A-LevelMathematicsComplex numbers9709/32 · Feb/Mar 2021 · Q89 Marks
The complex numbers u and v are defined by u = −4 + 2i and v = 3 + i.
- (a)Find u/v in the form x + iy, where x and y are real.[3]
- (b)Hence express u/v in the form re^(iθ), where r and θ are exact.[2]
- (c)In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and 2u + v respectively. State fully the geometrical relationship between OA and BC.[2]
- (d)Prove that angle AOB = ¾π.[2]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.
- [M1 OE] Multiply numerator and denominator by 3 – i
- [A1] Obtain numerator -10 + 10i or denominator 10
- [A1] Obtain final answer -1 + i
- [B1 FT] State or imply r = √2
- [B1 FT] State or imply that θ = ¾π
- [B1] State that OA and BC are parallel
- [B1] State that BC = 2OA
- [M1] Use angle AOB = arg u - arg v = arg (u/v)
- [A1] Obtain the given answer
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