A-LevelMathematicsComplex numbers9709/33 · Oct/Nov 2021 · Q1110 Marks
The complex number –√3 + i is denoted by u.
- (a)Express u in the form r e^(iθ), where r > 0 and −π < θ ≤ π, giving the exact values of r and θ.[2]
- (b)Hence show that u⁶ is real and state its value.[2]
- (c)(i)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities 0 ≤ arg(z – u) < ¼π and Re z < 2.[4]
- (c)(ii)Find the greatest value of |z| for points in the shaded region. Give your answer correct to 3 significant figures.[2]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.
- [B1] State or imply r = 2
- [B1] State or imply θ = 5π/6
- [M1] Use a correct method for finding the modulus or argument of u⁶
- [A1] Show correctly that u⁶ is real and has value -64
- [B1] Show half lines from the point representing -√3 + i
- [B1] Show correct half lines
- [B1] Show the line x = 2 in the first quadrant
- [B1] Shade the correct region
- [M1] Carry out a correct method to find the greatest value of |z|
- [A1] Obtain answer 5.14
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