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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 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>

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

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>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 $z^{2} + 2\mathbf{i}z - 1 - 6\mathbf{i} = 0$.

<p>Without using a calculator, find the square roots of $21-20~\mathbf{i}$.</p>

Without using a calculator, find the square roots of $21-20~\mathbf{i}$.

<p>Without using a calculator, find the square roots of $-15 + 8\mathbf{i}$.</p>

Without using a calculator, find the square roots of $-15 + 8\mathbf{i}$.

<ol class="abc-list"><li><div class="list-item-paragraph">The complex number $z$ is given by $z=x+y\mathbf{i}$, where $x$ and $y$ are non-zero real numbers. Given that $\left| z \right|=1$, find the possible values of $z$ for which $\frac{{{\left( {{z}^{2}} \right)}^{*}}}{z}$ is real.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Without the use of a calculator, find the roots of the equation ${{z}^{2}}=33+56\mathbf{i}$, expressing your answer in cartesian form $x+\mathbf{i}y$ where $x$ and $y$ are real.</div><p style="text-align: right;">[4]</p><div class="list-item-paragraph">Hence, find in cartesian form, the roots of the equation ${{w}^{2}}=-33+56\mathbf{i}$.</div><p style="text-align: right;">[2]</p></li></ol>

Conditions for Real and Purely Imaginary Values

2022 TMJC P1 Q9

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The roots of the equation ${{z}^{2}}=18\,\mathbf{i}$ are ${{z}_{1}}$ and ${{z}_{2}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{z}_{1}}$ and ${{z}_{2}}$ in cartesian form $x+y\mathbf{i}$, showing your working clearly.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The complex number $w$ is given by $-1-\mathbf{i}\sqrt{3}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the complex conjugate of $\frac{w}{{{z}^{2}}}$ in the form $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</div><p style="text-align: right;">[4]</p></li></ol>

The roots of the equation ${{z}^{2}}=18\,\mathbf{i}$ are ${{z}_{1}}$ and ${{z}_{2}}$.

The complex number $w$ is given by $-1-\mathbf{i}\sqrt{3}$.

Modulus, Argument, and Operations on Complex Numbers

Cartesian, Trigonometric and Exponential Forms

2023 MI P1 Q7

<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

Arithmetic with Complex Numbers in Cartesian Form

Solving Complex Equations by Substitution

N2004 P1 Q12

<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 $3+2\mathrm{i}$, $z$, $4-6\mathrm{i}$ are the first three terms in a geometric progression. Without using a calculator, find the two possible values of $z$.</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">The complex number $w$ is such that $w=a+b\mathrm{i}$, where $a$ and $b$ bare non-zero real numbers. The complex conjugate of $w$ is denoted by $w^*$. Given that $\frac{{{w}^{2}}}{w^*}$ is a real number, find the possible values of $w$ in terms of $a$ only.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence, find the exact possible arguments of $w$ if $a$ is positive.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

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