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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>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.

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

Using Complex Numbers to Deduce Trigonometric Values

Geometric Problems on the Argand Diagram

A square and the direction of its diagonal

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

Modulus, Argument, and Operations on Complex Numbers

A product giving an exact cosine value

<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 $C$.

2026 TJC P1 Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The equation $iz^3-1=0$ has 3 roots $z_1$, $z_2$ and $z_3$, which are represented by points $A$, $B$ and $C$ respectively on an Argand diagram. It is given that&nbsp;$z_1$ is purely imaginary and the real component of $z_2$ is positive.</p>
<ol class="abc-list"><li data-list-item-id="hci-q07-item01" data-list-item-path="hci-q07-item01"><div class="list-item-paragraph">Find the exact values of $z_1$, $z_2$ and $z_3$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q07-item02" data-list-item-path="hci-q07-item02"><div class="list-item-paragraph">Determine the modulus and the arguments of $z_1$, $z_2$ and $z_3$, where&nbsp;$-\pi&lt;\arg(z)\leq\pi$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q07-item03" data-list-item-path="hci-q07-item03"><div class="list-item-paragraph">Sketch $A$, $B$ and $C$ on an Argand diagram and state the geometrical shape of $ABC$.<span class="inline-marks">[3]</span></div></li></ol>

The equation $iz^3-1=0$ has 3 roots $z_1$, $z_2$ and $z_3$, which are represented by points $A$, $B$ and $C$ respectively on an Argand diagram. It is given that $z_1$ is purely imaginary and the real component of $z_2$ is positive.

Find the exact values of $z_1$, $z_2$ and $z_3$.

Determine the modulus and the arguments of $z_1$, $z_2$ and $z_3$, where $-\pi<\arg(z)\leq\pi$.

Sketch $A$, $B$ and $C$ on an Argand diagram and state the geometrical shape of $ABC$.

Finding Roots of Complex Numbers

2026 HCI P1 Q7

<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="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjMwMCwiaGVpZ2h0IjoyNjAsImVsZW1lbnRzIjpbeyJpZCI6ImgyLTIwMjUtcmktcHJvbW8tcTMtcmktcTMtc2VnbWVudC1ibG9jay0yLWNvbnRlbnQtZWxlbWVudC0wIiwiZGFzaCI6MiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlswLDBdLCJwb2ludDIiOlswLjQ1LDEuMzVdLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjAuNDUsInkiOjEuMzUsImlkIjoiaDItMjAyNS1yaS1wcm9tby1xMy1yaS1xMy1wb2ludC1ibG9jay0yLWNvbnRlbnQtZWxlbWVudC0xIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChBXFwpIiwicG9pbnRTaXplIjo0LCJwb2ludFN0eWxlIjoiY3Jvc3MiLCJsYWJlbE9mZnNldCI6WzUuMiwzLjddLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjAuMTIsInkiOjAuNzUsImlkIjoiaDItMjAyNS1yaS1wcm9tby1xMy1yaS1xMy10ZXh0LWJsb2NrLTItY29udGVudC1lbGVtZW50LTIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoclxcKSJ9LHsiaWQiOiJoMi0yMDI1LXJpLXByb21vLXEzLXJpLXEzLWFyYy1ibG9jay0yLWNvbnRlbnQtZWxlbWVudC0zIiwidHlwZSI6ImFyYyIsImNvbG9yIjoiIzAwMDAwMCIsImNlbnRlciI6WzAsMF0sInJhZGl1cyI6MC40MiwiZW5kQW5nbGUiOjcxLjU2NSwic3RhcnRBbmdsZSI6MCwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjowLjM3LCJ5IjowLjI3LCJpZCI6ImgyLTIwMjUtcmktcHJvbW8tcTMtcmktcTMtdGV4dC1ibG9jay0yLWNvbnRlbnQtZWxlbWVudC00IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKFxcdGhldGFcXCkifSx7ImlkIjoiNzk2OTc5YTctNmU2Yy00MGJjLWI5Y2MtZjQwOGE3NDJkMzdjIiwidHlwZSI6ImFyYyIsImNvbG9yIjoiIzAwMDAwMCIsImNlbnRlciI6WzAsMF0sInBvaW50MSI6WzAuMjksMF0sInBvaW50MiI6WzAuMjEsMC42XSwibGFzdEFycm93Ijp0cnVlLCJzdHJva2VXaWR0aCI6MS41fV0sInNob3dYQXhpcyI6dHJ1ZSwic2hvd1lBeGlzIjp0cnVlLCJ4QXhpc0xhYmVsIjoiUmUiLCJ5QXhpc0xhYmVsIjoiSW0iLCJib3VuZGluZ0JveCI6Wy0xLjQyLDIuMzUsMS43LC0xLjQyXSwib3JpZ2luTGFiZWwiOiJPIiwiZ3JpZFNldHRpbmdzIjp7Im1ham9yQ29sb3IiOiIjOTRhM2I4IiwibWFqb3JTdGVwWCI6MSwibWFqb3JTdGVwWSI6MSwibWlub3JDb2xvciI6IiNjYmQ1ZTEiLCJtYWpvck9wYWNpdHkiOjAuNDUsIm1pbm9yT3BhY2l0eSI6MC40NSwibWlub3JFbGVtZW50c1giOjQsIm1pbm9yRWxlbWVudHNZIjo0LCJtYWpvclN0cm9rZVdpZHRoIjowLjgsIm1pbm9yU3Ryb2tlV2lkdGgiOjAuNSwic2hvd01ham9yR3JpZExpbmVzIjp0cnVlLCJzaG93TWlub3JHcmlkTGluZXMiOnRydWV9LCJ4QXhpc1Bvc2l0aW9uIjowLCJ5QXhpc1Bvc2l0aW9uIjowLCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJ4VGlja0xhYmVsRXZlcnkiOjEsInlUaWNrTGFiZWxFdmVyeSI6MSwieEF4aXNMYWJlbE9mZnNldCI6Wy0xMS45LC0xNS41XSwieUF4aXNMYWJlbE9mZnNldCI6Wy0yMS4xLDIuN10sIm9yaWdpbkxhYmVsT2Zmc2V0IjpbLTUsLTJdfQ==" 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 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

