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>The complex number $z$ is given by \[z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.\]</p>
<p>Give all arguments as principal arguments in radians.</p>
<ol class="abc-list"><li data-list-item-id="q001-part-0" data-list-item-path="q001-part-0"><div class="list-item-paragraph">Express $z$ in the form $x+iy$, and find $|z|$ and $\arg z$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q001-part-1" data-list-item-path="q001-part-1"><div class="list-item-paragraph">The point $A$ represents $z$ on an Argand diagram. It is rotated clockwise through $\frac{\pi}{2}$ about the origin to the point $B$. Sketch $A$ and $B$, and state the complex number represented by $B$, both in terms of $z$ and in Cartesian form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $z$ is given by \[z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.\]

Give all arguments as principal arguments in radians.

Express $z$ in the form $x+iy$, and find $|z|$ and $\arg z$.

The point $A$ represents $z$ on an Argand diagram. It is rotated clockwise through $\frac{\pi}{2}$ about the origin to the point $B$. Sketch $A$ and $B$, and state the complex number represented by $B$, both in terms of $z$ and in Cartesian form.

Modulus and Argument of Complex Numbers

Geometrical Transformations of Complex Numbers

Arithmetic with Complex Numbers in Cartesian Form

Representing Complex Numbers on an Argand Diagram

A quotient followed by a clockwise rotation

Complex Numbers

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $z$ is such that&nbsp;$|z|=\sqrt{10}$ and $\arg(z)=-\tan^{-1}3$.</p>
<ol class="abc-list"><li data-list-item-id="tjc-q04-item01" data-list-item-path="tjc-q04-item01"><div class="list-item-paragraph">Find $z$ in the form $a+bi$, where&nbsp;$a$ and $b$ are integers.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="tjc-q04-item02" data-list-item-path="tjc-q04-item02"><div class="list-item-paragraph">In an Argand diagram, $O$ is the origin and the point $A$ represents $z$. The points $B$ and $C$ are such that&nbsp;$OABC$ is a rectangle described in an anti-clockwise sense and $OC=2OA$. Find the complex numbers represented by $B$ and $C$.<span class="inline-marks">[4]</span></div></li></ol>

<strong>Do not use a calculator in answering this question.</strong>

The complex number $z$ is such that $|z|=\sqrt{10}$ and $\arg(z)=-\tan^{-1}3$.

Find $z$ in the form $a+bi$, where $a$ and $b$ are integers.

In an Argand diagram, $O$ is the origin and the point $A$ represents $z$. The points $B$ and $C$ are such that $OABC$ is a rectangle described in an anti-clockwise sense and $OC=2OA$. Find the complex numbers represented by $B$ and $C$.

Geometric Problems on the Argand Diagram

2026 TJC P1 Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>It is given that&nbsp;$1-3i$ is a complex root of the equation $z^3+pz^2+(p+12)z+q=0$, where&nbsp;$p$ and $q$ are real numbers.</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q04-item01" data-list-item-path="nyjc-q04-item01"><div class="list-item-paragraph">State another complex root of the equation in the form $a+ib$, where&nbsp;$a$ and $b$ are real numbers and $b\ne0$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="nyjc-q04-item02" data-list-item-path="nyjc-q04-item02"><div class="list-item-paragraph">Find the values of $p$ and $q$, and find the third root.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="nyjc-q04-item03" data-list-item-path="nyjc-q04-item03"><div class="list-item-paragraph">The point $A$ represents the complex number $1-3i$ on an Argand diagram with respect to the origin $O$. The point $C$ represents the complex number $x+iy$, where&nbsp;$x$ and $y$ are real numbers and $x&gt;0$. Find the values of $x$ and $y$ such that&nbsp;$OA=AC$ and angle $OAC$ is $90^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $1-3i$ is a complex root of the equation $z^3+pz^2+(p+12)z+q=0$, where $p$ and $q$ are real numbers.

State another complex root of the equation in the form $a+ib$, where $a$ and $b$ are real numbers and $b\ne0$.

Find the values of $p$ and $q$, and find the third root.

The point $A$ represents the complex number $1-3i$ on an Argand diagram with respect to the origin $O$. The point $C$ represents the complex number $x+iy$, where $x$ and $y$ are real numbers and $x>0$. Find the values of $x$ and $y$ such that $OA=AC$ and angle $OAC$ is $90^\circ$.

