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PSLE

PSLE Foundation Mathematics

<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

Finding Roots of Equations with Complex Coefficients

Graphs of Conic Sections

Loci in the Argand Diagram

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

Complex Numbers

Graphing Techniques

<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

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Solve the simultaneous equations</div><div class="list-item-paragraph" style="text-align: center">$z=w+2\mathrm{i}-1$ and ${{z}^{2}}-\mathrm{i}w+\frac{5}{2}=0$,</div><div class="list-item-paragraph">giving $z$ and $w$ in the form $x+y\mathrm{i}$ where $x$ and $y$ are real.</div><p style="text-align: right;">[5]</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 $z=w-\frac{1}{w}$ where $w=2(\cos \theta +\mathrm{i}\sin \theta )$, $-\pi <\theta \le \pi $, express the real and imaginary parts $z$ in terms of $\theta $.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence show that the locus of $z$ on an Argand diagram lies on the curve with cartesian equation $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ where $a$ and $b$ are constants.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

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

Solving Simultaneous Complex Equations

Converting Parametric Equations to Cartesian Form

2016 MJC P1 Q10

<p>Given that the complex number $z=(1+\mathrm{i})t+\frac{1-\mathrm{i}}{t}$ is represented by the point $P$ on Argand diagram where $t$ is a non-zero real constant. Find the Cartesian equation of the locus of the point $P$.</p><p style="text-align: right;">[3]</p>

Given that the complex number $z=(1+\mathrm{i})t+\frac{1-\mathrm{i}}{t}$ is represented by the point $P$ on Argand diagram where $t$ is a non-zero real constant. Find the Cartesian equation of the locus of the point $P$.

2014 JJC P1 Q10(a)

<p>The complex number $z$ is given by $z=r{{e}^{\mathrm{i}\theta }}$, where&nbsp;$r&gt;0$ and $0\le \theta \le \frac{1}{2}\pi $.</p>
<ol class="i-list"><li data-list-item-id="item-1759739731008-ogf99zrez" data-list-item-path="item-1759739731008-ogf99zrez"><div class="list-item-paragraph">Given that&nbsp;$w=\left( 1-\mathrm{i}\sqrt{3} \right)z$, find $\left| w \right|$ in terms of $r$ and $\arg w$ in terms of $\theta $.<span class="inline-marks">[2]</span></div></li></ol>
<ol class="i-list" start="2"><li data-list-item-id="n2013-q8-repair-1" data-list-item-path="n2013-q8-repair-1"><div class="list-item-paragraph">Given that&nbsp;$r$ has a fixed value, draw an Argand diagram to show the locus of $z$ as&nbsp;$\theta$ varies. On the same diagram, show the corresponding locus of $w$. You should identify the modulus and argument of the end-points of each locus.<span class="inline-marks">[4]</span></div></li></ol>
<ol class="i-list" start="3"><li data-list-item-id="item-1759739783735-utcmd6rbl" data-list-item-path="item-1759739783735-utcmd6rbl"><div class="list-item-paragraph">Given that&nbsp;$\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} \right)=\pi $, find $\theta $.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $z$ is given by $z=r{{e}^{\mathrm{i}\theta }}$, where $r>0$ and $0\le \theta \le \frac{1}{2}\pi $.

Given that $r$ has a fixed value, draw an Argand diagram to show the locus of $z$ as $\theta$ varies. On the same diagram, show the corresponding locus of $w$. You should identify the modulus and argument of the end-points of each locus.

Geometrical Transformations of Complex Numbers

Modulus, Argument, and Operations on Complex Numbers

N2013 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5NjA4ODAtNGpuMWE4OWl4dTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg3NTcyNSwiZXhwIjoxNzkyODc2MDI1fQ.7YzWW4oj8o7PgMPP9g-yGksrKpgyK125QAqcKIMdMek" 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 $x=3$.

Properties of Trigonometric and Exponential Forms

Solving Complex Equations

Complex Roots of Quadratics with Real Coefficients

2020 DHS P2 Q4

<ol class="abc-list"><li data-list-item-id="n2014-repair-3" data-list-item-path="n2014-repair-3"><div class="list-item-paragraph">The complex number $z$ satisfies $|z+5-\mathrm i|=4$.</div><ol class="i-list"><li data-list-item-id="n2014-repair-4" data-list-item-path="n2014-repair-3/n2014-repair-4"><div class="list-item-paragraph">On an Argand diagram show the locus of $z$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="n2014-repair-5" data-list-item-path="n2014-repair-3/n2014-repair-5"><div class="list-item-paragraph">The complex number $z$ also satisfies $|z-6\mathrm i|=|z+10+4\mathrm i|$. Find exactly the possible values of $z$, giving your answers in the form $x+\mathrm iy$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="n2014-repair-6" data-list-item-path="n2014-repair-6"><div class="list-item-paragraph">It is given that&nbsp;$w=\sqrt{3}-\mathrm{i}$.</div><ol class="i-list"><li data-list-item-id="item-1759148136692-9zh4vdg7a" data-list-item-path="n2014-repair-6/item-1759148136692-9zh4vdg7a"><div class="list-item-paragraph">Without using a calculator, find an exact expression for&nbsp;${{w}^{6}}$. Give your answer in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where&nbsp;$r&gt;0$ and $0\le \theta &lt;2\pi .$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759148137360-iwf6meym7" data-list-item-path="n2014-repair-6/item-1759148137360-iwf6meym7"><div class="list-item-paragraph">Without using a calculator, find the three smallest positive whole number values for&nbsp;$n$ for which $\frac{{{w}^{n}}}{{{w}^{*}}}$ is a real number.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

The complex number $z$ satisfies $|z+5-\mathrm i|=4$.

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