Applying the Conjugate Root Theorem

2025 ACJC P1 Q10

<p>Let&nbsp;$z = -1 - \sqrt{3}i$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State the value of $|-1 - \sqrt{3}i|$ and $\arg(-1 - \sqrt{3}i)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Give the geometrical meanings of the statements "$|-1 - \sqrt{3}i|$" and "$\arg(-1 - \sqrt{3}i)$".</span></div></li></ol>

State the value of $|-1 - \sqrt{3}i|$ and $\arg(-1 - \sqrt{3}i)$.

Give the geometrical meanings of the statements "$|-1 - \sqrt{3}i|$" and "$\arg(-1 - \sqrt{3}i)$".

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>It is given that&nbsp;$z = \sqrt{3} - \mathrm{i}$ and $w = 2\mathrm{i}z$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the modulus and argument of $z$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the modulus and argument of $w$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>$O$, $A$, $B$ and $C$ represent the complex numbers $0$, $z$, $z + w$ and $w$ respectively.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Illustrate these four points on a single diagram.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Identify the shape of the quadrilateral $OABC$ and state its area.</span><span class="inline-marks">[2]</span></div></li></ol>

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

It is given that $z = \sqrt{3} - \mathrm{i}$ and $w = 2\mathrm{i}z$.

Deduce the modulus and argument of $w$.

$O$, $A$, $B$ and $C$ represent the complex numbers $0$, $z$, $z + w$ and $w$ respectively.

Illustrate these four points on a single diagram.

Identify the shape of the quadrilateral $OABC$ and state its area.

2025 YIJC P2 Q2

<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>Find the modulus and argument of $1+\sqrt{3}\,i$.</p>

Find the modulus and argument of $1+\sqrt{3}\,i$.

Worked Example 4

<p>Given $| z |=2$ and $\arg z=-\dfrac{2\pi }{3}$, find the real and imaginary parts of $z$ in exact form.</p>

Given $| z |=2$ and $\arg z=-\dfrac{2\pi }{3}$, find the real and imaginary parts of $z$ in exact form.

Forms of Complex Number Q3

<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>It is given that&nbsp;$z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where&nbsp;$b$ is real and positive.</p>
<ol class="i-list"><li data-list-item-id="item-1760957017876-s1hi3ko2r" data-list-item-path="item-1760957017876-s1hi3ko2r"><div class="list-item-paragraph">Given that&nbsp;$\left| \frac{z}{w} \right|=1$, show that&nbsp;$b=\sqrt{2}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760957018557-8rcw481un" data-list-item-path="item-1760957018557-8rcw481un"><div class="list-item-paragraph">Show that&nbsp;$\arg z+\arg(w^*)=13\pi/12$, where&nbsp;$w^*$ is the complex conjugate of $w$ and arguments are principal values.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760957019299-5idq5iawl" data-list-item-path="item-1760957019299-5idq5iawl"><div class="list-item-paragraph">Given that&nbsp;$\arg \left( p+z \right)-\arg \left( z \right)=-\frac{7\pi }{12}$, find the exact value of the real number $p$.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where $b$ is real and positive.

