Complex Numbers: exam review Exam review: second paper Worked solutions

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

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

   • M1 for collecting the real parts and the imaginary parts
   • A1 for \(16 - 6i\)

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 \(15 + 16i\)

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 \(2 - 3i\), giving denominator 13
   • A1 for \(\frac{9}{13} - \frac{20}{13}i\)

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

1. (a) Simplify \(i^{38}\). [1]

   • A1 for \(-1\) (the powers of \(i\) repeat every 4, so \(i^{38} = i^{2}\))

2. (b) Write \(\sqrt{-64}\) in the form \(ki\), where \(k\) is a real number. [1]

   • A1 for \(8i\)

3. (c) The complex number \(z = 5 - 12i\) is plotted on an Argand diagram. Write down the coordinates of the points that represent \(-z\) and \(z^*\). [2]

   • A1 for \(-z\) at (-5, 12)
   • A1 for \(z^*\) at (5, 12)

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

   • A1 for \(|z| = 13\)
   • A1 for \(zz^* = 169\)

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 \((a + bi)(3 - i) = 11 + 13i\). [3]

   • M1 for expanding the product and using \(i^2 = -1\)
   • A1 for the two equations \(b + 3 a = 11\) and \(- a + 3 b = 13\) (real parts, imaginary parts)
   • A1 for \(a = 2\) and \(b = 5\)

2. (b) Solve the equation \(3z + iz^* = 11 + 9i\), giving \(z\) in the form \(x + iy\). [4]

   • M1 for substituting \(z = x + iy\) and \(z^* = x - iy\)
   • A1 for \(iz^* = y + ix\), so \(3z + iz^* = (y + 3 x) + (x + 3 y)i\)
   • M1 for equating real parts and imaginary parts: \(y + 3 x = 11\), \(x + 3 y = 9\)
   • A1 for \(z = 3 + 2i\)

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

1. (a) Show that the discriminant of \(f(x) = x^2 + 4x + 13\) is negative, and state what this means for the graph of \(y = f(x)\). [2]

   • M1 for \(b^2 - 4ac = (4)^2 - 4(1)(13)\)
   • A1 for \(-36 < 0\), 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 \(-2 + 3i\) and \(-2 - 3i\)

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

   • A1 for \((-2, 9)\)

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

   • A1 for \(6\) (the points are \((-2, 3)\) and \((-2, -3)\))

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

1. (a) Let \(z = 3 - 7i\). 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 fourth quadrant, so \(\theta = 2\pi - \alpha\) with \(\alpha = \tan^{-1}\dfrac{7}{3}\)
   • A1 for \(r = \sqrt{58} = 7.62\)
   • A1 for \(\theta = 5.12\)

2. (b) Let \(w = 6\,\text{cis}\left(\frac{5 \pi}{3}\right)\). Write \(w\) in the form \(a + bi\), giving exact values. [2]

   • M1 for \(a = 6\cos\left(\frac{5 \pi}{3}\right)\) and \(b = 6\sin\left(\frac{5 \pi}{3}\right)\)
   • A1 for \(a = 3\) and \(b = - 3 \sqrt{3}\)

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

   • A1 for \(r = 7.62\), \(\theta = 2\pi - 5.12 = 1.17\)

Two complex numbers are given in modulus-argument form. Let \(z = 6\,\text{cis}\left(\frac{\pi}{3}\right)\) and \(w = 3\,\text{cis}\left(\frac{11 \pi}{6}\right)\). [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: \(18\) and \(\frac{13 \pi}{6}\) (this is not less than \(2\pi\), so subtract \(2\pi\))
   • A1 for \(18\,\text{cis}\left(\frac{\pi}{6}\right)\)

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 \(- \frac{3 \pi}{2}\) (this is negative, so add \(2\pi\))
   • A1 for \(2e^{\frac{\pi}{2}i}\)

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

   • M1 for \(z^2 = 36\,\text{cis}\left(\frac{2 \pi}{3}\right)\) (square the modulus, double the argument)
   • A1 for \(-18 + 18 \sqrt{3}i\)

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 \(z + 3i = 2z^* + 5\). [3]

   • M1 for writing \(z = x + iy\), so \(z^* = x - iy\), and equating real and imaginary parts: \(x = 2x + 5\) and \(y + 3 = -2y\)
   • A1 for \(x = -5\)
   • A1 for \(y = -1\)

2. (b) Solve the simultaneous equations \((2 - i)z + w = 8 + 3i\) and \(z - iw = 1 + 2i\), giving \(z\) and \(w\) in the form \(a + bi\). [3]

   • M1 for eliminating one unknown: from the first equation \(w = 8 + 3i - (2 - i)z\), substituted into the second to give \((2 + 2i)z = -2 + 10i\)
   • A1 for \(z = 2 + 3i\)
   • A1 for \(w = 1 - i\)

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

1. (a) Let \(z = 2 - 2 \sqrt{3}i\). 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 \(|z| = \sqrt{16}\)
   • A1 for \(|z| = 4\)
   • A1 for \(\arg z = - \frac{\pi}{3}\)

