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IB Mathematics: Applications and Interpretation HL · AHL 6: Complex Numbers · AHL 6.12

Complex numbers on the Argand diagram Problem set B Answers

Skills

    1. (a) \(z = -3 + 2 i,\ w = 4 + 3 i\)
    2. (b) \(z + w = 1 + 5 i\)
    1. (a) \(-w = -5 + i\)
    2. (b) \(z - w = -3 + 5 i\)
    1. (a) \(z = -4 + 3 i,\ \alpha = -1 - 2 i\)
    2. (b) \(z - \alpha = -3 + 5 i\)
    3. (c) \(5.83\)
    1. (a) \(|\sqrt{3} + i| = 2,\ \arg = \frac{\pi}{6}\)
    2. (b) P′ = \(-1 + \sqrt{3} i\); Q′ = \(-3 + 3 \sqrt{3} i\); R′ = \(-3 + 2 \sqrt{3} + \left(2 + 3 \sqrt{3}\right) i\)
    3. (c) A rotation of \(\frac{\pi}{6}\) anticlockwise about the origin, with an enlargement of scale factor \(2\) from the origin.
    1. (a) \(B = \frac{5}{2} - \frac{\sqrt{3}}{2} + \left(\frac{1}{2} + \frac{5 \sqrt{3}}{2}\right) i, \text{ the point } (\frac{5}{2} - \frac{\sqrt{3}}{2}, \frac{1}{2} + \frac{5 \sqrt{3}}{2})\)
    2. (b) \(C = \frac{\sqrt{3}}{2} + \frac{5}{2} + \left(\frac{1}{2} - \frac{5 \sqrt{3}}{2}\right) i, \text{ the point } (\frac{\sqrt{3}}{2} + \frac{5}{2}, \frac{1}{2} - \frac{5 \sqrt{3}}{2})\)
    1. (a) \(z = 6 - 6i\), \(|z| = 6\sqrt{2}\), \(\arg z = -\frac{\pi}{4}\); \(w = 3i\), \(|w| = 3\), \(\arg w = \frac{\pi}{2}\)
    2. (b) \(\left|\frac{z}{w}\right| = 2\sqrt{2}\), \(\arg\frac{z}{w} = -\frac{3\pi}{4}\)
    3. (c) A rotation of \(\frac{\pi}{2}\) clockwise about the origin and an enlargement of scale factor \(\frac{1}{3}\) from the origin; \(\frac{z}{w} = -2 - 2i\)
    1. (a) \(z - \alpha = 4 + 4i\)
    2. (b) \(4 - 4i\)
    3. (c) \(z' = 7 - 3i\)
    1. (a) \(w = 8 - 6i\)
    2. (b) A rotation of \(\frac{\pi}{2}\) clockwise and an enlargement of scale factor 3, both about the origin, then a translation of 5 units to the right.
    3. (c) \(z = -2 - i\)
    1. (a) \(|z| = 5.39,\ \arg z = 0.381\)
    2. (b) 0.919 radians
    3. (c) 8.57

In context

    1. (a) \(40 + 400 i\)
    2. (b) 402 N
    3. (c) \(200 + 280 i\)
    1. (a) \(9 - 8 i\)
    2. (b) 12.0 km
    1. (a) A rotation of \(\frac{\pi}{2}\) clockwise about the origin with scale factor \(2\).
    2. (b) A′ = \(-4 i\); B′ = \(4 - 4 i\); C′ = \(8\)
    1. (a) \(z - \alpha = 10\)
    2. (b) \(z' = 12.2 + 17.8i\)
    3. (c) 10 m

Stretch

    1. (a) \(R = -1 + 5 i, \text{ the point } (-1, 5)\)
    2. (b) \(Q = 4 + 6 i, \text{ the point } (4, 6)\)
    1. (a) \(q - p = 5 + 2 i\)
    2. (b) \(\frac{5}{2} - \sqrt{3} + \left(1 + \frac{5 \sqrt{3}}{2}\right) i\)
    3. (c) \(R = (\frac{1}{2} - \sqrt{3}, 2 + \frac{5 \sqrt{3}}{2})\)

jmaths.xyz · AI HL · AHL 6.12 Geometry on the Argand diagram