# Complex Numbers

###### 5 Essential Questions

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• Q1
• Q2
• Q3
• Q4
• Q5
##### 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
• (i)
• (ii)
• (iii)   • (i)
• (ii)
• (iii)   ##### 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
• (i)
• (ii)  • (i)
• (ii)  ##### 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   