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

   • M1 for \(z^5 = \left(4\right)^5 \text{cis}\left(5 \times - \frac{\pi}{3}\right)\)
   • A1 for \(1024\,\text{cis}\left(- \frac{5 \pi}{3}\right)\)
   • A1 for \(512 + 512 \sqrt{3}i\) (the argument \(- \frac{5 \pi}{3}\) is the same direction as \(\frac{\pi}{3}\))

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 \(n \times - \frac{\pi}{3} = k\pi\)
   • A1 for \(n = 3\) (check: \(z^{3} = -64\))

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

   • M1 for \(e^{2i} = \cos 2 + i\sin 2\), with the calculator in radians
   • A1 for \(2.75 + 2.73i\)
   • A1 for modulus \(3.87\)
   • A1 for argument \(0.781\)

5. (e) Let \(w = 2e^{\frac{\pi}{4}i}\). Write \(\dfrac{z}{w}\) in the form \(re^{i\theta}\), where \(-\pi < \theta \le \pi\). [2]

   • M1 for dividing the moduli (\(4\) and \(2\)) and subtracting the arguments
   • A1 for \(2e^{- \frac{7 \pi}{12}i}\)

The height above mean sea level, in metres, of two tidal components at a harbour, \(t\) hours after midnight, are \(h_1 = 2.4\cos(0.5t)\) and \(h_2 = 1.5\cos\left(0.5t + \dfrac{\pi}{6}\right)\). The total height is \(h_1 + h_2\). Use your calculator in radian mode. [16]

1. (a) Show that \(h_1 + h_2 = \text{Re}\left(e^{0.5ti}\left(2.4 + 1.5e^{\frac{\pi}{6}i}\right)\right)\). [2]

   • M1 for \(h_1 = \text{Re}\left(2.4e^{0.5ti}\right)\) and \(h_2 = \text{Re}\left(1.5e^{(0.5t + \frac{\pi}{6})i}\right)\)
   • A1 for \(e^{(0.5t + \frac{\pi}{6})i} = e^{0.5ti}e^{\frac{\pi}{6}i}\), taking out \(e^{0.5ti}\) (AG)

2. (b) Calculate \(2.4 + 1.5e^{\frac{\pi}{6}i}\), giving your answer in the form \(x + iy\). [2]

   • M1 for \(e^{\frac{\pi}{6}i} = \cos \frac{\pi}{6} + i\sin \frac{\pi}{6}\)
   • A1 for \(3.70 + 0.750i\)

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 \(R = \sqrt{3.70^2 + 0.750^2}\) or the calculator in radian mode
   • A1 for \(R = 3.77\)
   • A1 for \(\theta = 0.200\)

4. (d) Hence write \(h_1 + h_2\) in the form \(R\cos(0.5t + \theta)\), and write down the greatest value of \(h_1 + h_2\). [2]

   • A1 for \(3.77\cos(0.5t + 0.200)\)
   • A1 for greatest value \(3.77\) metres

5. (e) Find the first time \(t > 0\) at which \(h_1 + h_2 = 0\). [3]

   • M1 for \(3.77\cos(0.5t + 0.200) = 0\)
   • M1 for \(0.5t + 0.200 = \dfrac{\pi}{2}\) (smallest angle that gives \(t > 0\))
   • A1 for \(t = 2.74\)

6. (f) Find the first time \(t > 0\) at which \(h_1 + h_2\) is greatest. [2]

   • M1 for \(0.5t + 0.200 = 2\pi\) (cosine is greatest at \(2\pi\)); \(t > 0\)
   • A1 for \(t = 12.2\)

7. (g) At \(t = 1\), find the percentage of \(h_1 + h_2\) that is due to \(h_2\). Give your answer to 3 significant figures. [2]

   • M1 for \(h_2 = 1.5\cos(0.5\times 1 + \frac{\pi}{6}) = 0.780\) and \(h_1 + h_2 = 2.89\)
   • A1 for \(27.0\%\)

A drone's flight is plotted on an Argand diagram. The drone starts at the origin O, and its first waypoint A represents the complex number \(z = 3 + 5i\). [12]

1. (a) Find \(|z|\). [1]

   • A1 for \(\sqrt{9+25} = \sqrt{34}\)

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 \(b = (3 + 5i)\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)\) expanded with \(i^2 = -1\)
   • A1 for \(b = \frac{3}{2} - \frac{5 \sqrt{3}}{2} + \left(\frac{5}{2} + \frac{3 \sqrt{3}}{2}\right)i\)

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

   • M1 for \(b - z = - \frac{3}{2} - \frac{5 \sqrt{3}}{2} + \left(- \frac{5}{2} + \frac{3 \sqrt{3}}{2}\right)i\), so \(|b - z| = \sqrt{34}\)
   • 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 \(\frac{9}{2} - \frac{5 \sqrt{3}}{2} + \left(\frac{15}{2} + \frac{3 \sqrt{3}}{2}\right)i\)

5. (e) Triangle OAB is transformed by multiplying each of its vertices by \(3i\). Describe fully the transformation. [2]

