A-LevelMathematicsComplex numbers9709/31 · Oct/Nov 2021 · Q1012 Marks

The complex number 1 + 2i is denoted by u. The polynomial 2x³ + ax² + 4x + b, where a and b are real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.

(a)Find the values of a and b.[4]
(b)State a second complex root of this equation.[1]
(c)Find the real factors of p(x).[2]
(d)(i)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z - u| < √5 and arg z ≤ ¼π.[4]
(d)(ii)Find the least value of Im z for points in the shaded region. Give your answer in an exact form.[1]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[M1] Substitute `1 + 2i` in the polynomial and attempt expansions of `x²` and `x³`
[M1] Equate real and/or imaginary parts to zero
[A1] Obtain `a = -1`
[A1] Obtain `b = 15`
[B1] State second root `1 - 2i`
[B1] State the quadratic factor `x² - 2x + 5`
[B1] State the linear factor `2x + 3`
[B1] Show a circle with centre `1 + 2i`
[B1] Show circle passing through the origin
[B1] Show the half line `y = x` in the first quadrant (accept chord of circle)
[B1] Shade the correct region on a correct diagram
[B1] State answer `2 - √5`

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