A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p class="paragraph-header">Do not use a calculator in answering this question.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">One of the roots of the equation $\omega^4 - 2\omega^3 + 10\omega^2 + p\omega + q = 0$, where&nbsp;$p$ and $q$ are real, is $2 + 3\mathrm{i}$. Find the values of $p$ and $q$ and the other roots of the equation.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The complex numbers $u$ and $v$ are such that&nbsp;$u = -\sqrt{2} + \mathrm{i}\sqrt{2}$, and $\left| v \right| = 3$ and $\arg v = \theta$, where&nbsp;$0 &lt; \theta &lt; \frac{\pi}{4}$. The points $A$, $B$ and $C$ represents $u$, $v$ and $u + v$ respectively on an Argand diagram.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Find the modulus and argument of $u$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Sketch the points $A$, $B$ and $C$ on an Argand diagram.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-iii" data-list-item-path="item-b/item-b-iii"><div class="list-item-paragraph"><p>By finding the angle $OAC$ in terms of $\theta$ or otherwise, show that&nbsp;$\left| u + v \right|^2 = a + b\cos(\theta + K)$, where&nbsp;$a$, $b$ and $K$ are constants to be determined.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

Do not use a calculator in answering this question.

Equating Real and Imaginary Parts

Modulus and Argument of Complex Numbers

Applying the Conjugate Root Theorem

Geometric Problems on the Argand Diagram

Representing Complex Numbers on an Argand Diagram

2025 ACJC P1 Q10

Complex Numbers

<p>The complex number $w$ is given by $1 + 2\mathrm{i}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find $w^3$ in the form $x + \mathrm{i}y$, showing your working.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Given that&nbsp;$w$ is a root of the equation $az^3 + 5z^2 + 17z + b = 0$, find the values of the real numbers $a$ and $b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using these values of $a$ and $b$, find all the roots of this equation in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $w$ is given by $1 + 2\mathrm{i}$.

Arithmetic with Complex Numbers in Cartesian Form

N2013 P1 Q4

<p>A graphic calculator is not to be used in answering this question.</p>
<ol class="i-list"><li data-list-item-id="item-1770891051005-db9u4khhg" data-list-item-path="item-1770891051005-db9u4khhg"><div class="list-item-paragraph">It is given that&nbsp;${{z}_{1}}=1+\sqrt{3}\mathbf{i}$. Find the value of ${{z}_{1}}^{3}$, showing clearly how you obtain your answer.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1770891051641-3s5k7iqvi" data-list-item-path="item-1770891051641-3s5k7iqvi"><div class="list-item-paragraph">Given that&nbsp;$1+\sqrt{3}\mathbf{i}$ is a root of the equation $2{{z}^{3}}+a{{z}^{2}}+bz+4=0$, find the values of the real numbers $a$ and $b$.	<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1770891052242-oerr5ud5g" data-list-item-path="item-1770891052242-oerr5ud5g"><div class="list-item-paragraph"><p>For these values of $a$ and $b$, solve the equation in part <strong>(ii)</strong>, and show all the roots on an Argand diagram.</p><span class="inline-marks">[4]</span></div></li></ol>

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

N2008 P1 Q8

<p>If&nbsp;$\frac{a}{1+\mathbf{i}}-\frac{b}{2-\mathbf{i}}=5\mathbf{i}$, where&nbsp;$a$ and $b$ are real, find the values of $a$ and $b$.</p>

If $\frac{a}{1+\mathbf{i}}-\frac{b}{2-\mathbf{i}}=5\mathbf{i}$, where $a$ and $b$ are real, find the values of $a$ and $b$.

