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PSLE Foundation Mathematics

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the roots of the equation $x^2(1+\mathrm{i})-4x+(4-4\mathrm{i})=0$, giving your answers in cartesian form $a+\mathrm{i}b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The complex numbers $z$ and $w$ are such that&nbsp;$z=1+\mathrm{i}$ and $w=-1-\mathrm{i}\sqrt{3}$. Find the modulus and argument of $\frac{z}{w^*}$ exactly.<span class="inline-marks">[3]</span></div></li></ol>

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

Modulus, Argument, and Operations on Complex Numbers

Finding Roots of Equations with Complex Coefficients

Complex Numbers

<p><strong>A graphing calculator is not allowed for this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the square roots of $2\mathrm{i}$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $w^2 = 3 - 4\mathrm{i}$. Hence find the roots of the equation $z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0$, giving your answers in Cartesian form.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$w = \frac{1}{2 + \mathrm{i}\alpha}$ where&nbsp;$\alpha \in \mathbb{R}$, and $w^*$ denotes the complex conjugate of $w$, find $ww^*$ and hence prove that&nbsp;$w + w^* = 4ww^*$ ---(1).</div><div class="list-item-paragraph">The complex number $w$ is represented in an Argand diagram by the point $A$. By using (1) and considering $w = x + y\mathrm{i}$, where&nbsp;$x, y \in \mathbb{R}$, show that, $A$ lies on a circle as&nbsp;$\alpha$ varies. Find the Cartesian equation of this circle, stating clearly its centre and radius.<span class="inline-marks">[6]</span></div></li></ol>

**A graphing calculator is not allowed for this question.**

Find the square roots of $2\mathrm{i}$.

Solve the equation $w^2 = 3 - 4\mathrm{i}$. Hence find the roots of the equation $z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0$, giving your answers in Cartesian form.

Given that $w = \frac{1}{2 + \mathrm{i}\alpha}$ where $\alpha \in \mathbb{R}$, and $w^*$ denotes the complex conjugate of $w$, find $ww^*$ and hence prove that $w + w^* = 4ww^*$ ---(1).

Applying Complex Conjugate Identities

Finding Square Roots of Complex Numbers Algebraically

Graphs of Conic Sections

Loci in the Argand Diagram

2020 ASRJC P1 Q5(a) | 2024 SAJC P1 Q8b

Graphing Techniques

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>Find the roots of the equation $z^2-(1+2i)z+1+7i=0$, giving your answers in the cartesian form $a+ib$.<span class="inline-marks">[6]</span></p>

Find the roots of the equation $z^2-(1+2i)z+1+7i=0$, giving your answers in the cartesian form $a+ib$.

2025 RVHS P1 Q3

<p>Given that&nbsp;$1-3\mathbf{i}$ is a root of the equation $z^{3}-\left( 7+\mathbf{i} \right)z^{2}+\left( 31-8\mathbf{i} \right)z-31+33\mathbf{i}=0$, find the other roots of the equation.</p>

Given that $1-3\mathbf{i}$ is a root of the equation $z^{3}-\left( 7+\mathbf{i} \right)z^{2}+\left( 31-8\mathbf{i} \right)z-31+33\mathbf{i}=0$, find the other roots of the equation.

<p>Given that&nbsp;$1+2\mathbf{i}$ is a root of the equation $z^{3}-4z^{2}+(6+\mathbf{i})z-5+5\mathbf{i}=0$, find the other roots of the equation.</p>

Given that $1+2\mathbf{i}$ is a root of the equation $z^{3}-4z^{2}+(6+\mathbf{i})z-5+5\mathbf{i}=0$, find the other roots of the equation.

<p>The roots of the equation $x^{2}+(3\mathbf{i}-5)x+6-9\mathbf{i}=0$ are $\alpha$ and $\beta$, where&nbsp;$\alpha$ is a real number and $\beta$ is not a real number. Find $\alpha$ and $\beta$.</p>

The roots of the equation $x^{2}+(3\mathbf{i}-5)x+6-9\mathbf{i}=0$ are $\alpha$ and $\beta$, where $\alpha$ is a real number and $\beta$ is not a real number. Find $\alpha$ and $\beta$.

<p>If&nbsp;$z=\dfrac{2-i}{1+3i}$, find the real and imaginary parts of $z-\dfrac1z$.</p>
<p>Hence, find all complex numbers $w$ in Cartesian form such that&nbsp;$w-\dfrac1w=\dfrac1{10}+\dfrac{21}{10}i$.</p>

If $z=\dfrac{2-i}{1+3i}$, find the real and imaginary parts of $z-\dfrac1z$.

