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PSLE Foundation Mathematics

<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 $z=1-\sqrt3\,i$.

Using Complex Numbers to Deduce Trigonometric Values

Arithmetic with Complex Numbers in Cartesian Form

Modulus, Argument, and Operations on Complex Numbers

Cartesian, Trigonometric and Exponential Forms

Complex Numbers

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

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

Properties from Exponential or Trigonometric form Q5

<p>Without the use of a calculator, express $\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}$ and $1-\mathrm{i}$ in exponential form. Hence evaluate ${{\left( \frac{\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}}{1-\mathrm{i}} \right)}^{12}}$.</p>

Without the use of a calculator, express $\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}$ and $1-\mathrm{i}$ in exponential form. Hence evaluate ${{\left( \frac{\frac{1}{3}-\frac{\sqrt{3}}{3}\mathrm{i}}{1-\mathrm{i}} \right)}^{12}}$.

Properties of Trigonometric and Exponential Forms

Properties from Exponential or Trigonometric form Q4

<p>Express the following complex numbers in the form $z = r e^{\mathbf{i}\theta}$ where $r \ge 0$, and $-\pi < \theta \le \pi$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$3\mathrm{i}e^{-\mathbf{i}\pi}$</div></li><li><div class="list-item-paragraph">$-2e^{-\mathbf{i}\frac{\pi}{3}}$</div></li><li><div class="list-item-paragraph">$e^{1-\mathbf{i}\frac{\pi}{3}}$</div></li><li><div class="list-item-paragraph">$\left( \sqrt{3}-\mathbf{i} \right)e^{-\mathbf{i}\frac{\pi}{4}}$</div></li><li><div class="list-item-paragraph">$-2\left( 1+\sqrt{3}\mathbf{i} \right)e^{\mathbf{i}\frac{2\pi}{5}}$</div></li></ol>

Express the following complex numbers in the form $z = r e^{\mathbf{i}\theta}$ where $r \ge 0$, and $-\pi < \theta \le \pi$.

Forms of Complex Numbers Q8

<p>Express</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{e}^{\mathrm{i}\pi }}$ and</div></li><li><div class="list-item-paragraph">${{e}^{1+\mathrm{i}\frac{\pi }{6}}}$ in Cartesian form.</div></li></ol>

Forms of Complex Numbers Q7

<p>Evaluate</p>
<ol class="i-list"><li><div class="list-item-paragraph">$4{{e}^{-\mathrm{i}\frac{2\pi }{3}}}$</div></li><li><div class="list-item-paragraph">$2{{e}^{\mathrm{i}\pi }}$</div></li><li><div class="list-item-paragraph">$\sqrt{3}{{e}^{\mathrm{i}\frac{5\pi }{6}}}$ in the form of $x+y\mathrm{i}$ in exact form.</div></li></ol>

Forms of Complex Number Q4

<p>Express $-1+\sqrt{3}i$ in exact trigonometric form and exponential form.</p>

Express $-1+\sqrt{3}i$ in exact trigonometric form and exponential form.

Forms of Complex Number Q1

<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 $\arg w=\frac{\pi }{3}$.

Equating Real and Imaginary Parts

Solving Complex Equations

2021 NYJC P1 Q9

<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>

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

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 $v=\frac{{{z}^{2}}}{w}$.