<p><strong>Do not use a calculator in answering this question.</strong></p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="300" style="width:300px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-content"><span class="list-item-text">The point $P$ on the Argand diagram represents the complex number $w$ given by $w = u + \mathrm{i}v$, where&nbsp;$u$ and $v$ are real and positive.</span></div><ol class="i-list"><li data-list-item-id="item-a-4-i" data-list-item-path="item-a-4/item-a-4-i"><div class="list-item-content"><span class="list-item-text">Explain algebraically why $\arg(kw) = \arg(w)$ for any real constant $k &gt; 1$.</span><span class="inline-marks">[1]</span></div></li></ol><div class="list-item-after-content"><p>Points $Q$ and $R$ represent the complex numbers $kw$ and $\mathrm{i}kw$, where $k$ is a real constant and $k &gt; 1$.</p></div><div class="list-item-after-content"><ol class="i-list" start="2"><li data-list-item-id="item-1784229731176-fn0de3g18" data-list-item-path="item-1784229731176-fn0de3g18"><div class="list-item-paragraph"><p>On the same Argand diagram in the Printed Answer Booklet, plot the points $Q$ and $R$. Show clearly the geometrical relationship between the points $P$, $Q$ and $R$.</p><span class="inline-marks">[2]</span></div></li></ol></div></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-paragraph">In another Argand diagram, the points $A$, $B$ and $C$ represent the complex numbers $z$, $\mathrm{f}(z)$ and $\mathrm{f}(\mathrm{f}(z))$ respectively, where&nbsp;$\mathrm{f}(z) = z^2 - 2z$.</div><ol class="i-list"><li data-list-item-id="item-b-4-i" data-list-item-path="item-b-4/item-b-4-i"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\mathrm{f}(\mathrm{f}(z)) - \mathrm{f}(z) = z(z - 2)(z - 3)(z + 1)$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-4-text" data-list-item-path="item-b-4/item-b-4-text"><div class="list-item-paragraph">It is given that&nbsp;$ABC$ is a right-angled triangle, described in an anticlockwise sense, with a right angle at $B$, and $BC = mBA$, where&nbsp;$m$ is a positive real constant.</div><div class="list-item-paragraph"><p>By considering $\frac{\mathrm{f}(\mathrm{f}(z)) - \mathrm{f}(z)}{\mathrm{f}(z) - z}$, or otherwise, show that $(z - 2)(z + 1) = mi$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-4-iii" data-list-item-path="item-b-4/item-b-4-iii"><div class="list-item-content"><span class="list-item-text">In the case where&nbsp;$z = x + 2i$, where&nbsp;$x$ is a positive real number, find $x$ and $m$. Hence obtain the complex number represented by the point $B$.</span><span class="inline-marks">[5]</span></div></li></ol></li></ol>

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

The point $P$ on the Argand diagram represents the complex number $w$ given by $w = u + \mathrm{i}v$, where $u$ and $v$ are real and positive.

In another Argand diagram, the points $A$, $B$ and $C$ represent the complex numbers $z$, $\mathrm{f}(z)$ and $\mathrm{f}(\mathrm{f}(z))$ respectively, where $\mathrm{f}(z) = z^2 - 2z$.

Geometrical Transformations of Complex Numbers

2025 RI P1 Q8

<p>If&nbsp;$\frac{p}{2 - \mathbf{i}} + \frac{q}{1 + 3\mathbf{i}} = 7\mathbf{i}$, where&nbsp;$p$ and $q$ are real, find the values of $p$ and $q$.</p>

If $\frac{p}{2 - \mathbf{i}} + \frac{q}{1 + 3\mathbf{i}} = 7\mathbf{i}$, where $p$ and $q$ are real, find the values of $p$ and $q$.

<p>The complex numbers $z$ and $w$ where $w\ne 0$ satisfy the relation</p>
<p style="text-align: center">$2z=\left| w \right|+1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">It is given that $a$ is a real number and that $a$ and $z$ satisfy the equation $2{{z}^{3}}-5{{z}^{2}}+2z+\left( a+3 \right)\mathbf{i}=0$.</div><div class="list-item-paragraph">Explain, with justification, why $a=-3$ and that the only possible value of $z$is $2$.</div><p style="text-align: right;">[6]</p></li></ol>
<p>It is given that $\arg w=\frac{\pi }{3}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Express the complex number $w$ in the form $p+q\mathbf{i}$ where $p$ and $q$are in non-trigonometric form.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the least integer $n$ such that $\left| {{w}^{n}} \right|>\mathrm{20212021}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the least positive integer $k$ such that ${{w}^{k}}$ is a positive real number.</div><p style="text-align: right;">[2]</p></li></ol>

The complex numbers $z$ and $w$ where $w\ne 0$ satisfy the relation

It is given that $\arg w=\frac{\pi }{3}$.

Cartesian, Trigonometric and Exponential Forms

Properties of Trigonometric and Exponential Forms

Solving Complex Equations

2021 NYJC P1 Q9

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $z$ satisfies the equation</p>
<p style="text-align: center">${{z}^{2}}-\left( 4+\mathbf{i} \right)z+2\left( \mathbf{i}-t \right)=0$</p>
<p>where&nbsp;$t$ is a real number. It is given that one root is of the form $k-k\,\mathbf{i}$, where&nbsp;$k$ is real and positive. Find $t$ and $k$, and the other root of the equation.<span class="inline-marks">[7]</span></p>

The complex number $z$ satisfies the equation

${{z}^{2}}-\left( 4+\mathbf{i} \right)z+2\left( \mathbf{i}-t \right)=0$

where $t$ is a real number. It is given that one root is of the form $k-k\,\mathbf{i}$, where $k$ is real and positive. Find $t$ and $k$, and the other root of the equation.

