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PSLE

PSLE Foundation Mathematics

<p>Find all the complex numbers $z$ such that</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$z^4 = -16$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$z^3 = -27i$</div></li></ol>

Find all the complex numbers $z$ such that

Finding Roots of Complex Numbers

Solving Complex Equations

Obtaining the roots of a complex number

Complex Numbers

<p>Given that $z=r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, solve for $z$ in the form $x+y\mathrm{i}$ if $4+\mathrm{i}\left( 4z+1 \right)=2r{{e}^{\mathrm{i}\left( \pi +\theta \right)}}$.</p>

Given that $z=r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, solve for $z$ in the form $x+y\mathrm{i}$ if $4+\mathrm{i}\left( 4z+1 \right)=2r{{e}^{\mathrm{i}\left( \pi +\theta \right)}}$.

Properties of Trigonometric and Exponential Forms

Properties from Exponential or Trigonometric form Q8

<p>Given that $z=e^{i\theta}$, find the value of $\theta$ in radians if
$3+i(3z+1)=e^{i(\pi+\theta)}$ and $-\pi<\theta<\pi$.</p>

Given that $z=e^{i\theta}$, find the value of $\theta$ in radians if
$3+i(3z+1)=e^{i(\pi+\theta)}$ and $-\pi<\theta<\pi$.

Properties from Exponential or Trigonometric form Q6

<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.**

Using Complex Numbers to Deduce Trigonometric Values

Arithmetic with Complex Numbers in Cartesian Form

Modulus, Argument, and Operations on Complex Numbers

Solving Equations Involving Complex Conjugates

2024 RI P1 Q8

<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

Cartesian, Trigonometric and Exponential Forms

2021 NYJC P1 Q9

<p>It is given that $z=1+2\mathrm{i}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Without using a calculator, find the values of ${{z}^{2}}$ and $\frac{1}{{{z}^{3}}}$ in cartesian form $x+\mathrm{i}y$, showing your working.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The real numbers $p$ and $q$ are such that $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$ is real. Find, in terms of $p$, the value of $q$ and the value of $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$.</div><p style="text-align: right;">[3]</p></li></ol>

Conditions for Real and Purely Imaginary Values

N2014 P1 Q5

<ol class="abc-list"><li><div class="list-item-paragraph">Find the set of values of $\alpha $ for which the expression $\frac{2-\mathbf{i}\sin \,2\alpha }{1+2\,\mathbf{i}\,\sin \,2\alpha }$ is purely real.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">If&nbsp;$z$ and $w$ are different non-zero complex numbers and $\left| w \right|=1$, find $\left| \frac{3\left( w-z \right)}{1-{{z}^{*}}w} \right|$.<span class="inline-marks">[4]</span></div></li></ol>

Applying Complex Conjugate Identities

2023 DHS P1 Q4

<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 $O$.

Geometrical Transformations of Complex Numbers

Representing Complex Numbers on an Argand Diagram

2018 CJC P1 Q11

<ol class="abc-list"><li data-list-item-id="item-1760092351140-vei3oc3jj" data-list-item-path="item-1760092351140-vei3oc3jj"><ol class="i-list"><li data-list-item-id="subitem-1760092352340-xtvst6ooe" data-list-item-path="item-1760092351140-vei3oc3jj/subitem-1760092352340-xtvst6ooe"><div class="list-item-paragraph">Show that the cubic polynomial ${{x}^{3}}+p{{x}^{2}}+{{p}^{2}}x+q$ can be reduced to ${{y}^{3}}+\left( \frac{2{{p}^{2}}}{3} \right)y+\alpha $ by the substitution $x=y-\frac{p}{3}$, where&nbsp;$\alpha $ is to be determined in terms of $p$ and $q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760092359863-y6g5jygmy" data-list-item-path="item-1760092351140-vei3oc3jj/subitem-1760092359863-y6g5jygmy"><div class="list-item-paragraph">Given that&nbsp;$-3\mathbf{i}$ is a root of the equation ${{y}^{3}}+6y-9\mathbf{i}=0$, find the other two roots exactly in the form $a+b\mathbf{i}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760092363748-ppa9nnycn" data-list-item-path="item-1760092351140-vei3oc3jj/subitem-1760092363748-ppa9nnycn"><div class="list-item-paragraph">Hence find the exact roots of the equation ${{x}^{3}}+3{{x}^{2}}+9x+7-9\mathbf{i}=0$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760092373220-yir23pzie" data-list-item-path="item-1760092373220-yir23pzie"><div class="list-item-paragraph">Given that&nbsp;$z={{\mathrm{e}}^{\mathbf{i}\theta }}$, show that&nbsp;$1+z+{{z}^{2}}+{{z}^{3}}+...+{{z}^{n-1}}={{z}^{\frac{n-1}{2}}}\left( \frac{\sin \frac{n\theta }{2}}{\sin \frac{\theta }{2}} \right)$.<span class="inline-marks">[3]</span></div></li></ol>

