A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<p><strong>Do not use a calculator for this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2\mathbf{i}$, ${{z}_{2}}=2{{\mathrm{e}}^{\frac{\pi }{6}\,\mathbf{i}}}$ and ${{z}_{3}}=\frac{1}{7}\left( \cos \frac{2}{3}\pi +\mathbf{i}\sin \frac{2}{3}\pi \right)$. Find $\frac{1}{{{z}_{3}}}\left( {{z}_{2}}-{{z}_{1}} \right)$ in the form $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">The complex number $z$ satisfies the equation $2{{z}^{3}}+a{{z}^{2}}+18z-9=0$, where $a$ is a real number. It is given that one root is of the form $z=k\,\mathbf{i}$, where $k$ is real and negative. Find the values of $a$ and $k$, and the other roots of the equation.</span></div><p style="text-align: right;">[6]</p></li></ol>

**Do not use a calculator for this question.**

The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2\mathbf{i}$, ${{z}_{2}}=2{{\mathrm{e}}^{\frac{\pi }{6}\,\mathbf{i}}}$ and ${{z}_{3}}=\frac{1}{7}\left( \cos \frac{2}{3}\pi +\mathbf{i}\sin \frac{2}{3}\pi \right)$. Find $\frac{1}{{{z}_{3}}}\left( {{z}_{2}}-{{z}_{1}} \right)$ in the form $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.

The complex number $z$ satisfies the equation $2{{z}^{3}}+a{{z}^{2}}+18z-9=0$, where $a$ is a real number. It is given that one root is of the form $z=k\,\mathbf{i}$, where $k$ is real and negative. Find the values of $a$ and $k$, and the other roots of the equation.

Complex Numbers - General

2021 HCI P1 Q5

Complex Numbers

<p>Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.</p>

Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.

<p>The equation ${{z}^{4}}-4{{z}^{3}}+a{{z}^{2}}-20z+25=0,\mathrm{ }a\in \mathbb{R},$ has a root $k\mathbf{i}$, where&nbsp;$k$ is a real number.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain clearly why there is more than one possible value for&nbsp;$k$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the possible exact values of $k$ and show that&nbsp;$a=10$.</span></div><div class="list-item-paragraph">Hence find the roots of the above equation.<span class="inline-marks">[7]</span></div></li><li><div class="list-item-paragraph">Deduce the roots of the following equations in the form $x+\mathbf{i}y,$ where&nbsp;$x$ and $y$ are real,</div><ol class="abc-list"><li><div class="list-item-paragraph">${{w}^{4}}+4\mathrm{i}{{w}^{3}}-10{{w}^{2}}-20\mathrm{i}w+25=0,$</div></li><li><div class="list-item-paragraph">$25{{v}^{4}}-20{{v}^{3}}+10{{v}^{2}}-4v+1=0.$<span class="inline-marks">[4]</span></div></li></ol></li></ol>

The equation ${{z}^{4}}-4{{z}^{3}}+a{{z}^{2}}-20z+25=0,\mathrm{ }a\in \mathbb{R},$ has a root $k\mathbf{i}$, where $k$ is a real number.

Explain clearly why there is more than one possible value for $k$.

Find the possible exact values of $k$ and show that $a=10$.

Deduce the roots of the following equations in the form $x+\mathbf{i}y,$ where $x$ and $y$ are real,

2013 DHS P2 Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="i-list"><li><div class="list-item-paragraph">One of the roots of the equation $2{{z}^{3}}-9{{z}^{2}}+30z+b=0$ is $1+a\mathrm{i}$, where $a$ and $b$ are non-zero real numbers. Find the values of $a$ and $b$ are non-zero real numbers. Find the values of $a$ and $b$ and the roots of this equation.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Hence solve the equation $b{{z}^{3}}+30{{z}^{2}}-9z+2=0$.</div><p style="text-align: right;">[2]</p></li></ol>

**Do not use a calculator in answering this question.**

2021 YIJC P1 Q6

<p>Verify that $z=\mathrm{i}$ is a root of the equation</p>
<p style="text-align: center">${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-2z+5=0$.</p><p style="text-align: right;">[1]</p>
<p>Hence determine the other roots.</p><p style="text-align: right;">[4]</p>

Verify that $z=\mathrm{i}$ is a root of the equation

${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-2z+5=0$.