Finding Roots of Equations with Complex Coefficients

2022 VJC P1 Q5

<p>The complex number $z$ satisfies the equation</p>
<p style="text-align: center">${{z}^{2}}\left( 2+\mathbf{i} \right)-8\mathbf{i}\,z+t=0$</p>
<p>where $t$ is a real number. It is given that one of the root is of the form $k+k\,\mathbf{i}$, where $k$ is real and non-zero. Find $t$ and $k$, and the other root of the equation. </p><p style="text-align: right;">[8]</p>

${{z}^{2}}\left( 2+\mathbf{i} \right)-8\mathbf{i}\,z+t=0$

where $t$ is a real number. It is given that one of the root is of the form $k+k\,\mathbf{i}$, where $k$ is real and non-zero. Find $t$ and $k$, and the other root of the equation. 

N2020 P1 Q6

<p><strong>Do not use a graphing calculator in answering this question.</strong></p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$w=\mathrm{ }1\mathrm{ }+\mathrm{ }\mathbf{i}$ is a root of the equation</div><div class="list-item-paragraph" style="text-align: center">${{w}^{3}}+\mathrm{ }3{{w}^{2}}+pw+q=\mathrm{ }0$,</div><div class="list-item-paragraph">find the values of the real numbers $p$ and $q$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">For these values of $p$ and $q$, solve the equation in part <strong>(i)</strong>.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

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

2013 TPJC P2 Q3

<p>The complex number $z$ is given by $z=k+\mathbf{i}$ where $k$ is a non-zero real number.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the possible values of $k$ if $z=k+\mathbf{i}$ satisfies the equation ${{z}^{3}}-\mathbf{i}{{z}^{2}}-2z-4\mathbf{i}=0$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">For the complex number $z$ found in part <strong>(i)</strong> for which $k>0$, find the smallest integer value of $n$ such that $\left| {{z}^{n}} \right|>100$ and ${{z}^{n}}$ is real.</div><p style="text-align: right;">[3]</p></li></ol>

The complex number $z$ is given by $z=k+\mathbf{i}$ where $k$ is a non-zero real number.

Modulus, Argument, and Operations on Complex Numbers

2016 ACJC P1 Q8

<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>

Solving Complex Equations by Substitution

2021 YIJC P1 Q6

<ol class="abc-list"><li><div class="list-item-paragraph">One root of the equation ${{z}^{4}}+2{{z}^{3}}+a{{z}^{2}}+bz+50=0$, where&nbsp;$a$ and $b$ are real, is $z=1+3\mathbf{i}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$a=7$ and $b=30$ and find the other roots of the equation.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Deduce the roots of the equation ${{w}^{4}}-2\mathbf{i}{{w}^{3}}-7{{w}^{2}}+30\mathbf{i}w+50=0$<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2017 RI P1 Q9

<p><strong>Do not use a graphing calculator for this question.</strong></p>
<ol class="i-list"><li data-list-item-id="item-1759833015673-huc7wg2zj"><div class="list-item-paragraph">The complex number $a+\mathbf{i}$ is a root of the equation ${{z}^{2}}-2\sqrt{2}z+b=0$ where&nbsp;$a$ and $b$ are positive real constants. Find the exact values of $a$ and $b$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{f}(z)={{z}^{6}}-3{{z}^{4}}+11{{z}^{2}}-9$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759833030035-xrn9c8zyu"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}(z)=\mathrm{f}(-z)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759833038698-38vjrf4zc"><div class="list-item-paragraph">It is given that&nbsp;$a+\mathbf{i}$ is a root of the equation $\mathrm{f}(z)=0$. Using the results found in parts <strong>(i)</strong> and <strong>(ii)</strong>, show that&nbsp;$\mathrm{f}(z)$ can be factorised into $3$ quadratic factors with real coefficients, leaving your answer in exact form.<span class="inline-marks">[4]</span></div></li></ol>

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

It is given that $\mathrm{f}(z)={{z}^{6}}-3{{z}^{4}}+11{{z}^{2}}-9$.