Geometric Progression

Solving Complex Equations by Substitution

2021 ACJC P1 Q9

Sequence and Series

<p>Write down the real root of</p>
<p style="text-align: center">${{w}^{3}}+8=0$,</p>
<p>Where&nbsp;$w$ is a complex number. </p>
<p>Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form. </p>
<p>Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$. </p>
<p>Determine the roots of $z$ such that&nbsp;${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.</p>

Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form. 

Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$. 

Determine the roots of $z$ such that ${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.

2009 NJC P1 Q9 (b) [modified]

<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.

Complex Roots of Quadratics with Real Coefficients

2016 CJC P1 Q8

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the roots of the equation ${{w}^{2}}\left( 1-\mathrm{i} \right)+4w+\left( 10+10\mathrm{i} \right)=0$, giving your answers in cartesian form $a+\mathrm{i}b$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given that $z=-\sqrt{3}+\mathrm{i}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find an exact expression for ${{z}^{5}}$. Give your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the three smallest positive whole number values of $n$ for which $\frac{{{z}^{n}}}{\mathrm{i}{{z}^{*}}}$ is purely imaginary.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Given that $\left| 1+\frac{p}{{{z}^{*}}} \right|=\sqrt{7}$, find exactly the possible values of the real number $p$.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Finding Roots of Equations with Complex Coefficients

2018 RI P1 Q7

<p>Find the exact values of $x$ and $y$, where $x,y\in \mathbb{R}$, such that $2{{e}^{-\left( \frac{3+x+\mathbf{i}y}{\mathbf{i}} \right)}}=1-\mathbf{i}$ .</p><p style="text-align: right;">[4]</p>

Find the exact values of $x$ and $y$, where $x,y\in \mathbb{R}$, such that $2{{e}^{-\left( \frac{3+x+\mathbf{i}y}{\mathbf{i}} \right)}}=1-\mathbf{i}$ .

2017 ACJC P1 Q7(b)

<p>Given that ${{\left| \frac{2\mathbf{i}-{{z}^{*}}}{z}-1 \right|}^{2}}+z=-\mathbf{i}$, find $z$ in the form $x+\mathbf{i}y$.</p>
<p>State the exact values of $\left| z \right|$ and $\arg \left( z \right)$.</p><p style="text-align: right;">[5]</p>

Given that ${{\left| \frac{2\mathbf{i}-{{z}^{*}}}{z}-1 \right|}^{2}}+z=-\mathbf{i}$, find $z$ in the form $x+\mathbf{i}y$.

State the exact values of $\left| z \right|$ and $\arg \left( z \right)$.

Modulus and Argument of Complex Numbers

RI Complex Numbers Tutorial Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that that for any complex number $z=r{{\mathrm{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $,</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{z}{z-r}=\frac{1}{2}-\frac{1}{2}\mathbf{i}\cot \left( \frac{\theta }{2} \right)$.</div><p style="text-align: right;">[2]</p></li></ol>

Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that

Show that that for any complex number $z=r{{\mathrm{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $,

2014 SAJC P2 Q4 [Modified]

<p>For this question, you may use the result ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{n}}=(\cos n\theta +\mathbf{i}\sin n\theta )$.</p>
<p>Find the exact value(s) of $\theta $, where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$, given that</p>
<p style="text-align: center">${{\left( \cos \theta +\mathbf{i}\sin \theta  \right)}^{3}}+{{(\cos \theta +\mathbf{i}\sin \theta )}^{5}}=-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$.</p><p style="text-align: right;">[6]</p>

For this question, you may use the result ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{n}}=(\cos n\theta +\mathbf{i}\sin n\theta )$.

Find the exact value(s) of $\theta $, where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$, given that

${{\left( \cos \theta +\mathbf{i}\sin \theta  \right)}^{3}}+{{(\cos \theta +\mathbf{i}\sin \theta )}^{5}}=-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$.

