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H2 Math

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PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="item-a-6" data-list-item-path="item-a-6"><div class="list-item-content"><span class="list-item-text">It is given that the roots to the equation $-ix^3 + 5ix^2 + ax + b = 0$, where&nbsp;$a$ and $b$ are purely imaginary, are $2+i$, $2-i$ and $1$. Explain why the complex roots occur in conjugate pairs.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-6" data-list-item-path="item-b-6"><div class="list-item-paragraph">By using <strong>(a)</strong> and an appropriate substitution, find the roots of the equation $ix^3 + 5ix^2 + a^*x + b = 0$, where the complex conjugate of $a$ is denoted by $a^*$.<span class="inline-marks">[4]</span></div></li></ol>

It is given that the roots to the equation $-ix^3 + 5ix^2 + ax + b = 0$, where $a$ and $b$ are purely imaginary, are $2+i$, $2-i$ and $1$. Explain why the complex roots occur in conjugate pairs.

Applying the Conjugate Root Theorem

Solving Complex Equations by Substitution

2025 NYJC P2 Q2

Complex Numbers

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>It is given that&nbsp;$1+2\mathrm{i}$ is a complex root of the equation ${{z}^{3}}+p{{z}^{2}}+qz-10=0$, where&nbsp;$p$ and $q$ are real numbers.</p>
<ol class="abc-list"><li data-list-item-id="item-1762629547121-714t7woh5" data-list-item-path="item-1762629547121-714t7woh5"><div class="list-item-paragraph">State another complex root of the equation in the form $a+\mathrm{i}b$, where&nbsp;$a$ and $b$ are real numbers and 	$b\ne 0$. Justify your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1762629547692-iril9z6g1" data-list-item-path="item-1762629547692-iril9z6g1"><div class="list-item-paragraph">Find the values of $p$ and $q$, and find the third root.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1762629549049-hrgw35f8u" data-list-item-path="item-1762629549049-hrgw35f8u"><div class="list-item-paragraph">Hence find the roots of the equation $1+pw+q{{w}^{2}}-10{{w}^{3}}=0$.<span class="inline-marks">[2]</span></div></li></ol>

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

It is given that $1+2\mathrm{i}$ is a complex root of the equation ${{z}^{3}}+p{{z}^{2}}+qz-10=0$, where $p$ and $q$ are real numbers.

N2025 P1 Q12

<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>

Complex Roots of Quadratics with Real Coefficients

Solving Equations Involving Complex Conjugates

<ol class="i-list"><li><div class="list-item-paragraph">It is given that $-1+2\mathbf{i}$ is a root of the equation ${{z}^{3}}+2\left( 1+\mathbf{i} \right){{z}^{2}}+\left( 5+4\mathbf{i} \right)z+10\mathbf{i}=0$.</div><div class="list-item-paragraph">Explain why $-1-2\mathbf{i}$ may not be a root.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Without using a calculator, solve the equation ${{z}^{3}}+2\left( 1+\mathbf{i} \right){{z}^{2}}+\left( 5+4\mathbf{i} \right)z+10\mathbf{i}=0$, giving your answers in the form $a+\mathbf{i}\,b$, where $a$ and $b$ are exact values.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Hence solve $\mathbf{i}{{z}^{3}}+2\left( 1+\mathbf{i} \right){{z}^{2}}+\left( 4-5\mathbf{i} \right)z-10\mathbf{i}=0$.</div><p style="text-align: right;">[2]</p></li></ol>

Finding Roots of Equations with Complex Coefficients

2022 NYJC P1 Q5

<p>The complex numbers ${{z}_{1}}=2$ and ${{z}_{2}}=\,\,\mathbf{i}$ are roots of the cubic equation ${{z}^{3}}+A{{z}^{2}}+Bz+C=\mathrm{ }0$, where&nbsp;$A$, $B$ and $C$ are real constants.</p>
<ol class="i-list"><li data-list-item-id="item-1760435081766-y68877ii8" data-list-item-path="item-1760435081766-y68877ii8"><div class="list-item-paragraph">State the third root and find the values of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760435083506-ujakvald1" data-list-item-path="item-1760435083506-ujakvald1"><div class="list-item-paragraph">Hence, with a suitable substitution, find the roots of the equation</div><div class="list-item-paragraph" style="text-align: center">$2w^3-w^2+2w-1=0$.<span class="inline-marks">[3]</span></div></li></ol>

The complex numbers ${{z}_{1}}=2$ and ${{z}_{2}}=\,\,\mathbf{i}$ are roots of the cubic equation ${{z}^{3}}+A{{z}^{2}}+Bz+C=\mathrm{ }0$, where $A$, $B$ and $C$ are real constants.

