Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/complex-numbers/ahl-6-13/problems/

jmaths

IB Mathematics: Applications and Interpretation HL · AHL 6: Complex Numbers · AHL 6.13

Complex Numbers: exam review Exam review Answers

Section A: short questions

    1. (a) \(12 + 7i\)
    2. (b) \(17 + i\)
    3. (c) \(\frac{13}{10} + \frac{11}{10}i\)
    1. (a) \(-i\)
    2. (b) \(7i\)
    3. (c) \(-z\): (4, -3); \(z^*\): (-4, -3)
    4. (d) \(|z| = 5\), \(zz^* = 25\)
    1. (a) \(a = 3\), \(b = 2\)
    2. (b) \(z = 2 + i\)
    1. (a) \(-64\); the graph does not cross the \(x\)-axis
    2. (b) \(x = 3 + 4i\) and \(x = 3 - 4i\)
    3. (c) \((3, 16)\)
    4. (d) \(8\)
    1. (a) \(r = 5.39\), \(\theta = 4.33\)
    2. (b) \(w = - 2 \sqrt{3} - 2i\)
    3. (c) \(z^* = 5.39\,\text{cis}\,1.95\)
    1. (a) \(8\,\text{cis}\left(\frac{\pi}{6}\right)\)
    2. (b) \(2e^{\frac{3 \pi}{2}i}\)
    3. (c) \(8 - 8 \sqrt{3}i\)
    1. (a) \(z = -3 + \dfrac{1}{2}i\)
    2. (b) \(z = 5 - i\), \(w = -1 - 2i\)

Section B: long questions

    1. (a) \(|z| = 3 \sqrt{2}\), \(\arg z = \frac{3 \pi}{4}\)
    2. (b) \(z^5 = 972 - 972i\)
    3. (c) \(n = 4\)
    4. (d) modulus 3.03, argument 0.0932
    5. (e) \(\frac{3 \sqrt{2}}{2}e^{\frac{7 \pi}{12}i}\)
    1. (a) shown
    2. (b) \(10.5 + 4.33i\)
    3. (c) modulus 11.4, argument 0.391
    4. (d) \(11.4\cos(50t + 0.391)\); greatest value 11.4 volts
    5. (e) \(t = 0.0236\)
    6. (f) \(t = 0.118\)
    7. (g) 14.8%
    1. (a) \(2 \sqrt{5}\)
    2. (b) \(b = 2 - \sqrt{3} + \left(1 + 2 \sqrt{3}\right)i\)
    3. (c) \(|b - z| = 2 \sqrt{5}\); equilateral
    4. (d) \(6 - \sqrt{3} + \left(3 + 2 \sqrt{3}\right)i\)
    5. (e) a rotation of \(\dfrac{\pi}{2}\) anticlockwise about O, with an enlargement of scale factor 2
    6. (f) \(5 \sqrt{3} = 8.66\)
    1. (a) \(\dfrac{6 - k}{10} + \dfrac{2 + 3 k}{10}i\)
    2. (b) \(k = 6\)
    3. (c) \(k = 1\)
    4. (d) shown
    1. (a) \(r = \dfrac{1}{2}e^{\frac{\pi}{6}i}\)
    2. (b) \(u_3 = \dfrac{3}{4} + \dfrac{3\sqrt{3}}{4}i\)
    3. (c) \(|u_n| = 6\left(\dfrac{1}{2}\right)^{n-1}\); common ratio \(\dfrac{1}{2}\)
    4. (d) \(\dfrac{6}{1 - \frac{1}{2}} = 12\)
    5. (e) \(u_2^* = 3e^{-\frac{\pi}{6}i}\); \(u_2 u_2^* = 9\)
    1. (a) \(6i\)
    2. (b) a rotation of \(\dfrac{\pi}{3}\) anticlockwise about the origin, and an enlargement of scale factor 2 about the origin, followed by a translation by the vector \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\)
    3. (c) \(5 - \sqrt{3} + 2\sqrt{3}\,i\)
    4. (d) \(\left|\dfrac{Q}{P}\right| = 2\), \(\arg\dfrac{Q}{P} = \dfrac{\pi}{2}\); dividing by \(P\) rotates \(Q\) through \(\dfrac{\pi}{6}\) clockwise about the origin and divides its distance from the origin by 3
    5. (e) \(9\)

jmaths.xyz · AI HL · AHL 6.13 Unit review