A-LevelMathematicsComplex numbers9709/32 · Feb/Mar 2018 · Q911 Marks

The complex number 1 + 2i is denoted by u.

(i)(a)It is given that u is a root of the equation 2x³ – x² + 4x + k = 0, where k is a constant. Showing all working and without using a calculator, find the value of k.[3]
(i)(b)Showing all working and without using a calculator, find the other two roots of this equation.[4]
(ii)On an Argand diagram sketch the locus of points representing complex numbers z satisfying the equation |z - u| = 1. Determine the least value of arg z for points on this locus. Give your answer in radians correct to 2 decimal places.[4]

Mark scheme

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

[M1] Substitute x = 1 + 2i in the equation and attempt expansions of x² and x³
[M1] Use i² = −1 correctly at least once and solve for k
[A1] Obtain answer k = 15
[B1] State answer 1 - 2i
[M1] Carry out a complete method for finding a quadratic factor with zeros 1 + 2i and 1 – 2i
[A1] Obtain x² – 2x + 5
[A1] Obtain root –³⁄₂ , or equivalent, via division or inspection
[B1] Show a circle with centre 1 + 2i
[B1] Show a circle with radius 1
[M1] Carry out a complete method for calculating the least value of arg z
[A1] Obtain answer 0.64

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