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PSLE Foundation Mathematics

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>It is given that&nbsp;$1-3i$ is a complex root of the equation $z^3+pz^2+(p+12)z+q=0$, where&nbsp;$p$ and $q$ are real numbers.</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q04-item01" data-list-item-path="nyjc-q04-item01"><div class="list-item-paragraph">State another complex root of the equation in the form $a+ib$, where&nbsp;$a$ and $b$ are real numbers and $b\ne0$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="nyjc-q04-item02" data-list-item-path="nyjc-q04-item02"><div class="list-item-paragraph">Find the values of $p$ and $q$, and find the third root.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="nyjc-q04-item03" data-list-item-path="nyjc-q04-item03"><div class="list-item-paragraph">The point $A$ represents the complex number $1-3i$ on an Argand diagram with respect to the origin $O$. The point $C$ represents the complex number $x+iy$, where&nbsp;$x$ and $y$ are real numbers and $x&gt;0$. Find the values of $x$ and $y$ such that&nbsp;$OA=AC$ and angle $OAC$ is $90^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

<strong>Do not use a calculator in answering this question.</strong>

It is given that $1-3i$ is a complex root of the equation $z^3+pz^2+(p+12)z+q=0$, where $p$ and $q$ are real numbers.

State another complex root of the equation in the form $a+ib$, where $a$ and $b$ are real numbers and $b\ne0$.

Find the values of $p$ and $q$, and find the third root.

The point $A$ represents the complex number $1-3i$ on an Argand diagram with respect to the origin $O$. The point $C$ represents the complex number $x+iy$, where $x$ and $y$ are real numbers and $x>0$. Find the values of $x$ and $y$ such that $OA=AC$ and angle $OAC$ is $90^\circ$.

Geometrical Transformations of Complex Numbers

Applying the Conjugate Root Theorem

Geometric Problems on the Argand Diagram

Representing Complex Numbers on an Argand Diagram

2026 NYJC P1 Q4

Complex Numbers

<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

Modulus and Argument of Complex Numbers

2025 ACJC P1 Q10

<p>The complex number $w$ is given by $1 + 2\mathrm{i}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find $w^3$ in the form $x + \mathrm{i}y$, showing your working.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Given that&nbsp;$w$ is a root of the equation $az^3 + 5z^2 + 17z + b = 0$, find the values of the real numbers $a$ and $b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using these values of $a$ and $b$, find all the roots of this equation in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $w$ is given by $1 + 2\mathrm{i}$.

Arithmetic with Complex Numbers in Cartesian Form

N2013 P1 Q4

<p><strong>Conjugate Roots Theorem</strong></p>
<p>Given a <strong>Polynomial Equation</strong> with <strong>Real Coefficients</strong>:</p>
<p>\[a_0+a_1x+a_2x^2+a_3x^3+\dots+a_{n-1}x^{n-1}+a_nx^n=0 \text{ where}\hspace{0.5em} a_0, a_1, a_2, a_3, \dots, a_{n-1}, a_n \in \mathbb{R}\]</p>
<p>If&nbsp;$z$ is a complex root of the equation, then&nbsp;$z^*$ is also a complex root of the equation.</p>
<p><strong>Proof</strong></p>

Given a **Polynomial Equation** with **Real Coefficients**:

\[a_0+a_1x+a_2x^2+a_3x^3+\dots+a_{n-1}x^{n-1}+a_nx^n=0 \text{ where } a_0, a_1, a_2, a_3, \dots, a_{n-1}, a_n \in \mathbb{R}\]

If $z$ is a complex root of the equation, then $z^*$ is also a complex root of the equation.

<p>A graphic calculator is not to be used in answering this question.</p>
<ol class="i-list"><li data-list-item-id="item-1770891051005-db9u4khhg" data-list-item-path="item-1770891051005-db9u4khhg"><div class="list-item-paragraph">It is given that&nbsp;${{z}_{1}}=1+\sqrt{3}\mathbf{i}$. Find the value of ${{z}_{1}}^{3}$, showing clearly how you obtain your answer.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1770891051641-3s5k7iqvi" data-list-item-path="item-1770891051641-3s5k7iqvi"><div class="list-item-paragraph">Given that&nbsp;$1+\sqrt{3}\mathbf{i}$ is a root of the equation $2{{z}^{3}}+a{{z}^{2}}+bz+4=0$, find the values of the real numbers $a$ and $b$.	<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1770891052242-oerr5ud5g" data-list-item-path="item-1770891052242-oerr5ud5g"><div class="list-item-paragraph"><p>For these values of $a$ and $b$, solve the equation in part <strong>(ii)</strong>, and show all the roots on an Argand diagram.</p><span class="inline-marks">[4]</span></div></li></ol>

A graphic calculator is not to be used in answering this question.

N2008 P1 Q8

<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.

Solving Complex Equations by Substitution

2025 NYJC P2 Q2

<p>Given that the equation ${{z}^{3}}-4{{z}^{2}}+6z-4=0$ has a root $1-\mathbf{i}$, find the remaining roots of the equation.</p>

Given that the equation ${{z}^{3}}-4{{z}^{2}}+6z-4=0$ has a root $1-\mathbf{i}$, find the remaining roots of the equation.

