Mathematics 9709 · AS & A Level · Complex numbers

Complex numbers — practice question

For this question, a calculator must not be used. Write the complex number $1 - \sqrt{3}i$ as $u$.
(i)[2]

Find the modulus and the argument of $u$.

(ii)[2]

Show that $u^3 + 8 = 0$.

(iii)[4]

On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both inequalities $|z - u| \leq 2$ and $\operatorname{Re} z \geq 2$, where $\operatorname{Re} z$ is the real part of $z$.

(c(iii))[4]

On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both inequalities $|z - u| \leq 2$ and $\operatorname{Re} z \geq 2$, where $\operatorname{Re} z$ is the real part of $z$.

Worked solution & mark scheme

This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: State modulus $2$

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