Applying the Conjugate Root Theorem

2026 NYJC P1 Q4

<p>The point $A$ on the Argand diagram below represents the complex number $w_1$ with modulus $r$ and argument $\theta$. It is given that the complex numbers $w_2$ and $w_3$ satisfy the equations $w_2=-w_1$ and $w_3=iw_2$.</p>
<p>Let&nbsp;$B$, $C$ and $D$ represent the complex numbers $w_2$, $w_3$ and $w_2+w_3$ respectively.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="260" style="width:300px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-0" data-list-item-path="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-0"><div class="list-item-paragraph">On the copy of the Argand diagram in the Printed Answer Book, plot the points $B$, $C$ and $D$, indicating clearly the modulus and argument of $w_2$ and $w_3$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-1" data-list-item-path="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-1"><div class="list-item-paragraph">State, in radians, the angle $BOC$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-2" data-list-item-path="h2-2025-ri-promo-q3-ri-q3-item-block-3-item-2"><div class="list-item-paragraph">By considering the quadrilateral $OBDC$, find $\left|w_2+w_3\right|$ in terms of $r$ and $\arg(w_2+w_3)$ in terms of $\theta$.<span class="inline-marks">[2]</span></div></li></ol>

The point $A$ on the Argand diagram below represents the complex number $w_1$ with modulus $r$ and argument $\theta$. It is given that the complex numbers $w_2$ and $w_3$ satisfy the equations $w_2=-w_1$ and $w_3=iw_2$.

Let $B$, $C$ and $D$ represent the complex numbers $w_2$, $w_3$ and $w_2+w_3$ respectively.

On the copy of the Argand diagram in the Printed Answer Book, plot the points $B$, $C$ and $D$, indicating clearly the modulus and argument of $w_2$ and $w_3$.

By considering the quadrilateral $OBDC$, find $\left|w_2+w_3\right|$ in terms of $r$ and $\arg(w_2+w_3)$ in terms of $\theta$.

2025 RI Promo Q3

<p><strong>Do not use a caulculator in answering this question.</strong> </p>
<p>A complex number $z$ is given by $z = \frac{\sqrt{3} + \mathrm{i}}{\sqrt{3} - \mathrm{i}}$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Find $\vert{}z\vert{}$ and $\arg(z)$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The point $A$ represents the complex number $z$ on an Argand diagram. The point $A$ is rotated anticlockwise through $\frac{1}{2}\pi$ about the origin $O$ to the point $B$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Plot the points $A$ and $B$ on an Argand diagram. State in terms of $z$ the complex number that is represented by $B$, explaining your answer.<span class="inline-marks">[3]</span></div></li></ol>

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

A complex number $z$ is given by $z = \frac{\sqrt{3} + \mathrm{i}}{\sqrt{3} - \mathrm{i}}$.

Find $\vert{}z\vert{}$ and $\arg(z)$.

The point $A$ represents the complex number $z$ on an Argand diagram. The point $A$ is rotated anticlockwise through $\frac{1}{2}\pi$ about the origin $O$ to the point $B$.

Plot the points $A$ and $B$ on an Argand diagram. State in terms of $z$ the complex number that is represented by $B$, explaining your answer.

Modulus, Argument, and Operations on Complex Numbers

N2025 P1 Q3

<p>The point $A$ represents a fixed complex number $a$ where&nbsp;$\frac{\pi}{4} &lt; \arg(a) &lt; \frac{\pi}{2}$. The complex numbers $-ia$ and $-a$ are represented by the points $B$ and $C$ respectively.</p>
<p>On a single clearly labelled Argand diagram, mark the points $A$, $B$ and $C$.</p>
<p>Describe the geometrical relationship between</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$A$ and $B$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">$A$ and $C$.</span></div></li></ol>

The point $A$ represents a fixed complex number $a$ where $\frac{\pi}{4} < \arg(a) < \frac{\pi}{2}$. The complex numbers $-ia$ and $-a$ are represented by the points $B$ and $C$ respectively.

On a single clearly labelled Argand diagram, mark the points $A$, $B$ and $C$.

