + i\sin 3\), with the calculator in radians
   • A1 for \(3.02 + 0.282i\)
   • A1 for modulus \(3.03\)
   • A1 for argument \(0.0932\)

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]

   • M1 for dividing the moduli (\(3 \sqrt{2}\) and \(2\)) and subtracting the arguments
   • A1 for \(\frac{3 \sqrt{2}}{2}e^{\frac{7 \pi}{12}i}\)

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]

   • M1 for \(V_1 = \text{Re}\left(8e^{50ti}\right)\) and \(V_2 = \text{Re}\left(5e^{(50t + \frac{\pi}{3})i}\right)\)
   • A1 for \(e^{(50t + \frac{\pi}{3})i} = e^{50ti}e^{\frac{\pi}{3}i}\), taking out \(e^{50ti}\) (AG)

2. (b) Calculate \(8 + 5e^{\frac{\pi}{3}i}\), giving your answer in the form \(x + iy\). [2]

   • M1 for \(e^{\frac{\pi}{3}i} = \cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\)
   • A1 for \(10.5 + 4.33i\)

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]

   • M1 for \(R = \sqrt{10.5^2 + 4.33^2}\) or the calculator in radian mode
   • A1 for \(R = 11.4\)
   • A1 for \(\theta = 0.391\)

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]

   • A1 for \(11.4\cos(50t + 0.391)\)
   • A1 for greatest value \(11.4\) volts

5. (e) Find the first time \(t > 0\) at which \(V_1 + V_2 = 0\). [3]

   • M1 for \(11.4\cos(50t + 0.391) = 0\)
   • M1 for \(50t + 0.391 = \dfrac{\pi}{2}\) (smallest angle that gives \(t > 0\))
   • A1 for \(t = 0.0236\)

6. (f) Find the first time \(t > 0\) at which \(V_1 + V_2\) is greatest. [2]

   • M1 for \(50t + 0.391 = 2\pi\) (cosine is greatest at \(2\pi\)); \(t > 0\)
   • A1 for \(t = 0.118\)

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]

   • M1 for \(V_2 = 5\cos(50\times 0.005 + \frac{\pi}{3}) = 1.35\) and \(V_1 + V_2 = 9.10\)
   • A1 for \(14.8\%\)

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]

   • A1 for \(\sqrt{16+4} = 2 \sqrt{5}\)

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]

   • M1 for rotation through \(\dfrac{\pi}{3}\) with no change of distance: multiply by \(1\,\text{cis}\left(\dfrac{\pi}{3}\right)\)
   • M1 for \(b = (4 + 2i)\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)\) expanded with \(i^2 = -1\)
   • A1 for \(b = 2 - \sqrt{3} + \left(1 + 2 \sqrt{3}\right)i\)

3. (c) Find \(|b - z|\), and hence state the type of triangle OAB. [2]

   • M1 for \(b - z = -2 - \sqrt{3} + \left(-1 + 2 \sqrt{3}\right)i\), so \(|b - z| = 2 \sqrt{5}\)
   • A1 for \(OA = OB = AB\), so the triangle is equilateral

4. (d) The point C is the fourth vertex of the parallelogram OACB. Find the complex number that represents C. [2]

   • M1 for \(c = z + b\)
   • A1 for \(6 - \sqrt{3} + \left(3 + 2 \sqrt{3}\right)i\)

5. (e) Triangle OAB is transformed by multiplying each of its vertices by \(2i\). Describe fully the transformation. [2]

   • A1 for a rotation of \(\dfrac{\pi}{2}\) anticlockwise about O
   • A1 for an enlargement, scale factor 2, centre O (\(2i\) has modulus 2 and argument \(\dfrac{\pi}{2}\))

6. (f) Find the area of triangle OAB. [2]

   • M1 for area \(= \dfrac{1}{2}|z|^2\sin\dfrac{\pi}{3}\)
   • A1 for \(5 \sqrt{3} = 8.66\)

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]

   • M1 for multiplying top and bottom by \(3 + i\)
   • M1 for expanding the top with \(i^2 = -1\); the bottom is \(10\)
   • A1 for \(\dfrac{6 - k + (2 + 3 k)i}{10}\)

2. (b) Find the value of \(k\) for which \(\dfrac{z}{w}\) is purely imaginary. [2]

   • M1 for setting the real part to zero: \(6 - k = 0\)
   • A1 for \(k = 6\)

