A-LevelMathematicsComplex numbers9709/33 · May/June 2019 · Q89 Marks
Throughout this question the use of a calculator is not permitted. The complex number u is defined by u = 4i / (1-(√3)i)
- (i)Express u in the form x + iy, where x and y are real and exact.[3]
- (ii)Find the exact modulus and argument of u.[2]
- (iii)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z| < 2 and |z - u| < |z|.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [M1] Multiply numerator and denominator by 1 + √3i, or equivalent
- [A1] 4i - 4√3 and 3+1
- [A1] Obtain final answer -√3 + i
- [B1] State that the modulus of u is 2
- [B1] State that the argument of u is (5/6)π (or 150°)
- [B1] Show a circle with centre the origin and radius 2
- [B1 FT] Show u in a relatively correct position
- [B1 FT] Show the perpendicular bisector of the line joining u and the origin
- [B1] Shade the correct region
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