Describe the geometrical relationship between

2023 TJC NYJC VJC FM P1 Q6

<p>The complex number $z$ is such that&nbsp;$\left| z \right|=1$ and $\arg z=\theta $, where&nbsp;$0&lt;\theta &lt;\frac{\pi }{4}$.</p>
<ol class="i-list"><li data-list-item-id="item-1770890845752-ne9g4idph" data-list-item-path="item-1770890845752-ne9g4idph"><div class="list-item-paragraph">Mark a possible point $A$ on an Argand diagram. Hence, mark the points $B$ and $C$ representing ${{z}^{2}}$ and $z+{{z}^{2}}$ respectively on the same Argand diagram corresponding to point $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1770890856681-hvgbvm0mx" data-list-item-path="item-1770890856681-hvgbvm0mx"><div class="list-item-paragraph">State the geometrical shape of $OACB$.	<span class="inline-marks">[1]</span></div></li></ol>

The complex number $z$ is such that $\left| z \right|=1$ and $\arg z=\theta $, where $0<\theta <\frac{\pi }{4}$.

2017 IJC P2 Q1

<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 $\mathrm{f}(z) = z^2 - 2z$.

Equating Real and Imaginary Parts

2025 RI P1 Q8

<p>The points $P$, $Q$ and $R$ representing the complex numbers $p$, $q$ and $r$ on an Argand diagram are such that $\arg(p) = \alpha$ and $\arg(q) = \beta$, where $0 &lt; \alpha &lt; \beta &lt; \frac{\pi}{2}$, $\beta &gt; 2\alpha$ and $r = p + q$.</p>
<ol class="abc-list"><li data-list-item-id="item-a-3" data-list-item-path="item-a-3"><div class="list-item-content"><span class="list-item-text">If&nbsp;$|p|=|q|$, describe the shape of the quadrilateral of $OPRQ$. Hence find $\arg(r)$ in terms of $\alpha$ and $\beta$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3" data-list-item-path="item-b-3"><div class="list-item-paragraph">The point $Q'$, representing the complex number $q'$, is the reflection of the point $Q$ in $OP$. State the angle $\angle POQ'$.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph"><p>By leaving your answers in terms of $\alpha$, $\beta$ and $|q|$ where applicable, hence, or otherwise,</p></div><ol class="i-list"><li data-list-item-id="subitem-1784366782521-th33o3h40" data-list-item-path="item-b-3/subitem-1784366782521-th33o3h40"><div class="list-item-paragraph"><p>find the argument of the complex number $q'$,</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1784366783180-17jnovd1r" data-list-item-path="item-b-3/subitem-1784366783180-17jnovd1r"><div class="list-item-paragraph"><p>find the real and imaginary parts of $q'$ and write down $q'$ in $a+ib$ form.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

The points $P$, $Q$ and $R$ representing the complex numbers $p$, $q$ and $r$ on an Argand diagram are such that $\arg(p) = \alpha$ and $\arg(q) = \beta$, where $0 < \alpha < \beta < \frac{\pi}{2}$, $\beta > 2\alpha$ and $r = p + q$.

If $|p|=|q|$, describe the shape of the quadrilateral of $OPRQ$. Hence find $\arg(r)$ in terms of $\alpha$ and $\beta$.

2025 NYJC P1 Q6

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $w$ is such that&nbsp;$w=\mathrm{i}-\sqrt{3}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $|w|$ and $\mathrm{arg}(w)$ in exact form.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Represent $w$, $-w$ and $2-w$ in the same Argand diagram, illustrating clearly the geometrical relationship between them.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find $\mathrm{arg}(2-w)$ exactly.</span><span class="inline-marks">[2]</span></div></li></ol>

The complex number $w$ is such that $w=\mathrm{i}-\sqrt{3}$.

Find $|w|$ and $\mathrm{arg}(w)$ in exact form.

Represent $w$, $-w$ and $2-w$ in the same Argand diagram, illustrating clearly the geometrical relationship between them.

Hence, find $\mathrm{arg}(2-w)$ exactly.

2025 NJC P2 Q1

<p>Let ${{z}_{1}}=3+2\mathbf{i}$, ${{z}_{2}}=2-5\mathbf{i}$ be two complex numbers.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express ${{z}_{1}}$ and ${{z}_{2}}$ on the Argand diagram.</div></li><li><div class="list-item-paragraph">Hence, using the same Argand diagram,</div><ol class="abc-list"><li><div class="list-item-paragraph">${{z}_{1}}^*$,</div></li><li><div class="list-item-paragraph">${{z}_{1}}+{{z}_{2}}$,</div></li><li><div class="list-item-paragraph">$\mathbf{i}{{z}_{2}}$.</div></li></ol></li></ol>

Let ${{z}_{1}}=3+2\mathbf{i}$, ${{z}_{2}}=2-5\mathbf{i}$ be two complex numbers.

