A-LevelMathematicsComplex numbers9709/32 · Oct/Nov 2022 · Q58 Marks

Solve the equation z² – 6iz – 12 = 0, giving the answers in the form x + iy, where x and y are real and exact.

(a)Solve the equation z² – 6iz – 12 = 0, giving the answers in the form x + iy, where x and y are real and exact.[3]
(b)On a sketch of an Argand diagram with origin O, show points A and B representing the roots of the equation in part (a).[1]
(c)Find the exact modulus and argument of each root.[3]
(d)Hence show that the triangle OAB is equilateral.[1]

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 2022). The mark or mark type is shown in brackets.

[M1] (a) Use quadratic formula, or completing the square ((z-3i)² -3 = 0) and use i² = -1 to find a root
[A1] (a) Obtain a root, e.g. √3+3i
[A1] (a) Obtain the other root, e.g. -√3+3i
[B1 FT] (b) Show points representing the roots correctly
[B1 FT] (c) State modulus of either root is 2√3, or simplified exact equivalent
[M1] (c) Find the argument of one of their roots – get as far as tan⁻¹(...)
[A1] (c) Obtain correct arguments π/3 and 2π/3, or simplified exact equivalents
[B1] (d) Give a complete justification that the correct triangle is equilateral

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.