N2005 P2 Q1

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The point $P$ represents the complex number $z=-2+b\,\mathbf{i}$, where $b>0$. The line $OP$ makes an angle of ${{45}^{{}^\circ }}$ with the horizontal.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the value of $b$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence find the modulus and argument of $\frac{1}{z}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show on an Argand diagram the points $P$, $Q$ and $R$ representing the complex numbers $z$, $\frac{1}{z}$ and $z+\frac{1}{z}$ respectively. Indicate on your diagram the value of $\angle PRQ$.</div><p style="text-align: right;">[3]</p></li></ol>

The point $P$ represents the complex number $z=-2+b\,\mathbf{i}$, where $b>0$. The line $OP$ makes an angle of ${{45}^{{}^\circ }}$ with the horizontal.

2021 ASRJC JC2 MYA P1 Q3

<p>Find the roots of $z^{2} + 2z + 2 = 0$.</p>

<p>Simplify:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{\mathbf{i}}^{32}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{69}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{17}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{111}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{86}}$</div></li></ol>

<p>The complex numbers $z$ and $w$ have moduli $k$ and $3{{k}^{2}}$ respectively and arguments $\alpha $ and $4\alpha $ respectively, where $k$ is a positive constant and $-\frac{\pi }{7}<\alpha \le \frac{\pi }{7}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $\frac{{{z}^{3}}}{{{w}^{*}}}$ in the form $r{{\mathrm{e}}^{\mathbf{i}\theta }}$ where $r>0$ and $-\pi <\theta \le \pi $.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that $\alpha =\frac{\pi }{21}$. Find the integer values of $n$ such that ${{\left( \frac{{{z}^{3}}}{{{w}^{*}}} \right)}^{n}}$ is real.</span></div><p style="text-align: right;">[2]</p></li></ol>

The complex numbers $z$ and $w$ have moduli $k$ and $3{{k}^{2}}$ respectively and arguments $\alpha $ and $4\alpha $ respectively, where $k$ is a positive constant and $-\frac{\pi }{7}<\alpha \le \frac{\pi }{7}$.

Express $\frac{{{z}^{3}}}{{{w}^{*}}}$ in the form $r{{\mathrm{e}}^{\mathbf{i}\theta }}$ where $r>0$ and $-\pi <\theta \le \pi $.

It is given that $\alpha =\frac{\pi }{21}$. Find the integer values of $n$ such that ${{\left( \frac{{{z}^{3}}}{{{w}^{*}}} \right)}^{n}}$ is real.

2021 ASRJC P2 Q1

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $z$ is given by $z=1-\sqrt{3}\mathrm{i}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $1+z+{{z}^{2}}+{{z}^{3}}+{{z}^{4}}+{{z}^{5}}$ is purely imaginary.</span></div><p style="text-align: right;">[4]</p></li></ol>
<p>The complex number $w$ is given by $w=\frac{\mathrm{i}\sqrt{3}}{{{z}^{n}}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that $w$ cannot be a real number for all integer values of $n$.</span></div><p style="text-align: right;">[4]</p></li></ol>

The complex number $z$ is given by $z=1-\sqrt{3}\mathrm{i}$.

Express $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.

Show that $1+z+{{z}^{2}}+{{z}^{3}}+{{z}^{4}}+{{z}^{5}}$ is purely imaginary.

The complex number $w$ is given by $w=\frac{\mathrm{i}\sqrt{3}}{{{z}^{n}}}$.

Show that $w$ cannot be a real number for all integer values of $n$.

2022 TJC J2 MYE P1 Q6

<p>Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.</p>

Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.

<p>Find the modulus and argument of ${{\left( \frac{3-i}{1-2i} \right)}^{22}}$ in exact form and hence write the complex number in the exponential form.</p>

Find the modulus and argument of ${{\left( \frac{3-i}{1-2i} \right)}^{22}}$ in exact form and hence write the complex number in the exponential form.

Properties from Exponential or Trigonometric form Q5

<p>Find $\frac{1+2\mathbf{i}}{3-4\mathbf{i}}$.</p>

Find $\frac{1+2\mathbf{i}}{3-4\mathbf{i}}$.