Worked Example 6

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Showing your working, find the complex numbers $z$ and $w$ which satisfy the simultaneous equations</div><div class="list-item-paragraph" style="text-align: center">$2iz+\left( 1-2i \right)w=4$</div><div class="list-item-paragraph" style="text-align: center">$\left( 1+i \right)z+\left( 2+i \right)w=3$<span class="inline-marks">[6]</span></div></li><li><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761027063471-syhbzkkap7r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEwMjcwNjM0NzEtc3loYnpra2FwN3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg3NTcyNCwiZXhwIjoxNzkyODc2MDI0fQ.3N_DuI0ntGFin9_jeT89H79TexZgXbeUbE2vHFMEXOg" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><div class="list-item-paragraph">The points $A$ and $B$ on the Argand diagram represent the complex numbers ${{w}_{1}}$ and ${{w}_{2}}$ respectively.</div><ol class="i-list"><li><div class="list-item-paragraph">On the copy of the Argand diagram in the Printed Answer Booklet, plot the point $C$ to represent the complex number ${{w}_{3}}$, where&nbsp;${{w}_{3}}={{w}_{1}}+{{w}_{2}}$. Show clearly the geometrical relationship between the points $A$, $B$ and $C$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">The points ${A}'$, ${B}'$ and ${C}'$ represent the complex numbers $\mathbf{i}\,{{w}_{1}}$, $\mathbf{i}\,{{w}_{2}}$ and $\mathbf{i}\,{{w}_{3}}$ respectively.</div></li><li><div class="list-item-paragraph">On the same Argand diagram in the Printed Answer Booklet, plot the points ${A}'$, ${B}'$ and ${C}'$. State the transformation that maps the points $A$, $B$ and $C$ onto the points ${A}'$, ${B}'$ and ${C}'$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The transformation in part <strong>(ii)</strong> maps point ${A}'$ onto the point ${A}''$. Determine, with justification, whether ${A}''$ represents the complex conjugate of ${{w}_{1}}$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Solving Simultaneous Complex Equations

N2023 Specimen Paper P1 Q7

<ol class="abc-list"><li data-list-item-id="item-1760958379712-j660wkjfj" data-list-item-path="item-1760958379712-j660wkjfj"><div class="list-item-paragraph">Given that imaginary unit $\mathbf{i}$ is one of the roots of the equation ${{z}^{3}}-2{{z}^{2}}+az+1+3\mathbf{i}=0$, find the complex number ${a}$ and the other roots.</div></li><li data-list-item-id="item-1760958380522-umltyb9vc" data-list-item-path="item-1760958380522-umltyb9vc"><div class="list-item-paragraph"><p>The root that has the largest modulus in <strong>(a)</strong> is denoted as&nbsp;${w}$ and represented by the point ${A}$. The points ${B, C}$ and ${D}$ represent the complex numbers ${{w}^{*}}$, $\mathbf{i}w$ and ${{\mathbf{i}}^{2}}w$ respectively. On the same diagram, plot the points ${A, B, C}$ and ${D}$ and state each transformation that maps the point ${A}$ onto the point ${B, C}$ and ${D}$ respectively and state the angle ${OAC}$.</p></div></li></ol>

Finding Roots of Equations with Complex Coefficients

2014 HCI P1 Q3 [Modified]

<ol class="abc-list"><li><div class="list-item-paragraph">The complex numbers $z$ and $w$ satisfy the following equations.</div><div class="list-item-paragraph" style="text-align: center">$2z+\left| w \right|=2-4i$</div><div class="list-item-paragraph" style="text-align: center">$iz-w=2i$</div><div class="list-item-paragraph">Find $z$ and $w$, giving your answers in the form $a+ib$, where&nbsp;$a$ and $b$ are real numbers.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">It is given that&nbsp;${{z}_{1}}=2-i$ and ${{z}_{3}}=-3+2i$. On an Argand diagram, mark the points $A$ and $C$ representing ${{z}_{1}}$ and ${{z}_{3}}$ respectively.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">The points $B$ and $D$ on the Argand diagram represent complex numbers ${{z}_{2}}$ and ${{z}_{4}}$ respectively. Given that&nbsp;$ABCD$ is a square, labelled in an anti-clockwise direction, find ${{z}_{2}}$ and ${{z}_{4}}$.<span class="inline-marks">[4]</span></div></li></ol>

