 # Complex Numbers

Complex numbers are numbers with a real part and an imaginary part, $\text{i}$. Complex numbers are generally written in the form $a+b\text{i}$ where the imaginary number $\text{i}$ is equal to $\sqrt{-1}$.  Complex numbers are the name mathematicians give to the generalization of real numbers that allow us to take square roots of negative numbers. This allow us to solve certain equations that had no real solution.

##### 2017 NYJC P1 Q3

Do not use a calculator in answering this question.

(i)

Explain why the equation ${{z}^{3}}+a{{z}^{2}}+az+7=0$ cannot have more than two non-real roots, where $a$ is a real constant.



(i) Explain why the equation ${{z}^{3}}+a{{z}^{2}}+az+7=0$ cannot have more than two non-real roots, where $a$ is a real constant.



(ii)

Given that $z=-7$ is a root of the equation in (i), find the other roots, leaving your answers in the form $r{{e}^{\text{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi$ .



(ii) Given that $z=-7$ is a root of the equation in (i), find the other roots, leaving your answers in the form $r{{e}^{\text{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi$ .



(iii)

Hence, solve the equation $\text{i}{{z}^{3}}+8{{z}^{2}}-8\text{i}z-7=0$, leaving your answers in the form $r{{e}^{\text{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi$ .



(iii) Hence, solve the equation $\text{i}{{z}^{3}}+8{{z}^{2}}-8\text{i}z-7=0$, leaving your answers in the form $r{{e}^{\text{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi$ .



##### Suggested Handwritten and Video Solutions      ##### 2009 NJC P1 Q9 (b) Modified

Write down the real root of

${{w}^{3}}+8=0$,

Where $w$ is a complex number.
Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form.
Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$.
Determine the roots of $z$ such that ${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.

##### Suggested Handwritten and Video Solutions ##### 2014 JJC P1 Q10 (a)

Given that the complex number $z=(1+\text{i})t+\frac{1-\text{i}}{t}$ is represented by the point $P$ on Argand diagram where $t$ is a non-zero real constant. Find the Cartesian equation of the locus of the point $P$.



##### Suggested Handwritten and Video Solutions  ##### 2020 CJC P1 Q2

The complex number $z$ has modulus $2$ and argument $\frac{\pi }{8}$ . It is also given that $w=1+\text{i}$.

(i)

Given that $n$ is an integer, find $\frac{{{z}^{n}}}{w*}$ in terms of $n$, giving your answer in the form $r{{e}^{i\theta }}$ .



(i) Given that $n$ is an integer, find $\frac{{{z}^{n}}}{w*}$ in terms of $n$, giving your answer in the form $r{{e}^{i\theta }}$ .



(ii)

Hence, find the smallest two positive integers $n$ such that $\frac{{{z}^{n}}}{w*}$ is real and negative.



(ii) Hence, find the smallest two positive integers $n$ such that $\frac{{{z}^{n}}}{w*}$ is real and negative.



##### Suggested Handwritten and Video Solutions    ##### 2015 RI P1 Q6 (b) Modified

Prove that $\frac{1+\sin \frac{3\pi }{8}+\text{i}\cos \frac{3\pi }{8}}{1+\sin \frac{3\pi }{8}-\text{i}\cos \frac{3\pi }{8}}=\cos \frac{\pi }{8}+\text{i}\sin \frac{\pi }{8}$.

##### Suggested Handwritten and Video Solutions    