2021 CJC JC2 MYE P1 Q4 [Modified]

<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.

2019 TMJC BT Q4

<p>A complex number ${{z}_{1}}$ is given&nbsp;${{z}_{1}}=a+\left( a-3 \right)\,\,\mathbf{i}$, where&nbsp;$a$ is a positive real constant. It is given that&nbsp;$\arg {{z}_{1}}=\theta $, where&nbsp;$-\frac{\pi }{2}&lt;\theta &lt;0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760361074877-2nx4ecqeo"><div class="list-item-paragraph">Find, leaving your answers in terms of $\theta $,</div><ol class="i-list"><li data-list-item-id="subitem-1760361080712-g7rgn2vry"><div class="list-item-paragraph">$\arg \left( -2{{z}_{1}} \right)$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760361087064-fjhgfd80d"><div class="list-item-paragraph">$\arg \left( {{z}_{1}}-2a \right)$<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>Another complex number ${{z}_{2}}$ is given by ${{z}_{2}}=1+3\,\mathbf{i}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760361101127-56h8tqw57"><div class="list-item-paragraph">Without using a calculator, find the range of values of $a$ such that</div><div class="list-item-paragraph" style="text-align: center">$\frac{{{\left| {{z}_{1}}{{z}_{2}} \right|}^{2}}}{\operatorname{Im}\left( {{z}_{1}}{{z}_{2}} \right)}\le 10$.<span class="inline-marks">[5]</span></div></li></ol>

A complex number ${{z}_{1}}$ is given ${{z}_{1}}=a+\left( a-3 \right)\,\,\mathbf{i}$, where $a$ is a positive real constant. It is given that $\arg {{z}_{1}}=\theta $, where $-\frac{\pi }{2}<\theta <0$.

Another complex number ${{z}_{2}}$ is given by ${{z}_{2}}=1+3\,\mathbf{i}$.

Inequalities involving Rational Functions

2023 HCI P2 Q3

Inequalities

<ol class="abc-list"><li><div class="list-item-paragraph">Express ${{\left( 3-\mathrm{i} \right)}^{2}}$ in the form of $a+\mathrm{i}b$.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">Hence or otherwise find the roots of the equation</div><div class="list-item-paragraph" style="text-align: center">${{\left( z+\mathrm{i} \right)}^{2}}=-8+6\mathrm{i}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;${{z}_{1}}=2-4\mathbf{i}$ and ${{z}_{2}}=-2-\mathbf{i}$, find</div><ol class="i-list"><li><div class="list-item-paragraph">$\left| {{z}_{1}}-{{z}_{2}} \right|$.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\arg ({{z}_{1}}+{{z}_{2}})$.	<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Finding Square Roots of Complex Numbers Algebraically

Solving Complex Equations by Substitution

N2004 P1 Q12

<p><strong>Do not use a graphing calculator in answering this question.</strong></p>
<p>The complex number $w$ is a root to the equation</p>
<p style="text-align: center">${{z}^{4}}+a{{z}^{3}}+46{{z}^{2}}+bz+125=0$,</p>
<p>where $a$,$b\in \mathbb{R}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Prove that the product of the factors $z-w$ and $z-{{w}^{*}}$ is a quadratic polynomial with real coefficients.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $\left| w \right|=\sqrt{5}$ and $\arg (w)=-{{\tan }^{-1}}(2)$ where $-\frac{\pi }{2}<-{{\tan }^{-1}}(2)<0$, show that $w=1-2\mathrm{i}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find all the roots of the equation and the values $a$ and $b$.</div><p style="text-align: right;">[5]</p></li></ol>

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

The complex number $w$ is a root to the equation

${{z}^{4}}+a{{z}^{3}}+46{{z}^{2}}+bz+125=0$,

Properties of Complex Conjugates

Cartesian, Trigonometric and Exponential Forms

2020 HCI P1 Q8

<ol class="abc-list"><li><div class="list-item-paragraph">The complex numbers $3+2\mathrm{i}$, $z$, $4-6\mathrm{i}$ are the first three terms in a geometric progression. Without using a calculator, find the two possible values of $z$.</div><p style="text-align: right;">[4]</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">The complex number $w$ is such that $w=a+b\mathrm{i}$, where $a$ and $b$ bare non-zero real numbers. The complex conjugate of $w$ is denoted by $w^*$. Given that $\frac{{{w}^{2}}}{w^*}$ is a real number, find the possible values of $w$ in terms of $a$ only.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence, find the exact possible arguments of $w$ if $a$ is positive.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

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