Hence, find all complex numbers $w$ in Cartesian form such that $w-\dfrac1w=\dfrac1{10}+\dfrac{21}{10}i$.

Arithmetic with Complex Numbers in Cartesian Form

<p>If&nbsp;$z=\dfrac{1-\mathbf{i}}{2+\mathbf{i}}$, find the real and imaginary part of $z-\dfrac{1}{z}$.</p>
<p>Hence, find the complex number $w$ in Cartesian form such that&nbsp;$w-\dfrac{1}{w}=-\dfrac{3}{10}+\dfrac{21}{10}\mathbf{i}$.</p>

If $z=\dfrac{1-\mathbf{i}}{2+\mathbf{i}}$, find the real and imaginary part of $z-\dfrac{1}{z}$.

Hence, find the complex number $w$ in Cartesian form such that
$w-\dfrac{1}{w}=-\dfrac{3}{10}+\dfrac{21}{10}\mathbf{i}$.

<p>Find the square roots of the complex number $15-8\mathbf{i}$.<br>Hence, solve the quadratic equation $2{{z}^{2}}+\mathbf{i}z-2+\mathbf{i}=0$. </p>

Find the square roots of the complex number $15-8\mathbf{i}$.
Hence, solve the quadratic equation $2{{z}^{2}}+\mathbf{i}z-2+\mathbf{i}=0$. 

<p>Find the exact square roots of the complex number $6\mathbf{i}$.</p>
<p>Hence, solve the quadratic equation $z^{2} + 2\mathbf{i}z - 1 - 6\mathbf{i} = 0$.</p>

Find the exact square roots of the complex number $6\mathbf{i}$.

Hence, solve the quadratic equation $z^{2} + 2\mathbf{i}z - 1 - 6\mathbf{i} = 0$.

<ol class="abc-list"><li data-list-item-id="item-1760958379712-j660wkjfj" data-list-item-path="item-1760958379712-j660wkjfj"><div class="list-item-paragraph">Given that imaginary unit $\mathbf{i}$ is one of the roots of the equation ${{z}^{3}}-2{{z}^{2}}+az+1+3\mathbf{i}=0$, find the complex number ${a}$ and the other roots.</div></li><li data-list-item-id="item-1760958380522-umltyb9vc" data-list-item-path="item-1760958380522-umltyb9vc"><div class="list-item-paragraph"><p>The root that has the largest modulus in <strong>(a)</strong> is denoted as&nbsp;${w}$ and represented by the point ${A}$. The points ${B, C}$ and ${D}$ represent the complex numbers ${{w}^{*}}$, $\mathbf{i}w$ and ${{\mathbf{i}}^{2}}w$ respectively. On the same diagram, plot the points ${A, B, C}$ and ${D}$ and state each transformation that maps the point ${A}$ onto the point ${B, C}$ and ${D}$ respectively and state the angle ${OAC}$.</p></div></li></ol>

Geometrical Transformations of Complex Numbers

Geometric Problems on the Argand Diagram

Representing Complex Numbers on an Argand Diagram

2014 HCI P1 Q3 [Modified]

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $z$ satisfies the equation</p>
<p style="text-align: center">${{z}^{2}}-\left( 4+\mathbf{i} \right)z+2\left( \mathbf{i}-t \right)=0$</p>
<p>where&nbsp;$t$ is a real number. It is given that one root is of the form $k-k\,\mathbf{i}$, where&nbsp;$k$ is real and positive. Find $t$ and $k$, and the other root of the equation.<span class="inline-marks">[7]</span></p>

The complex number $z$ satisfies the equation

${{z}^{2}}-\left( 4+\mathbf{i} \right)z+2\left( \mathbf{i}-t \right)=0$

where $t$ is a real number. It is given that one root is of the form $k-k\,\mathbf{i}$, where $k$ is real and positive. Find $t$ and $k$, and the other root of the equation.

