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>It is given that \[u=\sqrt3+i,\qquad v=-1+\sqrt3\,i.\]</p>
<p>The points $P$, $Q$ and $R$ represent $u$, $v$ and $u+v$, respectively.</p>
<p>Give all arguments as principal arguments in radians.</p>
<ol class="abc-list"><li data-list-item-id="q002-part-0" data-list-item-path="q002-part-0"><div class="list-item-paragraph">Sketch $P$, $Q$ and $R$ on an Argand diagram with origin $O$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q002-part-1" data-list-item-path="q002-part-1"><div class="list-item-paragraph">Show that&nbsp;$OPRQ$ is a square.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q002-part-2" data-list-item-path="q002-part-2"><div class="list-item-paragraph">Hence, by considering the direction of the diagonal $OR$, show that \[\tan\frac{5\pi}{12}=2+\sqrt3.\]<span class="inline-marks">[3]</span></div></li></ol>

It is given that \[u=\sqrt3+i,\qquad v=-1+\sqrt3\,i.\]

The points $P$, $Q$ and $R$ represent $u$, $v$ and $u+v$, respectively.

Give all arguments as principal arguments in radians.

Hence, by considering the direction of the diagonal $OR$, show that \[\tan\frac{5\pi}{12}=2+\sqrt3.\]

Modulus and Argument of Complex Numbers

Using Complex Numbers to Deduce Trigonometric Values

Geometric Problems on the Argand Diagram

Representing Complex Numbers on an Argand Diagram

A square and the direction of its diagonal

Complex Numbers

<p>The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$.</p>
<p>Give all arguments as principal arguments in radians.</p>
<ol class="abc-list"><li data-list-item-id="q003-part-0" data-list-item-path="q003-part-0"><div class="list-item-paragraph">Find the modulus and principal argument of $wz$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q003-part-1" data-list-item-path="q003-part-1"><div class="list-item-paragraph">Express $w$ and $z$ in Cartesian form. Hence find the exact real and imaginary parts of $wz$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q003-part-2" data-list-item-path="q003-part-2"><div class="list-item-paragraph">Deduce that \[\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}.\]<span class="inline-marks">[2]</span></div></li></ol>

The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$.

Find the modulus and principal argument of $wz$.

Express $w$ and $z$ in Cartesian form. Hence find the exact real and imaginary parts of $wz$.

Deduce that \[\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}.\]

Arithmetic with Complex Numbers in Cartesian Form

Modulus, Argument, and Operations on Complex Numbers

A product giving an exact cosine value

<p>The complex number $w$ has modulus $\sqrt2$ and argument $-3\pi/4$, and the complex number $z$ has modulus $2$ and argument $-\pi/3$. Given that&nbsp;$w=-1-i$ and $z=1-\sqrt3\,i$.</p>
<ol class="abc-list"><li data-list-item-id="item-1784107715811-wpdirzdf3" data-list-item-path="item-1784107715811-wpdirzdf3"><div class="list-item-paragraph">Find the exact values of the modulus and argument of $wz$.</div></li><li data-list-item-id="item-1784107728124-98v5ysayo" data-list-item-path="item-1784107728124-98v5ysayo"><div class="list-item-paragraph">By first expressing $w$ and $z$ in the form $x+y\mathbf{i}$, find the exact real and imaginary part of $wz$.</div></li><li data-list-item-id="item-1784107803405-q1ru6n5ma" data-list-item-path="item-1784107803405-q1ru6n5ma"><div class="list-item-paragraph">Hence show that&nbsp;$\sin \frac{\pi }{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$ .</div></li></ol>

The complex number $w$ has modulus $\sqrt2$ and argument $-3\pi/4$, and the complex number $z$ has modulus $2$ and argument $-\pi/3$. Given that $w=-1-i$ and $z=1-\sqrt3\,i$.

Cartesian, Trigonometric and Exponential Forms

<p>The complex number $w$ has modulus $\sqrt{2}$ and argument $-\frac{3\pi}{4}$, and the complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$. Find the exact values of the modulus and argument of $wz$. By first expressing $w$ and $z$ in the form $x+y\mathrm{i}$, find the exact real and imaginary part of $wz$. Hence show that $\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$.</p>

The complex number $w$ has modulus $\sqrt{2}$ and argument $-\frac{3\pi}{4}$, and the complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$. Find the exact values of the modulus and argument of $wz$. By first expressing $w$ and $z$ in the form $x+y\mathrm{i}$, find the exact real and imaginary part of $wz$. Hence show that $\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$.