2013 SRJC P1 Q4

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="i-list"><li><div class="list-item-paragraph">One of the roots of the equation $2{{z}^{3}}-9{{z}^{2}}+30z+b=0$ is $1+a\mathrm{i}$, where $a$ and $b$ are non-zero real numbers. Find the values of $a$ and $b$ are non-zero real numbers. Find the values of $a$ and $b$ and the roots of this equation.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Hence solve the equation $b{{z}^{3}}+30{{z}^{2}}-9z+2=0$.</div><p style="text-align: right;">[2]</p></li></ol>

Equating Real and Imaginary Parts

2021 YIJC P1 Q6

<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>

Modulus and Argument of Complex Numbers

Finding Square Roots of Complex Numbers Algebraically

Arithmetic with Complex Numbers in Cartesian Form

N2004 P1 Q12

<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

Properties of Trigonometric and Exponential Forms

Solving Complex Equations

2021 ACJC P1 Q9

Sequence and Series

<ol class="abc-list"><li><div class="list-item-paragraph">The complex numbers $z$ and $w$ satisfy the simultaneous equations</div><div class="list-item-paragraph" style="text-align: center">$\left| z \right|-{{w}^{*}}=-3-\sqrt{2}\mathbf{i}$ and ${{w}^{*}}+w+5z=1+20\mathbf{i}$,</div><div class="list-item-paragraph">where ${{w}^{*}}$ is the complex conjugate of $w$. Find the value of $z$ and the corresponding value of $w$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">It is given that $8\mathbf{i}$ is a root of the equation $\mathbf{i}{{z}^{3}}+\left( 8-2\mathbf{i} \right){{z}^{2}}+az+40=0$ where $a$ is a complex number.</div><ol class="i-list"><li><div class="list-item-paragraph">Find $a$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, find the other roots of the equation, leaving your answer in the form $a+b\mathbf{i}$ where $a$ and $b$ are real constants.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce the number of real roots in the equation, ${{z}^{3}}-\left( 8-2\mathbf{i} \right){{z}^{2}}+a\mathbf{i}z+40=0$ has.</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

Solving Simultaneous Complex Equations

2019 NYJC P1 Q9

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Two complex numbers are $w=1-\sqrt{3}\mathrm{i}$ and $z=\sqrt{2}\left( \cos \frac{3}{4}\pi -\mathrm{i}\sin \frac{3}{4}\pi \right)$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find ${{w}^{2}}{{z}^{*}}$ in the form $r\left( \cos \theta +\mathrm{i}\sin \theta \right)$, where $r>0$ and $-\pi <\theta \le \pi $.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Show that there is no integer value of $n$ for which the real part of ${{w}^{n}}$ is zero.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">One of the roots of the equation $3{{z}^{3}}+13{{z}^{2}}+20z+14=0$ is $-1+\mathrm{i}$. Find the other roots of the equation in cartesian form, $p+\mathrm{i}q$, showing your working.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find the roots of the equation ${{w}^{3}}+13{{w}^{2}}+60w+126=0$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Modulus, Argument, and Operations on Complex Numbers

2021 MI P1 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">One root of the equation ${{z}^{4}}+2{{z}^{3}}+a{{z}^{2}}+bz+50=0$, where&nbsp;$a$ and $b$ are real, is $z=1+3\mathbf{i}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$a=7$ and $b=30$ and find the other roots of the equation.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Deduce the roots of the equation ${{w}^{4}}-2\mathbf{i}{{w}^{3}}-7{{w}^{2}}+30\mathbf{i}w+50=0$<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2017 RI P1 Q9

<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 ${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.

