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<p>Find the roots of ${{z}^{2}}+z+2=0$.</p>
<p>Hence, solve</p>
<ol class="i-list"><li data-list-item-id="item-1762581706526-h8s5i1hwk" data-list-item-path="item-1762581706526-h8s5i1hwk"><div class="list-item-paragraph">${{(3w-1)}^{2}}+(1-3w)+2=0$</div></li><li data-list-item-id="item-1762581707074-mz7bmbcgx" data-list-item-path="item-1762581707074-mz7bmbcgx"><div class="list-item-paragraph">$(w^*)^2-w^*+2=0$, where&nbsp;$w^*$ is the complex conjugate of $w$.</div></li><li data-list-item-id="item-1762581707898-luhk6ylp0" data-list-item-path="item-1762581707898-luhk6ylp0"><div class="list-item-paragraph">$\frac{1}{{{w}^{2}}}-\frac{1}{w}+2=0$.</div></li></ol>

Solving Complex Equations by Substitution

Complex Roots of Quadratics with Real Coefficients

Solving Equations Involving Complex Conjugates

Complex Numbers

<p>Find the roots of $z^{2} - 4z + 13 = 0$.</p>

<p>Find the roots of $z^{2} + 2z + 2 = 0$.</p>

<p>Find the roots of $z^{2} - 2z + 5 = 0$.</p>

<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>

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

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

Modulus, Argument, and Operations on Complex Numbers

Cartesian, Trigonometric and Exponential Forms

Properties of Trigonometric and Exponential Forms

Solving Complex Equations

2016 CJC P1 Q8

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5NjA4ODAtNGpuMWE4OWl4dTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg3NTY5NywiZXhwIjoxNzkyODc1OTk3fQ.5bXcLaUiJiKtnsHB4fSuR9lQkQ2JUuEMz_BSS8qaNUo" 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>Find the roots of ${{z}^{2}}+2z+5=0$.</p>
<p>Hence, solve</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{(1+w)}^{2}}+2(1+w)+5=0$</div></li><li><div class="list-item-paragraph">${{\left( {{w}^{}} \right)}^{2}}-2{{w}^{}}+5=0$</div></li><li><div class="list-item-paragraph">$\frac{1}{{{w}^{2}}}-\frac{2}{w}+5=0$.</div></li></ol>

${{\left( {{w}^{*}} \right)}^{2}}-2{{w}^{*}}+5=0$

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