Proving Trigonometric Ratios by Complex Numbers Q3

<p>Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$. Deduce that&nbsp;$\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.</p>

Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$. Deduce that $\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.

Proving Trigonometric Ratios by Complex Numbers Q2

<p>Evaluate the product $(1+i)(1+\sqrt{3}i)$. Deduce that $\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.</p>

Evaluate the product $(1+i)(1+\sqrt{3}i)$. Deduce that $\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.

Proving Trigonometric Ratios by Complex Numbers Q1

<p>It is given that $-2+2i$ is a root of the equation</p>
<p style="text-align: center">${{z}^{3}}+a{{z}^{2}}+bz-16\sqrt{2}=0$,</p>
<p>where $a$ and $b$ are real numbers.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$ and the other two roots. Leave your answers in the exact form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">In an Argand diagram with origin $O$, the three roots are represented by points $A$, $B$ and $C$ where $A$ represents $-2+2i$ and $C$ represents the real root. Label these points on an Argand diagram, indicating clearly the modulus and argument of each root.</div><div class="list-item-paragraph">State also a geometrical relationship between $A$ and $B$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence, prove that</div><div class="list-item-paragraph" style="text-align: center">$\tan \frac{3\pi }{8}=1+\sqrt{2}$.</div><p style="text-align: right;">[3]</p></li></ol>

It is given that $-2+2i$ is a root of the equation

Applying the Conjugate Root Theorem

2025 TMJC P1 Q9

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-1760958279806-xq2aenzsd" data-list-item-path="item-1760958279806-xq2aenzsd"><div class="list-item-paragraph">The complex number $w$ is such that&nbsp;$w=a+ib$, where&nbsp;$a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by ${{w}^{*}}$. Given that</div><div class="list-item-paragraph" style="text-align: center">$w{{w}^{*}}=4-2i+2i{{w}^{*}}$,</div><div class="list-item-paragraph">find the two possible values of $w$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760958280459-pkr2zagji" data-list-item-path="item-1760958280459-pkr2zagji"><div class="list-item-paragraph">The complex number $z$ is given by $z=\frac{1-\sqrt{3}i}{-1+i}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760958315657-b2fvtu8co" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958315657-b2fvtu8co"><div class="list-item-paragraph">Find $\arg (z)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760958316353-3o0qjy3xy" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958316353-3o0qjy3xy"><div class="list-item-paragraph">Find $z$ in cartesian form $x+iy$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760958317155-wrr9fl1ge" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958317155-wrr9fl1ge"><div class="list-item-paragraph">Hence find the value of $\tan \frac{\pi }{12}$ in the form $c+d\sqrt{3}$, where&nbsp;$c$ and $d$ are integers to be found.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

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

Solving Complex Equations

Solving Equations Involving Complex Conjugates

2024 RI P1 Q8

<p>The complex numbers ${{z}_{1}}$ and ${{z}_{2}}$ are given by $\sqrt{3}+\mathbf{i}$ and $-1+\sqrt{3}\mathbf{i}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the modulus and argument of ${{z}_{1}}$ and ${{z}_{2}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">In an Argand diagram with origin $O$, the complex numbers ${{z}_{1}}$ and ${{z}_{2}}$ are represented by the points $P$ and $Q$ respectively. With the aid of an Argand diagram, show that $\angle POQ=\frac{\pi }{2}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find ${{z}_{1}}+{{z}_{2}}$ in the form $a+b\,\mathbf{i}$, where $a$,$b\in \mathbb{R}$. The complex number ${{z}_{1}}+{{z}_{2}}$ is represented by the point $R$ in the Argand diagram. Hence use the Argand diagram to find the exact argument of ${{z}_{1}}+{{z}_{2}}$ and show that $\tan \left( \frac{5\pi }{12} \right)=\frac{\sqrt{3}+1}{\sqrt{3}-1}$.</div><p style="text-align: right;">[3]</p></li></ol>

The complex numbers ${{z}_{1}}$ and ${{z}_{2}}$ are given by $\sqrt{3}+\mathbf{i}$ and $-1+\sqrt{3}\mathbf{i}$ respectively.