De Moivre's Theorem

2022 DHS P1 Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Using ${{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $ and the trigonometry formulas, show that $\left( {{\mathrm{e}}^{\mathbf{i}{{\theta }_{1}}}} \right)\left( {{\mathrm{e}}^{\mathbf{i}{{\theta }_{2}}}} \right)={{\mathrm{e}}^{\mathbf{i}\left( {{\theta }_{1}}+{{\theta }_{2}} \right)}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the least positive integer value of $n$ such that the complex number ${{\left( \mathbf{i}{{\mathrm{e}}^{\mathbf{i}\frac{\pi }{5}}} \right)}^{n}}$ is purely imaginary.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">The graph of $\mathrm{f}\left( x \right)={{x}^{3}}-7{{x}^{2}}+17x-15$ is shown below. It cuts the $x$-axis at $x=3$.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1764583960880-4jn1a89ixu9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5NjA4ODAtNGpuMWE4OWl4dTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg3NTg0MCwiZXhwIjoxNzkyODc2MTQwfQ.v2h-_msZS21QI_7sf6d5oBwFrVo1EhdQAfj50FcJ3Js" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">Without the use of a graphing calculator, show that the roots of the equation of $\mathrm{f}\left( x \right)=0$ are $m$, ${{z}_{0}}$ and ${{z}_{1}}$ where $m$ is a real constant and ${{z}_{0}}$ and ${{z}_{1}}$ are complex constants to be determined.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The complex number $w$ is such that \[w={{z}_{0}}+\lambda \left( {{z}_{0}}-{{z}_{1}} \right)\] where $\lambda $ is a real number. By considering an Argand diagram or otherwise, find the least value of $\left| w \right|$ when $\lambda $ varies.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The graph of $\mathrm{f}\left( x \right)={{x}^{3}}-7{{x}^{2}}+17x-15$ is shown below. It cuts the $x$-axis at $x=3$.

Loci in the Argand Diagram

2020 DHS P2 Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The equation ${{z}^{3}}+4{{z}^{2}}+8z+8=0$ has a root $z=-2$. Find the other two roots ${{z}_{1}}$ and ${{z}_{2}}$ where $-\pi <\arg {{z}_{2}}<\arg {{z}_{1}}\le \pi $.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the modulus and argument of $w$, where $w=\frac{{{z}_{1}}}{{{z}_{2}}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the set of positive integers $n$ for which ${{w}^{n}}$ is real, and show that, for these values of $n$, ${{w}^{n}}$ is $1$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Express ${{w}^{100}}-{{\left( {{w}^{*}} \right)}^{100}}$ in the form $k\mathbf{i}$, giving the exact value of $k$ in non-trigonometrical form.</span></div><p style="text-align: right;">[2]</p></li></ol>

The equation ${{z}^{3}}+4{{z}^{2}}+8z+8=0$ has a root $z=-2$. Find the other two roots ${{z}_{1}}$ and ${{z}_{2}}$ where $-\pi <\arg {{z}_{2}}<\arg {{z}_{1}}\le \pi $.

Find the modulus and argument of $w$, where $w=\frac{{{z}_{1}}}{{{z}_{2}}}$.

Find the set of positive integers $n$ for which ${{w}^{n}}$ is real, and show that, for these values of $n$, ${{w}^{n}}$ is $1$.

Express ${{w}^{100}}-{{\left( {{w}^{*}} \right)}^{100}}$ in the form $k\mathbf{i}$, giving the exact value of $k$ in non-trigonometrical form.

2018 NYJC P2 Q3

<p>Given that $z={{e}^{\mathrm{i}\theta}}$, find the value of $\theta$ in radian $3+\mathrm{i}(3z+1)={{e}^{\mathrm{i}(\pi+\theta)}}$ and $-\pi<\theta<\pi$.</p>

Given that $z={{e}^{\mathrm{i}\theta}}$, find the value of $\theta$ in radian $3+\mathrm{i}(3z+1)={{e}^{\mathrm{i}(\pi+\theta)}}$ and $-\pi<\theta<\pi$.

Solving Equation involving Exponential and Cartesian Form

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that $\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Given that $\left| z \right|=1$, find $\left| \frac{z-3\,\mathbf{i}}{1+3\,\mathbf{i}\,z} \right|$.</span></div><p style="text-align: right;">[3]</p></li></ol>

Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that $\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.

Given that $\left| z \right|=1$, find $\left| \frac{z-3\,\mathbf{i}}{1+3\,\mathbf{i}\,z} \right|$.

Math Question Bank