A-LevelMathematicsComplex numbers9709/31 · May/June 2020 · Q1012 Marks

The complex number u is defined by u = 3i / (a + 2i), where a is real.

(a)(i)Express u in the Cartesian form x + iy, where x and y are in terms of a.[3]
(a)(ii)Find the exact value of a for which arg u* = π/3.[3]
(b)(i)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z – 2i| < |z − 1 − i| and |z – 2 – i| ≤ 2.[4]
(b)(ii)Calculate the least value of arg z for points in this region.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[M1] (a)(i) Multiply numerator and denominator by a – 2i, or equivalent
[A1] (a)(i) Use i² =-1 at least once
[A1] (a)(i) Obtain answer (6/(a²+4)) + (3a/(a²+4))i
[B1] (a)(ii) Either state that arg u = -π/3 or express u* in terms of a (FT on u)
[M1] (a)(ii) Use correct method to form an equation in a
[A1] (a)(ii) Obtain answer a = -2√3
[B1] (b)(i) Show the perpendicular bisector of points representing 2i and 1 + i
[B1] (b)(i) Show the point representing 2+i
[B1FT] (b)(i) Show a circle with radius 2 and centre 2 + i (FT on the position of the point for 2 + i)
[B1] (b)(i) Shade the correct region
[B1] (b)(ii) State or imply the critical point 2 + 3i
[B1] (b)(ii) Obtain answer 56.3° or 0.983 radians

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.