Mathematics 9709 · AS & A Level · Complex numbers

Complex numbers — practice question

Let the complex number $1 + 2i$ be called $u$. The polynomial $p(x)$ is $2x^3 + ax^2 + 4x + b$, where $a$ and $b$ are real constants. It is known that $u$ is a root of the equation $p(x) = 0$.
(a)[4]

Determine the values of $a$ and $b$.

(b)[1]

State one further complex root of this equation.

(c)[2]

Find the real factors into which $p(x)$ can be split.

(d(i))[4]

On an Argand diagram sketch, shade the set of points representing complex numbers $z$ that satisfy $|z - u| \leq \sqrt{5}$ and $\arg z \leq \frac{1}{4}\pi$.

(d(ii))[1]

Find the least value of $\operatorname{Im} z$ for points in the shaded region. Give your answer exactly.

Worked solution & mark scheme

This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Replace $x$ with $1+2i$ in the polynomial and expand $x^2$ and $x^3$.” …

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