   • A1 for a rotation of \(\dfrac{\pi}{2}\) anticlockwise about O
   • A1 for an enlargement, scale factor 3, centre O (\(3i\) has modulus 3 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 \(\frac{17 \sqrt{3}}{2} = 14.7\)

Let \(z = 3 + ki\) and \(w = 5 - i\), 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 \(5 + i\)
   • M1 for expanding the top with \(i^2 = -1\); the bottom is \(26\)
   • A1 for \(\dfrac{15 - k + (3 + 5 k)i}{26}\)

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

   • M1 for setting the real part to zero: \(15 - k = 0\)
   • A1 for \(k = 15\)

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 \(15 - k = 3 + 5 k\)
   • A1 for \(k = 2\)

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

   • M1 for \(z^2 = x^2 - y^2 + 2xyi\)
   • M1 for \((z^*)^2 = x^2 - y^2 - 2xyi\)
   • A1 for \(z^2 - (z^*)^2 = 4xyi\), which has no real part

A sequence of complex numbers \(u_1, u_2, u_3, \dots\) is geometric with first term \(u_1 = 10\) and common ratio \(r = \dfrac{-1 + i}{2}\). [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 \(\frac{\sqrt{2}}{2}\) and the argument \(\frac{3\pi}{4}\) (second quadrant)
   • A1 for \(r = \dfrac{\sqrt{2}}{2}e^{\frac{3\pi}{4}i}\)

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

   • M1 for \(u_3 = u_1 r^2 = 10 \times \dfrac{1}{2} e^{\frac{3\pi}{2}i} = 5e^{-\frac{\pi}{2}i}\)
   • A1 for \(u_3 = -5i\)

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 \(|u_n| = |u_1||r|^{n-1} = 10\left(\dfrac{\sqrt{2}}{2}\right)^{n-1}\), or \(\left|\dfrac{u_{n+1}}{u_n}\right| = |r|\)
   • R1 for a constant ratio \(\dfrac{\sqrt{2}}{2}\), so the moduli are geometric

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

   • M1 for \(S_\infty = \dfrac{|u_1| = 10}{1 - \dfrac{\sqrt{2}}{2}}\), valid because \(-1 < \dfrac{\sqrt{2}}{2} < 1\)
   • A1 for \(\dfrac{10}{1 - \frac{\sqrt{2}}{2}} = 20 + 10\sqrt{2}\)

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

   • M1 for \(u_2 = -5 + 5i, \text{ which has modulus } 5\sqrt{2} \text{ and argument } \dfrac{3\pi}{4}\); the conjugate keeps the modulus and changes the sign of the argument
   • A1 for \(u_2^* = 5\sqrt{2}e^{-\frac{3\pi}{4}i}\), and \(u_2 u_2^* = |u_2|^2 = 50\)

In a plane drawn as an Argand diagram, a beacon is at the point \(B\) representing \(\alpha = -1 + 3i\) and a boat is at the point \(T\) representing \(z = 4 - i\). [14]

1. (a) The boat turns through \(\dfrac{\pi}{2}\) anticlockwise about \(B\). Find the complex number that represents its new position. [3]

   • M1 for \(z - \alpha = 5 - 4i\), the vector from \(B\) to \(T\)
   • M1 for \((z - \alpha)\,\text{cis}\left(\frac{\pi}{2}\right) = i(5 - 4i) = 4 + 5i\), then adding \(\alpha\)
   • A1 for \(3 + 8i\)

2. (b) The boat's navigation software moves points using the mapping \(w = az + b\), where \(a = 3 + 3i\) and \(b = -2 + 3i\). Describe fully the geometric transformation that this mapping applies, in the order in which the steps happen. [3]

   • A1 for \(|a| = 3\sqrt{2}\) and \(\arg a = \dfrac{\pi}{4}\): a rotation through \(\dfrac{\pi}{4}\) anticlockwise and an enlargement of scale factor \(3\sqrt{2}\), both about the origin
   • A1 for this step happening first
   • A1 for a translation by \(b = -2 + 3i\), that is \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\), second

3. (c) Find the image of the point \(1 + 2i\) under this mapping. [2]

   • M1 for \((3 + 3i)(1 + 2i) + (-2 + 3i)\)
   • A1 for \(-5 + 12i\)

4. (d) The point \(Q\) is \(8\,\text{cis}\left(\dfrac{5\pi}{6}\right)\) and the point \(P\) is \(2\,\text{cis}\left(\dfrac{\pi}{4}\right)\). 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 \(\left|\dfrac{Q}{P}\right| = \dfrac{8}{2} = 4\)
   • M1 for subtracting the arguments: \(\dfrac{5\pi}{6} - \dfrac{\pi}{4}\)
   • A1 for \(\dfrac{7\pi}{12}\)
   • A1 for a rotation of \(\dfrac{\pi}{4}\) clockwise and a division of the distance by 2, both about the origin

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

   • M1 for \(\dfrac{1}{2} \times 8 \times 2 \times \sin\left(\dfrac{5\pi}{6} - \dfrac{\pi}{4}\right)\) in radian mode
   • A1 for \(2\sqrt{2} + 2\sqrt{6} = 7.73\)