2022 HCI P1 Q8

<ol class="abc-list"><li><div class="list-item-paragraph">It is given that $z=1+\sqrt{3}\,\mathbf{i}$ is a root of the equation $3{{z}^{3}}+a{{z}^{2}}+bz-8=0$, where $a$ and $b$ are real numbers. Find the exact values of $a$ and $b$ and hence solve the equation completely, giving all the roots in exact form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Using Euler’s formula that ${{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $, show that ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{4}}=\cos 4\theta +\mathbf{i}\sin 4\theta $.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using result shown in <strong>(i)</strong>, by comparing the real parts, show that $\cos 4\theta ={{\cos }^{4}}\theta -6{{\cos }^{2}}\theta {{\sin }^{2}}\theta +{{\sin }^{4}}\theta $.</div><div class="list-item-paragraph">Obtain an expression for $\sin 4\theta $ in terms of $\cos \theta $ and $\sin \theta $. Hence show that $\tan 4\theta =\frac{4\tan \theta -4{{\tan }^{3}}\theta }{1-6{{\tan }^{2}}\theta +{{\tan }^{4}}\theta }$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence, find the possible values of $\tan \theta $ given that $\tan 4\theta =4$, leaving your answers correct to $3$ significant figures.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

De Moivre's Theorem

2021 SAJC P1 Q7

<p>Do not use a calculator in answering this question.</p>
<ol class="abc-list"><li data-list-item-id="item-1759655845946-6r6erg2qr" data-list-item-path="item-1759655845946-6r6erg2qr"><div class="list-item-paragraph">Verify that&nbsp;$-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="n2016-p1-q7-b" data-list-item-path="n2016-p1-q7-b"><div class="list-item-paragraph">The equation $z^3-5z^2+16z+k=0$, where&nbsp;$k$ is a real constant, has a root $z=1+a\mathrm i$, where&nbsp;$a$ is a positive real constant. Find the values of $a$ and $k$, showing your working.<span class="inline-marks">[5]</span></div></li></ol>

The equation $z^3-5z^2+16z+k=0$, where $k$ is a real constant, has a root $z=1+a\mathrm i$, where $a$ is a positive real constant. Find the values of $a$ and $k$, showing your working.

N2016 P1 Q7

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-1759608940556-j1pi5k8v5" data-list-item-path="item-1759608940556-j1pi5k8v5"><div class="list-item-paragraph">The complex number $z$ has modulus $1$ and argument $\frac{\pi }{6}$. The complex number $w$ is such that&nbsp;$w=1+\sqrt{3}\,\mathbf{i}$. Indicate the points $A$, $B$ and $C$ representing the complex numbers $z$, $w$ and $z+w$ on an Argand diagram. Hence find ${{\left| z+w \right|}^{2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759608941215-w93ngix6c" data-list-item-path="item-1759608941215-w93ngix6c"><div class="list-item-paragraph">It is given that&nbsp;$x=1+\sqrt{3}\,\mathbf{i}$ is one of the roots of the equation $3{{x}^{3}}+a{{x}^{2}}+16x+b=0$, where&nbsp;$a$ and $b$ are real. Find the other roots of the equation and the values of $a$ and $b$.<span class="inline-marks">[5]</span></div></li></ol>

2025 ACJC LCP2B Q2

<p>Show that the equation</p>
<p style="text-align: center">${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-8z+8=0$</p>
<p>has a root of the form $k\mathrm{i}$, where&nbsp;$k$ is real.<span class="inline-marks">[3]</span></p>
<p>Hence solve the equation</p>
<p style="text-align: center">${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-8z+8=0$.<span class="inline-marks">[3]</span></p>

has a root of the form $k\mathrm{i}$, where $k$ is real.

${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-8z+8=0$.

N2006 P1 Q6

<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.

2021 HCI P1 Q5

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">One of the roots of the equation ${{x}^{3}}+p{{x}^{2}}+6x+q=0$ where $p$ and $q$ are real, is $5-\mathrm{i}$. Find the other roots of the equation and the values of $p$ and $q$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text"><strong>Do not use a calculator in answering this part.</strong></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The complex number $z$ has modulus $2\sqrt{3}$ and argument $-\frac{\pi }{3}$. Find $z-2\sqrt{3}$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $. Give $r$ and $\theta $ in exact form.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the three smallest positive whole number values of $n$ for which $\frac{{{\left( z-2\sqrt{3} \right)}^{*}}}{{{\left( z-2\sqrt{3} \right)}^{n}}}$ is a real number.</span></div><p style="text-align: right;">[3]</p></li></ol></li></ol>

One of the roots of the equation ${{x}^{3}}+p{{x}^{2}}+6x+q=0$ where $p$ and $q$ are real, is $5-\mathrm{i}$. Find the other roots of the equation and the values of $p$ and $q$.

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

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