Geometrical Transformations of Complex Numbers

Representing Complex Numbers on an Argand Diagram

2023 EJC P1 Q10

<p>The complex numbers, ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given by ${{z}_{1}}={{\left( 1-\sqrt{3}\mathbf{i} \right)}^{2}}$, ${{z}_{2}}={{\left[ \sqrt{2}\left( \cos \frac{\pi }{4}+\mathbf{i}\sin \frac{\pi }{4} \right) \right]}^{6}}$ and ${{z}_{3}}=-1+\sqrt{3}\,\mathbf{i}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using an algebraic method, find $\frac{{{z}_{2}}}{{{z}_{1}}}$ in the form $r{{\mathrm{e}}^{\mathbf{i}\theta }}$, where&nbsp;$r&gt;0$ and $\theta $ is an exact real constant such that&nbsp;$-\pi &lt;\theta \le \pi $.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find $\frac{{{z}_{2}}}{{{z}_{1}}}+{{z}_{3}}$ in the form $p{{\mathrm{e}}^{\mathbf{i}\alpha }}$, where both&nbsp;$r$ and $\theta $ are exact real constants such that&nbsp;$r&gt;0$ and $-\pi &lt;\theta \le \pi $.<span class="inline-marks">[3]</span></div></li></ol>

The complex numbers, ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given by ${{z}_{1}}={{\left( 1-\sqrt{3}\mathbf{i} \right)}^{2}}$, ${{z}_{2}}={{\left[ \sqrt{2}\left( \cos \frac{\pi }{4}+\mathbf{i}\sin \frac{\pi }{4} \right) \right]}^{6}}$ and ${{z}_{3}}=-1+\sqrt{3}\,\mathbf{i}$.

2022 ASRJC P1 Q2

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

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

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

Finding Square Roots of Complex Numbers Algebraically

2023 MI 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-1760350733187-ft0ymlzq4" data-list-item-path="item-1760350733187-ft0ymlzq4"><div class="list-item-paragraph">It is given that&nbsp;$w=1+\mathrm{i}$ and $z=-\sqrt{3}-\mathrm{i}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760350743890-loro32v9r" data-list-item-path="item-1760350733187-ft0ymlzq4/subitem-1760350743890-loro32v9r"><div class="list-item-paragraph">Find $\frac{{{w}^{3}}}{z}$ in the form $r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, where&nbsp;$r&gt;0$ and $-\pi &lt;\theta \le \pi $.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760350744572-egh1i0ozn" data-list-item-path="item-1760350733187-ft0ymlzq4/subitem-1760350744572-egh1i0ozn"><div class="list-item-paragraph">Find the smallest positive integer value of $n$, such that&nbsp;${{\left( \frac{{{w}^{3}}}{z} \right)}^{n}}$ is real.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760350764080-yskxvfuvi" data-list-item-path="item-1760350764080-yskxvfuvi"><div class="list-item-paragraph">The complex number $q$ is given by $\frac{{{\mathrm{e}}^{\mathrm{i}\theta }}}{{{\mathrm{e}}^{-\mathrm{i}\theta }}-\mathrm{i}}$, where&nbsp;$0 &lt; \theta &lt; \pi $.</div><div class="list-item-paragraph">Show that the real part of $q$ is $\frac{1}{2}\left( 1-2\sin \theta  \right)$.<span class="inline-marks">[4]</span></div></li></ol>

De Moivre's Theorem

2022 JPJC MYE P2 Q6

<p>Express the following complex numbers in the form $z=r{{e}^{\mathrm{i}\theta }}$, where&nbsp;$r\ge 0$ and $-\pi &lt;\theta \le \pi $,</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{\cos \theta +\mathrm{i}\sin \theta }{\cos \theta -\mathrm{i}\sin \theta }$</div></li><li><div class="list-item-paragraph">${{e}^{2+\mathrm{i}\theta }}$</div></li><li><div class="list-item-paragraph">$\frac{1}{1+{{e}^{\mathrm{i}\theta }}}$</div></li><li><div class="list-item-paragraph">$1+\cos \left( 2\theta \right)-\mathrm{i sin}\left( 2\theta \right)$</div></li></ol>

Express the following complex numbers in the form $z=r{{e}^{\mathrm{i}\theta }}$, where $r\ge 0$ and $-\pi <\theta \le \pi $,