Finding Roots of Complex Numbers

2009 NJC P1 Q9 (b) [modified]

<p><strong>Do not use a calculator in answering this question.</strong></p>
<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">$w-2z=9$, </div><div class="list-item-paragraph" style="text-align: center">$3w-wz^*=17-30\mathrm{i}$.</div><div class="list-item-paragraph">Find $w$ and $z$ in the form $a+b\mathrm{i}$, where $a$ and $b$ are real and $\operatorname{Re}(z)<0$.</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">Given that $-\mathrm{i}$ is a root of the equation</div><div class="list-item-paragraph" style="text-align: center">${{z}^{3}}+k{{z}^{2}}+\left( 8+2\sqrt{2}\mathrm{i} \right)z+8\mathrm{i}=0$,</div><div class="list-item-paragraph">where $k$ is a constant to be determined, find the other roots, leaving your answers in exact cartesian form $x+y\mathrm{i}$, showing your working.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence solve the equation $\mathrm{i}{{z}^{3}}+k{{z}^{2}}+\left( 2\sqrt{2}-8\mathrm{i} \right)z-8\mathrm{i}=0$, leaving your answers in exact cartesian form.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Let ${{z}_{0}}$ be the root in (i) such that $\arg ({{z}_{0}})>0$. Find the smallest positive integer value of $n$ such that${{(\mathrm{i}{{z}_{0}})}^{n}}$ is a purely imaginary number.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

2019 ACJC P1 Q8

<ol class="i-list"><li><div class="list-item-paragraph">Show that for any complex number $z=r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$, and $-\pi <\theta \le \pi $,</div><div class="list-item-paragraph" style="text-align: center">$\frac{z}{z-r}=\frac{1}{2}-\frac{1}{2}\left( \cot \frac{\theta }{2} \right)\mathrm{i}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $z=2{{\mathrm{e}}^{\mathrm{i}\left( \frac{\pi }{3} \right)}}$ is a root of the equation ${{z}^{2}}-2z+4=0$. State, in similar form, the other root of the equation.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using parts <strong>(i)</strong> and <strong>(ii)</strong>, solve the equation $\frac{4{{w}^{2}}}{{{\left( w-1 \right)}^{2}}}-\frac{4w}{w-1}+4=0$.</div><p style="text-align: right;">[4]</p></li></ol>

2017 SAJC P1 Q7

<p><strong>Do not use a calculator in answering this question</strong></p>
<p>It is given that&nbsp;${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are the roots of the equation</p>
<p style="text-align: center">$2{{z}^{3}}+p{{z}^{2}}+qz-4=0$</p>
<p>such that&nbsp;$\arg {{z}_{1}}&lt;\arg {{z}_{2}}&lt;\arg {{z}_{3}}$ and ${{z}_{1}}=1-\mathbf{i}\sqrt{3}$. Find the values of the real numbers $p$ and $q$ and the remaining roots of the equation.<span class="inline-marks">[6]</span></p>
<p>Hence solve the equation $2iz^3+pz^2-qiz+4=0$, leaving your answers in exact Cartesian form.<span class="inline-marks">[2]</span></p>

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

It is given that ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are the roots of the equation

such that $\arg {{z}_{1}}<\arg {{z}_{2}}<\arg {{z}_{3}}$ and ${{z}_{1}}=1-\mathbf{i}\sqrt{3}$. Find the values of the real numbers $p$ and $q$ and the remaining roots of the equation.

Hence solve the equation $2iz^3+pz^2-qiz+4=0$, leaving your answers in exact Cartesian form.

2025 RI Complex Numbers Assignment Q3

<p>One of the roots of the equation ${{z}^{4}}-2{{z}^{3}}+a{{z}^{2}}-8z+40=0$ is $b\mathrm{i}$, where $a$ and $b$ are positive real constants.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $a$ and $b$ and hence find the other roots of the equation.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Deduce that roots of the equation ${{w}^{4}}+2{{w}^{3}}+a{{w}^{2}}+8w+40=0$.</span></div><p style="text-align: right;">[3]</p></li></ol>

One of the roots of the equation ${{z}^{4}}-2{{z}^{3}}+a{{z}^{2}}-8z+40=0$ is $b\mathrm{i}$, where $a$ and $b$ are positive real constants.

Find the values of $a$ and $b$ and hence find the other roots of the equation.

Deduce that roots of the equation ${{w}^{4}}+2{{w}^{3}}+a{{w}^{2}}+8w+40=0$.

2019 MI P2 Q5 (a)

<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">Given that $-\mathbf{i}$ is a root of the equation ${{z}^{3}}+k{{z}^{2}}+\left( 8+2\sqrt{2}\,\mathbf{i} \right)z+8\mathbf{i}=0$, where $k$ is a constant to be determined, find the other roots, leaving your answers in exact cartesian form $x+y\mathbf{i}$, showing your working.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence solve the equation $\mathbf{i}{{z}^{3}}+k{{z}^{2}}+\left( 2\sqrt{2}-8\mathbf{i} \right)z-8\mathbf{i}=0$, leaving your answers in exact cartesian form.</span></div><p style="text-align: right;">[2]</p></li></ol>

Given that $-\mathbf{i}$ is a root of the equation ${{z}^{3}}+k{{z}^{2}}+\left( 8+2\sqrt{2}\,\mathbf{i} \right)z+8\mathbf{i}=0$, where $k$ is a constant to be determined, find the other roots, leaving your answers in exact cartesian form $x+y\mathbf{i}$, showing your working.

