A graphic calculator is not to be used in answering this question. Two complex numbers $p$ and $q$ a

Question

A graphic calculator is not to be used in answering this question.

Two complex numbers $p$ and $q$ are given by $p=1-\mathrm{i}$ and $q=-1+\sqrt{3}\mathrm{i}$, and $z=\frac{p}{q}$.

Express $z$ in the form $x+y\mathrm{i}$, where $x$ and $y$ are exact real values to be determined.[2]
By considering the moduli and arguments of $p$ and $q$, find the exact values of $\left| z \right|$ and $\arg z$, where $-\pi <\arg z\le \pi $.[4]
Hence, show that $\sin \left( \frac{11}{12}\pi \right)=\frac{\sqrt{6}-\sqrt{2}}{4}$.[3]

Accepted Answer

(i) $x=-\frac{1+\sqrt3}{4}$, $y=\frac{1-\sqrt3}{4}$ (ii) $|z|=\frac{\sqrt2}{2}$, $\arg(z)=-\frac{11\pi}{12}$ (iii) $\sin\left(\frac{11\pi}{12}\right)=\frac{\sqrt6-\sqrt2}{4}$

2013 TJC P1 Q5

2013 TJC P1 Q5

Video Solution:

Solution:

Solution locked