For the whole of this question, calculator use is not allowed. The complex number with modulus $1$ and argument $\tfrac{1}{3}\pi$ is called $w$. The complex number $1 + 2i$ is called $u$. The complex number $v$ satisfies $|v| = 2|u|$ and $\arg v = \arg u + \tfrac{1}{3}\pi$.
(i)[1]
Write $w$ in the form $x + iy$, where $x$ and $y$ are exact real numbers.
(ii)[2]
Sketch an Argand diagram that shows the points for $u$ and $v$.
(iii)[4]
Explain why $v$ may be written as $2uw$. Hence determine $v$, expressing your answer in the form $a + ib$, where $a$ and $b$ are exact real numbers.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Hence $w=\frac{1}{2}+\frac{\sqrt{3}}{2}i$” …