Worked Example 1

<ol class="abc-list"><li><div class="list-item-paragraph">The complex number $z$ is given by $z=x+y\mathbf{i}$, where $x$ and $y$ are non-zero real numbers. Given that $\left| z \right|=1$, find the possible values of $z$ for which $\frac{{{\left( {{z}^{2}} \right)}^{*}}}{z}$ is real.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Without the use of a calculator, find the roots of the equation ${{z}^{2}}=33+56\mathbf{i}$, expressing your answer in cartesian form $x+\mathbf{i}y$ where $x$ and $y$ are real.</div><p style="text-align: right;">[4]</p><div class="list-item-paragraph">Hence, find in cartesian form, the roots of the equation ${{w}^{2}}=-33+56\mathbf{i}$.</div><p style="text-align: right;">[2]</p></li></ol>

2022 TMJC P1 Q9

<p>Given that the equation ${{z}^{3}}-4{{z}^{2}}+6z-4=0$ has a root $1-\mathbf{i}$, find the remaining roots of the equation.</p>

Given that the equation ${{z}^{3}}-4{{z}^{2}}+6z-4=0$ has a root $1-\mathbf{i}$, find the remaining roots of the equation.

Worked Example 2

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The roots of the equation ${{z}^{2}}=18\,\mathbf{i}$ are ${{z}_{1}}$ and ${{z}_{2}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{z}_{1}}$ and ${{z}_{2}}$ in cartesian form $x+y\mathbf{i}$, showing your working clearly.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The complex number $w$ is given by $-1-\mathbf{i}\sqrt{3}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the complex conjugate of $\frac{w}{{{z}^{2}}}$ in the form $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</div><p style="text-align: right;">[4]</p></li></ol>

The roots of the equation ${{z}^{2}}=18\,\mathbf{i}$ are ${{z}_{1}}$ and ${{z}_{2}}$.

The complex number $w$ is given by $-1-\mathbf{i}\sqrt{3}$.

2023 MI P1 Q7

<p>The complex number $z$ is given by $z=r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $0\le \theta \le \frac{1}{2}\pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that $w=\left( 1-\mathrm{i}\sqrt{3} \right)z$, find $\left| w \right|$ in terms of $r$ and $\arg w$ in terms of $\theta $.</div><p style="text-align: right;">[2]</p></li></ol>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Given that $\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} \right)=\pi $, find $\theta $.</div><p style="text-align: right;">[3]</p></li></ol>

The complex number $z$ is given by $z=r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $0\le \theta \le \frac{1}{2}\pi $.

N2013 P1 Q8

<p style="margin-bottom: 1.5rem">A number of the form $z=x+y\mathbf{i}$, where&nbsp;$x$, $y\in \mathbb{R}$ is called a <strong>complex number</strong>. <br>Real part, $\mathbf{Re}\left( \mathbf{z} \right)=x$ ; Imaginary part, $\mathbf{Im}\left( \mathbf{z} \right)=y$. Find $\mathbf{Re}\left( \mathbf{z} \right)$and $\mathbf{Im}\left( \mathbf{z} \right)$ in the above cases. </p>
<p>Simplify:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{\mathbf{i}}^{3}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{4}}$</div></li><li><div class="list-item-paragraph">${{\mathbf{i}}^{5}}$</div></li></ol>

A number of the form $z=x+y\mathbf{i}$, where $x$, $y\in \mathbb{R}$ is called a **complex number**. 
Real part, $\mathbf{Re}\left( \mathbf{z} \right)=x$ ; Imaginary part, $\mathbf{Im}\left( \mathbf{z} \right)=y$. Find $\mathbf{Re}\left( \mathbf{z} \right)$and $\mathbf{Im}\left( \mathbf{z} \right)$ in the above cases. 

<p>Find the roots of $z^{2} - 2z + 5 = 0$.</p>

<p>Let&nbsp;$z = x + y\mathbf{i}$ and $k$ is a real constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{({{z}^{*}})}^{*}}=z$</div></li><li><div class="list-item-paragraph">$z+{{z}^{*}}=2\mathrm{Re}(z)$</div></li><li><div class="list-item-paragraph">$z-z*=\left(2\mathrm{Im}(z)\right)i$</div></li><li><div class="list-item-paragraph">$z{{z}^{*}}={{x}^{2}}+{{y}^{2}}$</div></li><li><div class="list-item-paragraph">${{({{z}{1}}\pm {{z}{2}})}^{}}={{z}{1}}^{}\pm {{z}{2}}^{*}$</div></li><li><div class="list-item-paragraph">${{(kz)}^{}}=k{{z}^{}}$</div></li><li><div class="list-item-paragraph">${{({{z}_{1}}{{z}_{2}})}^{*}}={{z}_{1}}^{*}{{z}_{2}}^{*}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)}^{*}}=\frac{{{z}_{1}}^{*}}{{{z}_{2}}^{*}}$</div></li></ol>

Let $z = x + y\mathbf{i}$ and $k$ is a real constant.

Math Question Bank