Equating Real and Imaginary Parts

2022 VJC P1 Q5

<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>

Applying the Conjugate Root Theorem

Solving Complex Equations by Substitution

2022 NYJC P1 Q5

<p>The complex number $z$ satisfies the equation</p>
<p style="text-align: center">${{z}^{2}}\left( 2+\mathbf{i} \right)-8\mathbf{i}\,z+t=0$</p>
<p>where $t$ is a real number. It is given that one of the root is of the form $k+k\,\mathbf{i}$, where $k$ is real and non-zero. Find $t$ and $k$, and the other root of the equation. </p><p style="text-align: right;">[8]</p>

${{z}^{2}}\left( 2+\mathbf{i} \right)-8\mathbf{i}\,z+t=0$

where $t$ is a real number. It is given that one of the root is of the form $k+k\,\mathbf{i}$, where $k$ is real and non-zero. Find $t$ and $k$, and the other root of the equation. 

N2020 P1 Q6

<ol class="abc-list"><li><div class="list-item-paragraph">If one of the roots of the equation ${{z}^{3}}+2z+k=0$ is $\mathbf{i}$, find $k$. Hence find the other two roots.</div><p style="text-align: right;">[4]</p></li></ol>

2013 RI P2 Q4

<p>The complex number $z$ is given by $z=k+\mathbf{i}$ where $k$ is a non-zero real number.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the possible values of $k$ if $z=k+\mathbf{i}$ satisfies the equation ${{z}^{3}}-\mathbf{i}{{z}^{2}}-2z-4\mathbf{i}=0$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">For the complex number $z$ found in part <strong>(i)</strong> for which $k>0$, find the smallest integer value of $n$ such that $\left| {{z}^{n}} \right|>100$ and ${{z}^{n}}$ is real.</div><p style="text-align: right;">[3]</p></li></ol>

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

Properties of Trigonometric and Exponential Forms

2016 ACJC P1 Q8

<p><strong>Do not use a graphing calculator in answering this question.</strong></p>
<ol class="i-list"><li data-list-item-id="item-1760101721975-j2pukmwnp" data-list-item-path="item-1760101721975-j2pukmwnp"><div class="list-item-paragraph">The roots of the equation ${{z}^{2}}=-8\mathbf{i}$ are ${{z}_{1}}$ and ${{z}_{2}}$. Find ${{z}_{1}}$ and ${{z}_{2}}$ in cartesian form $x+\mathbf{i}y$, showing your working.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760101722500-lqv1u4jsi" data-list-item-path="item-1760101722500-lqv1u4jsi"><div class="list-item-paragraph">Hence, or otherwise, find in cartesian form the roots ${{w}_{1}}$ and ${{w}_{2}}$ of the equation</div><div class="list-item-paragraph" style="text-align: center">${{w}^{2}}+4w+\left( 4+2\mathbf{i} \right)=0$<span class="inline-marks">[3]</span></div></li></ol>
<ol class="i-list" start="3"><li data-list-item-id="n2011-q10-repair-1" data-list-item-path="n2011-q10-repair-1"><div class="list-item-paragraph">Using a single Argand diagram, sketch the loci</div><ol class="abc-list"><li data-list-item-id="n2011-q10-repair-2" data-list-item-path="n2011-q10-repair-1/n2011-q10-repair-2"><div class="list-item-paragraph">$|z-z_1|=|z-z_2|$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="n2011-q10-repair-3" data-list-item-path="n2011-q10-repair-1/n2011-q10-repair-3"><div class="list-item-paragraph">$|z-w_1|=|z-w_2|$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="n2011-q10-repair-4" data-list-item-path="n2011-q10-repair-4"><div class="list-item-paragraph">Give a reason why there are no points which lie on both of these loci.<span class="inline-marks">[1]</span></div></li></ol>

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

Using a single Argand diagram, sketch the loci

Give a reason why there are no points which lie on both of these loci.

N2011 P1 Q10

<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">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

<p>Do not use a calculator in answering this question.</p>
<ol class="abc-list"><li data-list-item-id="item-1759655845946-6r6erg2qr" data-list-item-path="item-1759655845946-6r6erg2qr"><div class="list-item-paragraph">Verify that&nbsp;$-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="n2016-p1-q7-b" data-list-item-path="n2016-p1-q7-b"><div class="list-item-paragraph">The equation $z^3-5z^2+16z+k=0$, where&nbsp;$k$ is a real constant, has a root $z=1+a\mathrm i$, where&nbsp;$a$ is a positive real constant. Find the values of $a$ and $k$, showing your working.<span class="inline-marks">[5]</span></div></li></ol>

Do not use a calculator in answering this question.

The equation $z^3-5z^2+16z+k=0$, where $k$ is a real constant, has a root $z=1+a\mathrm i$, where $a$ is a positive real constant. Find the values of $a$ and $k$, showing your working.

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