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

Complex Numbers: exam review Exam review: second paper Answers

Section A: short questions

    1. (a) \(16 - 6i\)
    2. (b) \(15 + 16i\)
    3. (c) \(\frac{9}{13} - \frac{20}{13}i\)
    1. (a) \(-1\)
    2. (b) \(8i\)
    3. (c) \(-z\): (-5, 12); \(z^*\): (5, 12)
    4. (d) \(|z| = 13\), \(zz^* = 169\)
    1. (a) \(a = 2\), \(b = 5\)
    2. (b) \(z = 3 + 2i\)
    1. (a) \(-36\); the graph does not cross the \(x\)-axis
    2. (b) \(x = -2 + 3i\) and \(x = -2 - 3i\)
    3. (c) \((-2, 9)\)
    4. (d) \(6\)
    1. (a) \(r = 7.62\), \(\theta = 5.12\)
    2. (b) \(w = 3 - 3 \sqrt{3}i\)
    3. (c) \(z^* = 7.62\,\text{cis}\,1.17\)
    1. (a) \(18\,\text{cis}\left(\frac{\pi}{6}\right)\)
    2. (b) \(2e^{\frac{\pi}{2}i}\)
    3. (c) \(-18 + 18 \sqrt{3}i\)
    1. (a) \(z = -5 - i\)
    2. (b) \(z = 2 + 3i\), \(w = 1 - i\)

Section B: long questions

    1. (a) \(|z| = 4\), \(\arg z = - \frac{\pi}{3}\)
    2. (b) \(z^5 = 512 + 512 \sqrt{3}i\)
    3. (c) \(n = 3\)
    4. (d) modulus 3.87, argument 0.781
    5. (e) \(2e^{- \frac{7 \pi}{12}i}\)
    1. (a) shown
    2. (b) \(3.70 + 0.750i\)
    3. (c) modulus 3.77, argument 0.200
    4. (d) \(3.77\cos(0.5t + 0.200)\); greatest value 3.77 metres
    5. (e) \(t = 2.74\)
    6. (f) \(t = 12.2\)
    7. (g) 27.0%
    1. (a) \(\sqrt{34}\)
    2. (b) \(b = \frac{3}{2} - \frac{5 \sqrt{3}}{2} + \left(\frac{5}{2} + \frac{3 \sqrt{3}}{2}\right)i\)
    3. (c) \(|b - z| = \sqrt{34}\); equilateral
    4. (d) \(\frac{9}{2} - \frac{5 \sqrt{3}}{2} + \left(\frac{15}{2} + \frac{3 \sqrt{3}}{2}\right)i\)
    5. (e) a rotation of \(\dfrac{\pi}{2}\) anticlockwise about O, with an enlargement of scale factor 3
    6. (f) \(\frac{17 \sqrt{3}}{2} = 14.7\)
    1. (a) \(\dfrac{15 - k}{26} + \dfrac{3 + 5 k}{26}i\)
    2. (b) \(k = 15\)
    3. (c) \(k = 2\)
    4. (d) shown
    1. (a) \(r = \dfrac{\sqrt{2}}{2}e^{\frac{3\pi}{4}i}\)
    2. (b) \(u_3 = -5i\)
    3. (c) \(|u_n| = 10\left(\dfrac{\sqrt{2}}{2}\right)^{n-1}\); common ratio \(\dfrac{\sqrt{2}}{2}\)
    4. (d) \(\dfrac{10}{1 - \frac{\sqrt{2}}{2}} = 20 + 10\sqrt{2}\)
    5. (e) \(u_2^* = 5\sqrt{2}e^{-\frac{3\pi}{4}i}\); \(u_2 u_2^* = 50\)
    1. (a) \(3 + 8i\)
    2. (b) a rotation of \(\dfrac{\pi}{4}\) anticlockwise about the origin, and an enlargement of scale factor \(3\sqrt{2}\) about the origin, followed by a translation by the vector \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\)
    3. (c) \(-5 + 12i\)
    4. (d) \(\left|\dfrac{Q}{P}\right| = 4\), \(\arg\dfrac{Q}{P} = \dfrac{7\pi}{12}\); dividing by \(P\) rotates \(Q\) through \(\dfrac{\pi}{4}\) clockwise about the origin and divides its distance from the origin by 2
    5. (e) \(2\sqrt{2} + 2\sqrt{6} = 7.73\)

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