Expressing Various Complex Numbers in Exponential Form

<p>Express the following complex numbers in the form $z=r{{e}^{\mathrm{i}\theta }}$, where&nbsp;$r\ge 0$ and $-\pi &lt;\theta \le \pi $,</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\cos \theta -\mathrm{i sin}\theta $</div></li><li><div class="list-item-paragraph">$\sin \theta -\mathrm{i cos}\theta $</div></li><li><div class="list-item-paragraph">$-\sin \theta +\mathrm{i cos}\theta $</div></li><li><div class="list-item-paragraph">$-\sin \theta -\mathrm{i cos}\theta $</div></li></ol>

Expressing sin θ + i cos θ as r e^(iθ) and other variations

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The complex number $q$ is given by $q=\frac{{{e}^{i\theta }}}{1-{{e}^{i\theta }}}$, where&nbsp;$0 &lt; \theta \le 2\pi $.</div><ol class="i-list"><li><div class="list-item-paragraph">find the real part of $q$,</div></li><li><div class="list-item-paragraph">show that the imaginary part of $q$ is $\frac{1}{2}\cot \left( \frac{1}{2}\theta \right)$.</div></li></ol></li></ol>

N1998 P2 Q12

<ol class="abc-list"><li data-list-item-id="item-1760257139205-71hk9ns32" data-list-item-path="item-1760257139205-71hk9ns32"><div class="list-item-paragraph">Write down the modulus and argument of the complex number $w$, where&nbsp;$w=1+\mathbf{i}\sqrt{3}$. Hence find the set of values of the positive integer $n$ for which ${{w}^{n}}$is real, and show that, for these values of $n$, the value of ${{w}^{n}}$ is either ${{2}^{n}}$or $-{{2}^{n}}$.</div><div class="list-item-paragraph">Express ${{w}^{100}}-{{\left( w^* \right)}^{100}}$ in the form $k\mathbf{i}$, where&nbsp;$k$ is real, giving the exact value of $k$ in non-trigonometrical form.</div></li></ol>

Properties of Complex Conjugates

N1993 P2 Q12

<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 $O$.

2018 CJC P1 Q11

<p>Show that&nbsp;$\frac{1+\mathrm{i}\tan \theta }{1-\mathrm{i}\tan \theta }={{\mathrm{e}}^{\mathrm{i}\left( 2\theta \right)}}$ where&nbsp;$-\frac{\pi }{2}&lt;\theta &lt;\frac{\pi }{2}$.</p>
<p>Hence find the value of ${{\left( \frac{1-\mathrm{i}\sqrt{3}}{1+\mathrm{i}\sqrt{3}} \right)}^{11}}$, giving your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\alpha }}$ where&nbsp;$r&gt;0$ and $-\pi &lt;\alpha \le \pi $.<span class="inline-marks">[5]</span></p>

Show that $\frac{1+\mathrm{i}\tan \theta }{1-\mathrm{i}\tan \theta }={{\mathrm{e}}^{\mathrm{i}\left( 2\theta \right)}}$ where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

Hence find the value of ${{\left( \frac{1-\mathrm{i}\sqrt{3}}{1+\mathrm{i}\sqrt{3}} \right)}^{11}}$, giving your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\alpha }}$ where $r>0$ and $-\pi <\alpha \le \pi $.

2013 MJC P1 Q11

<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

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

Modulus and Argument of Complex Numbers

Applying the Conjugate Root Theorem

2020 HCI 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">It is given that complex numbers ${{z}_{1}}$ and ${{z}_{2}}$ are the roots of the equation ${{z}^{2}}-6z+36=0$ such that $\arg ({{z}_{1}})>\arg ({{z}_{2}})$. Find exact expressions of ${{z}_{1}}$ and ${{z}_{2}}$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the complex number $\frac{{{z}_{1}}^{4}}{\mathrm{i}{{z}_{2}}}$ in exact polar form.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Find the smallest positive integer $n$ such that ${{z}_{2}}^{n}$ is a positive real number.</p><p style="text-align: right;">[2]</p>

Find the smallest positive integer $n$ such that ${{z}_{2}}^{n}$ is a positive real number.

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