Worked Example 2

<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>Given that the equation $z^{4}-3z^{3}+8z^{2}-7z+5=0$ has a root $1-2\mathbf{i}$, solve the equation.</p>

Given that the equation $z^{4}-3z^{3}+8z^{2}-7z+5=0$ has a root $1-2\mathbf{i}$, solve the equation.

<p>Given that&nbsp;$-1-\mathbf{i}$ is a root of the equation $z^{3}-z^{2}-4z+p=0$, find the value of $p$, and the other two roots.</p>

Given that $-1-\mathbf{i}$ is a root of the equation $z^{3}-z^{2}-4z+p=0$, find the value of $p$, and the other two roots.

<p>Given that the equation $4z^{3}-11z^{2}+25=0$ has a root $2+\mathbf{i}$, find the remaining roots of the equation.</p>

Given that the equation $4z^{3}-11z^{2}+25=0$ has a root $2+\mathbf{i}$, find the remaining roots of the equation.

<p>Given that the equation ${{z}^{3}}-4{{z}^{2}}+6z-4=0$ has a root $1-\mathbf{i}$, find the remaining roots of the equation.</p>

<p>It is given that $-2+2i$ is a root of the equation</p>
<p style="text-align: center">${{z}^{3}}+a{{z}^{2}}+bz-16\sqrt{2}=0$,</p>
<p>where $a$ and $b$ are real numbers.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$ and the other two roots. Leave your answers in the exact form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">In an Argand diagram with origin $O$, the three roots are represented by points $A$, $B$ and $C$ where $A$ represents $-2+2i$ and $C$ represents the real root. Label these points on an Argand diagram, indicating clearly the modulus and argument of each root.</div><div class="list-item-paragraph">State also a geometrical relationship between $A$ and $B$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence, prove that</div><div class="list-item-paragraph" style="text-align: center">$\tan \frac{3\pi }{8}=1+\sqrt{2}$.</div><p style="text-align: right;">[3]</p></li></ol>

It is given that $-2+2i$ is a root of the equation

Using Complex Numbers to Deduce Trigonometric Values

2025 TMJC P1 Q9

<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><strong>Do not use a graphing calculator in answering this question.</strong></p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$w=\mathrm{ }1\mathrm{ }+\mathrm{ }\mathbf{i}$ is a root of the equation</div><div class="list-item-paragraph" style="text-align: center">${{w}^{3}}+\mathrm{ }3{{w}^{2}}+pw+q=\mathrm{ }0$,</div><div class="list-item-paragraph">find the values of the real numbers $p$ and $q$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">For these values of $p$ and $q$, solve the equation in part <strong>(i)</strong>.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

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

2013 TPJC P2 Q3

<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="abc-list"><li><div class="list-item-paragraph">One root of the equation $2{{z}^{3}}-5{{z}^{2}}+\alpha z-5=0$, where&nbsp;$\alpha \in \mathbb{R}$, is $z=1-2\mathbf{i}$.</div><div class="list-item-paragraph">Find the value of $\alpha $ and the other roots.<span class="inline-marks">[5]</span></div></li></ol>

2023 RI Promo Q8

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-1760434940391-ewp8cwfwk"><div class="list-item-paragraph">One of the roots of the equation $3{{x}^{3}}+p{{x}^{2}}+qx+3=0$, where&nbsp;$p$ and $q$ are real, is $1+\sqrt{2}\,\mathbf{i}$.</div><div class="list-item-paragraph">Find the other roots and the values of  $p$ and $q$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760434941013-gdr3de9g9"><div class="list-item-paragraph">The complex numbers $w$ and $z$, with $\left| w \right|&lt;\left| z \right|$, satisfy the simultaneous equations</div><div class="list-item-paragraph" style="text-align: center">$w{{z}^{*}}=3-\mathbf{i}$ and $w+{{z}^{*}}=2-\mathbf{i}$.</div><div class="list-item-paragraph">Find $w$ and $z$.<span class="inline-marks">[5]</span></div></li></ol>

Solving Simultaneous Complex Equations

2023 VJC Promo Q7

<ol class="i-list"><li data-list-item-id="item-1760418529048-pv772tg9x"><div class="list-item-paragraph">Two complex numbers $w$ and $u$ satisfy the equations</div><div class="list-item-paragraph" style="text-align: center">${{w}^{*}}-2\mathbf{i}u=8$ and $(2\mathbf{i}-1)w+2{{u}^{*}}=4$.</div><div class="list-item-paragraph">Find $w$ and $u$, giving your answers in the form $x+\mathbf{i}y$, where&nbsp;$x$ and $y$ are real.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760418530234-myl4mvt7t"><div class="list-item-paragraph">Given that both&nbsp;$w$ and $u$ are roots of the equation $\left( {{z}^{2}}-4z+c \right)\left( {{z}^{2}}-dz+10 \right)=0$, find the values of the real numbers $c$ and $d$.<span class="inline-marks">[3]</span></div></li></ol>

2022 NJC 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>

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