Geometrical Transformations of Complex Numbers

2019 TMJC BT Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex numbers $z$ and $w$ are given by $z=\frac{{{\left( 1+\mathbf{i} \right)}^{4}}}{{{\left( 1-\mathbf{i} \right)}^{2}}}$ and $w=\frac{8}{{{\left( \sqrt{3}+\mathbf{i} \right)}^{2}}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express $z$ and $w$ in polar form $r\left( \cos \theta +\mathbf{i}\sin \theta \right)$, where $r>0$ and $-\pi <\theta \le \pi $. Give $r$ and $\theta $ in exact form.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Given that ${{z}^{2}}$, $w$ and ${{w}^{*}}$ are the roots of the equation ${{x}^{3}}+b{{x}^{2}}+cx+d=0$ where $b$, $c$ and $d$ are real values, find the equation.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch on an Argand diagram with origin $O$, the points $P$, $Q$ and $R$ representing the complex numbers $z$, $w$ and $z+w$ respectively.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By considering the quadrilateral $OPRQ$ and the argument of $z+w$, deduce that</div><div class="list-item-paragraph" style="text-align: center">$\tan \frac{5\pi }{12}=2+\sqrt{3}$.</div><p style="text-align: right;">[3]</p></li></ol>

The complex numbers $z$ and $w$ are given by $z=\frac{{{\left( 1+\mathbf{i} \right)}^{4}}}{{{\left( 1-\mathbf{i} \right)}^{2}}}$ and $w=\frac{8}{{{\left( \sqrt{3}+\mathbf{i} \right)}^{2}}}$.

2019 TJC P1 Q8

<p>The complex numbers $w$ and $z$ are given by $w=1+\mathbf{i}$ and $z=-\frac{\sqrt{6}}{2}+\frac{\sqrt{2}}{2}\mathbf{i}$.</p>
<ol class="i-list"><li data-list-item-id="item-1759580779183-57121hy97" data-list-item-path="item-1759580779183-57121hy97"><div class="list-item-paragraph">Find the modulus and argument of $w$ and $z$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759580779814-ovgv7so02" data-list-item-path="item-1759580779814-ovgv7so02"><div class="list-item-paragraph">On the same Argand diagram, sketch the points $P$, $Q$, $R$ representing the complex numbers $z$, $w$ and $z+w$ respectively. State the geometrical shape of the quadrilateral $OPRQ$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759580781044-cky71t1cl" data-list-item-path="item-1759580781044-cky71t1cl"><div class="list-item-paragraph">Referring to the Argand diagram in part <strong>(ii)</strong>, find $\arg \left( w+z \right)$ and show that&nbsp;$\tan \left( \frac{11}{24}\pi \right)=\frac{a+\sqrt{2}}{\sqrt{6}+b}$ where&nbsp;$a$ and $b$ are constants to be determined.<span class="inline-marks">[2]</span></div></li></ol>

The complex numbers $w$ and $z$ are given by $w=1+\mathbf{i}$ and $z=-\frac{\sqrt{6}}{2}+\frac{\sqrt{2}}{2}\mathbf{i}$.

Find the modulus and argument of $w$ and $z$.

On the same Argand diagram, sketch the points $P$, $Q$, $R$ representing the complex numbers $z$, $w$ and $z+w$ respectively. State the geometrical shape of the quadrilateral $OPRQ$.

2025 RI Complex Numbers Assignment Q4

<p>A graphic calculator is not to be used in answering this question.</p>
<p>Two complex numbers $p$ and $q$ are given by $p=1-\mathrm{i}$ and $q=-1+\sqrt{3}\mathrm{i}$, and $z=\frac{p}{q}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $z$ in the form $x+y\mathrm{i}$, where $x$ and $y$ are exact real values to be determined.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">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 $.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, show that $\sin \left( \frac{11}{12}\pi \right)=\frac{\sqrt{6}-\sqrt{2}}{4}$.</span></div><p style="text-align: right;">[3]</p></li></ol>

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.

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 $.

Hence, show that $\sin \left( \frac{11}{12}\pi \right)=\frac{\sqrt{6}-\sqrt{2}}{4}$.