2025 TJC 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.

Using Complex Numbers to Deduce Trigonometric Values

2019 TMJC BT Q4

<p>In an Argand diagram, the points $A$, $B$, $C$, $D$ represent the complex numbers $a$, $b$, $c$, $d$ respectively.</p>
<p>Given that&nbsp;$ABCD$ is a square described in an anticlockwise sense, with $a=-4+6\mathbf{i}$ and $c=12-8\mathbf{i}$, find $b$ and $d$.</p>

In an Argand diagram, the points $A$, $B$, $C$, $D$ represent the complex numbers $a$, $b$, $c$, $d$ respectively.

Given that $ABCD$ is a square described in an anticlockwise sense, with $a=-4+6\mathbf{i}$ and $c=12-8\mathbf{i}$, find $b$ and $d$.

Geometrical effects of conjugation, negation, addition, subtraction, and multiplication by i Q2

<p>The points $A$, $B$, $C$ and $D$ represent the complex numbers $-3-\mathbf{i}$, ${{z}_{1}}$, $7+5\mathbf{i}$ and ${{z}_{2}}$ respectively.</p>
<p>Given that&nbsp;$ABCD$ is a square, labelled in an anti-clockwise direction. Find ${{z}_{1}}$ and ${{z}_{2}}$.</p>

The points $A$, $B$, $C$ and $D$ represent the complex numbers $-3-\mathbf{i}$, ${{z}_{1}}$, $7+5\mathbf{i}$ and ${{z}_{2}}$ respectively.

Given that $ABCD$ is a square, labelled in an anti-clockwise direction. Find ${{z}_{1}}$ and ${{z}_{2}}$.

Geometrical effects of conjugation, negation, addition, subtraction, and multiplication by $\mathbf{i}$

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex numbers $z$ and $w$ have the same modulus $r$, and have arguments $\alpha $ and $\beta $ respectively.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$z+w=2r\cos \frac{\alpha -\beta }{2}\left( \cos \frac{\alpha +\beta }{2}+\mathbf{i}\,\sin \frac{\alpha +\beta }{2} \right)$ and hence write down expressions for&nbsp;$\left| z+w \right|$ and for&nbsp;$\arg \left( z+w \right)$.<span class="inline-marks">[4]</span></div></li></ol>
<p>It is now given that&nbsp;$z=2{{\mathrm{e}}^{\mathbf{i}\,\frac{\pi }{12}}}$ and $w=2{{\mathrm{e}}^{\mathbf{i}\,\frac{5\pi }{12}}}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the values of $\left| z+w \right|$ and $\arg \left( z+w \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Let&nbsp;$v$ be the complex number such that&nbsp;$v=\frac{{{z}^{2}}}{w}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find $\left| v \right|$ and $\arg \left( v \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Let&nbsp;$O$ be the origin and $P$ and $Q$ be points representing $z+w$ and $v$ respectively on an Argand diagram. Find the area of triangle $OPQ$.<span class="inline-marks">[2]</span></div></li></ol>

The complex numbers $z$ and $w$ have the same modulus $r$, and have arguments $\alpha $ and $\beta $ respectively.

It is now given that $z=2{{\mathrm{e}}^{\mathbf{i}\,\frac{\pi }{12}}}$ and $w=2{{\mathrm{e}}^{\mathbf{i}\,\frac{5\pi }{12}}}$.

Let $v$ be the complex number such that $v=\frac{{{z}^{2}}}{w}$.