3. (c) Find the value of \(k\) for which the real part and the imaginary part of \(\dfrac{z}{w}\) are equal. [2]

   • M1 for \(6 - k = 2 + 3 k\)
   • A1 for \(k = 1\)

4. (d) Show that \((z^*)^2 = (z^2)^*\) for \(z = x + iy\), where \(x\) and \(y\) are real. [3]

   • M1 for \((z^*)^2 = (x - iy)^2 = x^2 - y^2 - 2xyi\)
   • M1 for \(z^2 = x^2 - y^2 + 2xyi\), so \((z^2)^* = x^2 - y^2 - 2xyi\)
   • A1 for the two expressions equal, so \((z^*)^2 = (z^2)^*\)

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]

   • M1 for the modulus \(\frac{1}{2}\) and the argument \(\frac{\pi}{6}\) (first quadrant, \(\tan\theta = \frac{1}{\sqrt{3}}\))
   • A1 for \(r = \dfrac{1}{2}e^{\frac{\pi}{6}i}\)

2. (b) Find \(u_3\) in the form \(a + bi\), giving exact values. [2]

   • M1 for \(u_3 = u_1 r^2 = 6\left(\dfrac{1}{2}\right)^2 e^{\frac{\pi}{3}i} = \dfrac{3}{2}\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)\)
   • A1 for \(u_3 = \dfrac{3}{4} + \dfrac{3\sqrt{3}}{4}i\)

3. (c) Show that the moduli \(|u_1|, |u_2|, |u_3|, \dots\) form a geometric sequence, and write down its common ratio. [2]

   • M1 for \(|u_n| = |u_1||r|^{n-1} = 6\left(\dfrac{1}{2}\right)^{n-1}\), or \(\left|\dfrac{u_{n+1}}{u_n}\right| = |r|\)
   • R1 for a constant ratio \(\dfrac{1}{2}\), so the moduli are geometric

4. (d) Find the sum of the moduli of all the terms of the sequence. [2]

   • M1 for \(S_\infty = \dfrac{|u_1| = 6}{1 - \dfrac{1}{2}}\), valid because \(-1 < \dfrac{1}{2} < 1\)
   • A1 for \(\dfrac{6}{1 - \frac{1}{2}} = 12\)

5. (e) Write \(u_2^*\) in the form \(Re^{i\theta}\) with \(-\pi < \theta \le \pi\), and hence find \(u_2 u_2^*\). [2]

   • M1 for \(u_2 = \dfrac{3\sqrt{3}}{2} + \dfrac{3}{2}i, \text{ which has modulus } 3 \text{ and argument } \dfrac{\pi}{6}\); the conjugate keeps the modulus and changes the sign of the argument
   • A1 for \(u_2^* = 3e^{-\frac{\pi}{6}i}\), and \(u_2 u_2^* = |u_2|^2 = 9\)

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]

   • M1 for \(z - \alpha = 4 + i\), the vector from \(S\) to \(R\)
   • M1 for \((z - \alpha)\,\text{cis}\left(\frac{\pi}{2}\right) = i(4 + i) = -1 + 4i\), then adding \(\alpha\)
   • A1 for \(6i\)

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]

   • A1 for \(|a| = 2\) and \(\arg a = \dfrac{\pi}{3}\): a rotation through \(\dfrac{\pi}{3}\) anticlockwise and an enlargement of scale factor 2, both about the origin
   • A1 for this step happening first
   • A1 for a translation by \(b = 3 - i\), that is \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\), second

3. (c) Find the image of the point \(2 + i\) under this mapping. [2]

   • M1 for \((1 + \sqrt{3}i)(2 + i) + 3 - i\)
   • A1 for \(5 - \sqrt{3} + 2\sqrt{3}\,i\)

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]

   • A1 for \(\left|\dfrac{Q}{P}\right| = \dfrac{6}{3} = 2\)
   • M1 for subtracting the arguments: \(\dfrac{2\pi}{3} - \dfrac{\pi}{6}\)
   • A1 for \(\dfrac{\pi}{2}\)
   • A1 for a rotation of \(\dfrac{\pi}{6}\) clockwise and a division of the distance by 3, both about the origin

5. (e) Find the area of triangle \(OPQ\), where \(O\) is the origin. [2]

   • M1 for \(\dfrac{1}{2} \times 6 \times 3 \times \sin\left(\dfrac{2\pi}{3} - \dfrac{\pi}{6}\right)\) in radian mode
   • A1 for \(9\)