2013 TJC P1 Q5

<p>The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1}{2}\left( 1-\mathbf{i} \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, find the value of ${{a}^{2}}b$ in the form $x+\mathbf{i}y$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">The diagram below shows an isosceles right triangle $ABC$, where the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $c$ respectively. Find the exact value of $c$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759222096991-8gm7y3hzzp6.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMjIwOTY5OTEtOGdtN3kzaHp6cDYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg3NTc5OSwiZXhwIjoxNzkyODc2MDk5fQ.9OVF1xcSKwgiiU0txUOvT4oNMGv7OENLW7zxoaxfE9w" width="400" class="aligncenter" /></figure>

The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1}{2}\left( 1-\mathbf{i} \right)$.

Without using a calculator, find the value of ${{a}^{2}}b$ in the form $x+\mathbf{i}y$.

By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$.

Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$.

The diagram below shows an isosceles right triangle $ABC$, where the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $c$ respectively. Find the exact value of $c$.

2016 JJC P2 Q4

<p>The complex number $w$ has modulus $\sqrt{2}$ and argument $\frac{1}{4}\pi $ and the complex number $z$ has modulus $\sqrt{2}$ and argument $\frac{5}{6}\pi $.</p>
<ol class="i-list"><li data-list-item-id="item-1759147513427-h1r79xrc3" data-list-item-path="item-1759147513427-h1r79xrc3"><div class="list-item-paragraph">By first expressing $w$ and $z$ in the form $x+\mathrm{i}y$, find the exact real and imaginary parts of $w+z$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759147514274-z1k5k1tzu" data-list-item-path="item-1759147514274-z1k5k1tzu"><div class="list-item-paragraph">On the first Argand diagram, sketch the points $P$, $Q$, $R$ representing the complex numbers $z$,$w$ and $z+w$ respectively. State the geometrical shape of the quadrilateral $OPRQ$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759147514883-wqldgrm58" data-list-item-path="item-1759147514883-wqldgrm58"><div class="list-item-paragraph"><p>Referring to the Argand diagram in part <strong>(ii)</strong>, find $\arg (w+z)$ and show that&nbsp;$\tan \left( \frac{11}{24}\pi \right)=\frac{a+\sqrt{2}}{\sqrt{6}+b}$ where&nbsp;$a$ and $b$ are constants to be determined.</p><span class="inline-marks">[2]</span></div></li></ol>

The complex number $w$ has modulus $\sqrt{2}$ and argument $\frac{1}{4}\pi $ and the complex number $z$ has modulus $\sqrt{2}$ and argument $\frac{5}{6}\pi $.

By first expressing $w$ and $z$ in the form $x+\mathrm{i}y$, find the exact real and imaginary parts of $w+z$.

On the first Argand diagram, sketch the points $P$, $Q$, $R$ representing the complex numbers $z$,$w$ and $z+w$ respectively. State the geometrical shape of the quadrilateral $OPRQ$.

2019 DHS P2 Q4

<p>A calculator is not to be used in answering this question.</p>
<p>The complex numbers $a$ and $b$ are given by $\frac{1+\mathbf{i}}{1-\mathbf{i}}$ and $\frac{\surd 2}{1-\mathbf{i}}$ respectively.</p>
<p>Find the moduli and arguments of $a$ and $b$.<span class="inline-marks">[3]</span></p>
<p>In an Argand diagram, the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $a+b$ respectively. The origin is denoted by $O$. By considering the quadrilateral $OACB$ and the argument of $a+b$, show that&nbsp;$\tan \left( \frac{3\pi}{8} \right)=1+\surd 2$.<span class="inline-marks">[3]</span></p>

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

The complex numbers $a$ and $b$ are given by $\frac{1+\mathbf{i}}{1-\mathbf{i}}$ and $\frac{\surd 2}{1-\mathbf{i}}$ respectively.

Find the moduli and arguments of $a$ and $b$.

In an Argand diagram, the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $a+b$ respectively. The origin is denoted by $O$. By considering the quadrilateral $OACB$ and the argument of $a+b$, show that $\tan \left( \frac{3\pi}{8} \right)=1+\surd 2$.

2015 CJC P2 Q3

<p>Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$.</p>
<p>Deduce that&nbsp;$\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.</p>

Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$.

Deduce that $\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.

<p>Evaluate the product $(1+\mathbf{i})(1+\sqrt{3}\mathbf{i})$.</p>
<p>Hence, deduce that&nbsp;$\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.</p>

Evaluate the product $(1+\mathbf{i})(1+\sqrt{3}\mathbf{i})$.

Hence, deduce that $\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.

Math Question Bank