Hence solve the equation $\mathbf{i}{{z}^{3}}+k{{z}^{2}}+\left( 2\sqrt{2}-8\mathbf{i} \right)z-8\mathbf{i}=0$, leaving your answers in exact cartesian form.

RI Complex Numbers Tutorial Q7

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The complex numbers $z$ and $w$ satisfy the simultaneous equations</span></div><div class="list-item-paragraph" style="text-align: center">$z+{{w}^{*}}+5\mathbf{i}=10$ and ${{\left| w \right|}^{2}}=z+18+\mathbf{i}$.</div><div class="list-item-paragraph">Find $z$ and $w$.<span class="inline-marks">[4]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">It is given that&nbsp;$2+\mathbf{i}$ is a root of the equation ${{z}^{2}}-5z+7+\mathbf{i}=0$. Find the second root of the equation in Cartesian form, showing your working clearly.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}+5w+7\mathbf{i}-1=0$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The complex number $z$ is given by $z=-a+a\,\mathbf{i}$, where&nbsp;$a$ is a positive real number.</div><ol class="i-list"><li><div class="list-item-paragraph">It is given that&nbsp;$w=-{{\frac{\sqrt{2}z}{{{z}^{4}}}}^{*}}$. Express $w$ in the form $r{{\mathrm{e}}^{i\theta }}$, in terms of $a$, where&nbsp;$r&gt;0$ and $-\pi &lt;\theta \le \pi $.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the two smallest positive whole number values of $n$ such that&nbsp;$\operatorname{Re}\,({{w}^{n}})=0$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The complex numbers $z$ and $w$ satisfy the simultaneous equations

The complex number $z$ is given by $z=-a+a\,\mathbf{i}$, where $a$ is a positive real number.

2017 CJC P2 Q4

<ol class="i-list"><li data-list-item-id="item-1759136124046-w0h486src" data-list-item-path="item-1759136124046-w0h486src"><div class="list-item-paragraph">Find the roots of the equation $\mathbf{i}{{z}^{2}}-\left( 5+\mathbf{i} \right)z+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where&nbsp;$a$, $b\in \mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759136124549-52l3hviq4" data-list-item-path="item-1759136124549-52l3hviq4"><div class="list-item-paragraph">Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}-\left( 1-5\mathbf{i} \right)w+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where&nbsp;$a$, $b\in \mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759136125232-uc8yzfzdp" data-list-item-path="item-1759136125232-uc8yzfzdp"><div class="list-item-paragraph"><p>Given that the roots found in part <strong>(i)</strong> are also roots of the equation $P\left( z \right)=0$, where&nbsp;$P\left( z \right)$ is a polynomial of degree $4$ with real coefficients, find $P\left( z \right)$.</p><span class="inline-marks">[3]</span></div></li></ol>

Find the roots of the equation $\mathbf{i}{{z}^{2}}-\left( 5+\mathbf{i} \right)z+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.

Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}-\left( 1-5\mathbf{i} \right)w+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.

2022 ACJC P2 Q3

<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">Explain why the equation ${{z}^{3}}+a{{z}^{2}}+az+7=0$ cannot have more than two non-real roots, where $a$ is a real constant.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Given that $z=-7$ is a root of the equation in (i), find the other roots, leaving your answers in the form $r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $ .</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the equation $\mathrm{i}{{z}^{3}}+8{{z}^{2}}-8\mathrm{i}z-7=0$, leaving your answers in the form $r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $ .</span></div><p style="text-align: right;">[2]</p></li></ol>

Explain why the equation ${{z}^{3}}+a{{z}^{2}}+az+7=0$ cannot have more than two non-real roots, where $a$ is a real constant.

Given that $z=-7$ is a root of the equation in (i), find the other roots, leaving your answers in the form $r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $ .

Hence, solve the equation $\mathrm{i}{{z}^{3}}+8{{z}^{2}}-8\mathrm{i}z-7=0$, leaving your answers in the form $r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $ .

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