A-LevelMathematicsComplex numbers9709/32 · Oct/Nov 2017 · Q78 Marks
Throughout this question the use of a calculator is not permitted. The complex number 1 – (√3)i is denoted by u.
- (i)Find the modulus and argument of u.[2]
- (ii)Show that u³ + 8 = 0.[2]
- (iii)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying both the inequalities |z – u| ≤ 2 and Re z ≥ 2, where Re z denotes the real part of z.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.
- [1] State modulus 2
- [1] State argument -π/3 or -60° (5π/3 or 300°)
- [1] EITHER: Expand (1-(√3)i)³ completely and process i² and i³ OR: u³ = 2³(cos(-π) + i sin(-π)) or equivalent: follow their answers to (i)
- [1] Verify that the given relation is satisfied
- [1] Show a circle with centre 1-(√3)i in a relatively correct position
- [1] Show a circle with radius 2 passing through the origin
- [1] Show the line Re z = 2
- [1] Shade the correct region
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