Cartesian, Trigonometric and Exponential Forms

Properties of Trigonometric and Exponential Forms

2023 EJC P1 Q10

<p>On an Argand diagram, the points $P$ and $Q$ represent the complex numbers $p$ and $q$ respectively where $p=\cos \theta +\mathbf{i}\sin \theta $, $0&lt;\theta &lt;\frac{\pi }{4}$ and $q=2{{\mathrm{e}}^{\mathbf{i}\,\,\phi }}$, $\frac{\pi }{4}&lt;\,\phi &lt;\frac{\pi }{2}$ and $\theta +\,\phi &gt;\frac{\pi }{2}$. Find the moduli and arguments of the complex numbers $a$, $b$,$c$, $d$ and $\mathrm{e}$, where $a=\mathbf{i}\,p$, $b={{q}^{2}}$, $c=\frac{1}{q}$ and $d=p+{{p}^{*}}$ and $e=pq$.</p>
<p>On your Argand diagram, sketch and label the points $P$, $Q$, $A$, $B$, $C$,$D$ and $E$ representing the complex numbers $p$, $q$, $a$, $b$, $c$, $d$ and $e$ respectively.</p>

On an Argand diagram, the points $P$ and $Q$ represent the complex numbers $p$ and $q$ respectively where $p=\cos \theta +\mathbf{i}\sin \theta $, $0<\theta <\frac{\pi }{4}$ and $q=2{{\mathrm{e}}^{\mathbf{i}\,\,\phi }}$, $\frac{\pi }{4}<\,\phi <\frac{\pi }{2}$ and $\theta +\,\phi >\frac{\pi }{2}$. Find the moduli and arguments of the complex numbers $a$, $b$,$c$, $d$ and $\mathrm{e}$, where $a=\mathbf{i}\,p$, $b={{q}^{2}}$, $c=\frac{1}{q}$ and $d=p+{{p}^{*}}$ and $e=pq$.

On your Argand diagram, sketch and label the points $P$, $Q$, $A$, $B$, $C$,$D$ and $E$ representing the complex numbers $p$, $q$, $a$, $b$, $c$, $d$ and $e$ respectively.

Properties of Complex Conjugates

ACJC Complex Numbers Tutorial Q13

<p>It is given that&nbsp;$z=-\frac{1}{2}$ is a root of the equation</p>
<p style="text-align: center">$8{{z}^{3}}+\left( 4-4\sqrt{2} \right){{z}^{2}}+\left( 2-2\sqrt{2} \right)z+1=0$</p>
<p>The roots of the equation are denoted by ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$, where&nbsp;$\arg \left( {{z}_{1}} \right)&lt;\arg \left( {{z}_{2}} \right)&lt;\arg \left( {{z}_{3}} \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ in the exact form $r{{\mathrm{e}}^{\mathrm{i}\,\theta }}$, where&nbsp;$r&gt;0$ and $-\pi &lt;\theta \le \pi $.<span class="inline-marks">[6]</span></div></li></ol>
<p>The complex number $w$ has modulus $\sqrt{2}$ and argument $\frac{\pi }{24}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the modulus and argument of ${{z}_{4}}$, where&nbsp;${{z}_{4}}=\frac{{{w}^{2}}}{{{z}_{1}}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The complex numbers ${{z}_{2}}$ , ${{z}_{3}}$ and ${{z}_{4}}$ are represented by the points ${{Z}_{2}}$, ${{Z}_{3}}$ and ${{Z}_{4}}$ respectively in an Argand diagram with origin $O$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Mark, on an Argand diagram, the points ${{Z}_{2}}$, ${{Z}_{3}}$ and ${{Z}_{4}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">By considering $\sin \left( A-B \right)$ with suitable values of $A$ and $B$, show that&nbsp;$\sin \left( \frac{\pi }{12} \right)=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1 \right)$.</div><div class="list-item-paragraph">Hence or otherwise, find the exact area of the quadrilateral $O{{Z}_{3}}{{Z}_{4}}{{Z}_{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $z=-\frac{1}{2}$ is a root of the equation

$8{{z}^{3}}+\left( 4-4\sqrt{2} \right){{z}^{2}}+\left( 2-2\sqrt{2} \right)z+1=0$

The roots of the equation are denoted by ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$, where $\arg \left( {{z}_{1}} \right)<\arg \left( {{z}_{2}} \right)<\arg \left( {{z}_{3}} \right)$.

The complex number $w$ has modulus $\sqrt{2}$ and argument $\frac{\pi }{24}$.

The complex numbers ${{z}_{2}}$ , ${{z}_{3}}$ and ${{z}_{4}}$ are represented by the points ${{Z}_{2}}$, ${{Z}_{3}}$ and ${{Z}_{4}}$ respectively in an Argand diagram with origin $O$.

Math Question Bank