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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 A Answers

Skills

    1. (a) \(z = 1 + 4 i,\ w = 5 - 2 i\)
    2. (b) \(z + w = 6 + 2 i\)
    1. (a) \(-w = -3 - 2 i\)
    2. (b) \(z - w = -4 + 3 i\)
    1. (a) \(z = 2 + 5 i,\ \alpha = 6 + i\)
    2. (b) \(z - \alpha = -4 + 4 i\)
    3. (c) \(5.66\)
    1. (a) \(|1 + i| = \sqrt{2},\ \arg = \frac{\pi}{4}\)
    2. (b) P′ = \(2 + 2 i\); Q′ = \(4 + 4 i\); R′ = \(1 + 7 i\)
    3. (c) A rotation of \(\frac{\pi}{4}\) anticlockwise about the origin, with an enlargement of scale factor \(\sqrt{2}\) from the origin.
    1. (a) \(B = 3 - \sqrt{3} + \left(1 + 3 \sqrt{3}\right) i, \text{ the point } (3 - \sqrt{3}, 1 + 3 \sqrt{3})\)
    2. (b) \(C = \sqrt{3} + 3 + \left(1 - 3 \sqrt{3}\right) i, \text{ the point } (\sqrt{3} + 3, 1 - 3 \sqrt{3})\)
    1. (a) \(z = -4 + 4i\), \(|z| = 4\sqrt{2}\), \(\arg z = \frac{3\pi}{4}\); \(w = 2i\), \(|w| = 2\), \(\arg w = \frac{\pi}{2}\)
    2. (b) \(\left|\frac{z}{w}\right| = 2\sqrt{2}\), \(\arg\frac{z}{w} = \frac{\pi}{4}\)
    3. (c) A rotation of \(\frac{\pi}{2}\) clockwise about the origin and an enlargement of scale factor \(\frac{1}{2}\) from the origin; \(\frac{z}{w} = 2 + 2i\)
    1. (a) \(z - \alpha = 5 + 3i\)
    2. (b) \(3 - 5i\)
    3. (c) \(z' = 4 - 3i\)
    1. (a) \(w = 8 - 2i\)
    2. (b) A rotation of \(\frac{\pi}{2}\) clockwise and an enlargement of scale factor 2, both about the origin, then a translation of 4 units to the right.
    3. (c) \(z = -2 + i\)
    1. (a) \(|z| = 4.47,\ \arg z = 1.11\)
    2. (b) 0.707 radians
    3. (c) 8.72

In context

    1. (a) \(450 - 150 i\)
    2. (b) 474 N
    3. (c) \(150 + 350 i\)
    1. (a) \(8 - 6 i\)
    2. (b) 10 km
    1. (a) A rotation of \(\frac{\pi}{2}\) anticlockwise about the origin with scale factor \(3\).
    2. (b) A′ = \(3 i\); B′ = \(9 i\); C′ = \(-6 + 3 i\)
    1. (a) \(z - \alpha = 5 + 3i\)
    2. (b) \(z' = 4.89 + 7.52i\)
    3. (c) 5.83 m

Stretch

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

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