A-LevelMathematicsComplex numbers9709/31 · May/June 2018 · Q78 Marks

Showing all working and without using a calculator, solve the equation z² + (2√6)z + 8 = 0, giving your answers in the form x + iy, where x and y are real and exact.

(i)Showing all working and without using a calculator, solve the equation z² + (2√6)z + 8 = 0, giving your answers in the form x + iy, where x and y are real and exact.[3]
(ii)Sketch an Argand diagram showing the points representing the roots.[1]
(iii)The points representing the roots are A and B, and O is the origin. Find angle AOB.[3]
(iv)Prove that triangle AOB is equilateral.[1]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[M1] Use quadratic formula, or completing the square, or the substitution z = x + iy to find a root, using i² = −1
[A1] Obtain a root, e.g. -√6-√2i
[A1] Obtain the other root, e.g. -√6-√2i
[B1ft] Represent both roots in relatively correct positions
[B1ft] State or imply correct value of a relevant length or angle, e.g. OA, OB, AB, angle between OA or OB and the real axis
[M1] Carry out a complete method for finding angle OAB
[A1] Obtain AOB = 60° correctly
[B1] Give a complete justification of the given statement

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