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PSLE Foundation Mathematics

<p>The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-42" data-list-item-path="p1421-item-42"><div class="list-item-paragraph">Find $z^*w^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-44" data-list-item-path="p1421-item-44"><div class="list-item-paragraph">Find $(zz^*)(ww^*)$, explaining how you regroup the factors.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-46" data-list-item-path="p1421-item-46"><div class="list-item-paragraph"><p>Find $\operatorname{Re}(zw)$ and $\operatorname{Im}(zw)$.</p><span class="inline-marks">[2]</span></div></li></ol>

The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.

Applying Complex Conjugate Identities

Using a product and its conjugate

Complex Numbers

<p>A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-26" data-list-item-path="p1421-item-26"><div class="list-item-paragraph">Find $z$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-28" data-list-item-path="p1421-item-28"><div class="list-item-paragraph">Find $z-z^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-30" data-list-item-path="p1421-item-30"><div class="list-item-paragraph">Use the given sum and product to find $\dfrac1z+\dfrac1{z^*}$, without evaluating the two reciprocals separately.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)>0$.

Solving Equations Involving Complex Conjugates

Recovering a number from its sum and product

<p>A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-71" data-list-item-path="p1421-item-71"><div class="list-item-paragraph">Find $(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-73" data-list-item-path="p1421-item-73"><div class="list-item-paragraph">Find $z^3+(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-75" data-list-item-path="p1421-item-75"><div class="list-item-paragraph">Find $(z^*)^6$ in Cartesian form.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.

Using conjugates of powers

<p>Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-59" data-list-item-path="p1421-item-59"><div class="list-item-paragraph">$\dfrac{w^*}{z^*}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-61" data-list-item-path="p1421-item-61"><div class="list-item-paragraph">$\left(\dfrac{z+w}{z-w}\right)^*$. Explain why the denominator is nonzero.<span class="inline-marks">[4]</span></div></li></ol>

Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.

Conjugating ratios without finding the numbers

<p>The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-7" data-list-item-path="p1421-item-7"><div class="list-item-paragraph">$(3z-2w^*)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-9" data-list-item-path="p1421-item-9"><div class="list-item-paragraph">$(z^*+w)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-11" data-list-item-path="p1421-item-11"><div class="list-item-paragraph">$(iz)^*$ and $iz^*$. Explain why these values differ.<span class="inline-marks">[3]</span></div></li></ol>

The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.

Conjugating a linear combination

<p>Let&nbsp;$z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-91" data-list-item-path="p1421-item-91"><div class="list-item-paragraph">Show that&nbsp;$A^*=z^*w$, and hence that&nbsp;$zw^*+z^*w$ is real.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-93" data-list-item-path="p1421-item-93"><div class="list-item-paragraph">Show that&nbsp;$zw^*-z^*w$ has zero real part. Explain why it need not be a nonzero purely imaginary number.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-95" data-list-item-path="p1421-item-95"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{zw^*-z^*w}{i}$ is real.<span class="inline-marks">[2]</span></div></li></ol>

Let $z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.

Properties of Complex Conjugates

Recognising real and imaginary expressions

<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).

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

Graphing Techniques

<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>Given that&nbsp;${z}$ and ${w}$ are complex numbers, prove that</p>
<p style="text-align: center">${{{\left| z+w \right|}^{2}}-{{\left| z+w^{*}\right|}^{2}}=4 \, \mathrm{Im}\, z}$ ${\mathrm{Im}\, w}$,</p>
<p>where&nbsp;${w^*}$ denotes the complex conjugate of ${w}$.</p>

Given that ${z}$ and ${w}$ are complex numbers, prove that

${{{\left| z+w \right|}^{2}}-{{\left| z+w^{*}\right|}^{2}}=4 \, \mathrm{Im}\, z}$ ${\mathrm{Im}\, w}$,

where ${w^*}$ denotes the complex conjugate of ${w}$.

Complex Numbers Q5

<p style="text-align: center">${z={{\left( a+b\mathbf{i} \right)}^{4n}}+{{\left( b+a\mathbf{i} \right)}^{4n}}}$, ${a\in \mathbb{R}}$, ${b\in \mathbb{R}}$, ${n\in \mathbb{N}}$.</p>
<p>Show that&nbsp;${z}$ is a real number.</p>

${z={{\left( a+b\mathbf{i} \right)}^{4n}}+{{\left( b+a\mathbf{i} \right)}^{4n}}}$, ${a\in \mathbb{R}}$, ${b\in \mathbb{R}}$, ${n\in \mathbb{N}}$.

Modulus, Argument, and Operations on Complex Numbers

<p style="text-align: center">${z={{\left( 5+2\mathbf{i} \right)}^{n}}+{{\left( 5-2\mathbf{i} \right)}^{n}}}$, ${n\in \mathbb{N}}$</p>
<p>Show clearly that&nbsp;${z}$ is a real number.</p>

${z={{\left( 5+2\mathbf{i} \right)}^{n}}+{{\left( 5-2\mathbf{i} \right)}^{n}}}$, ${n\in \mathbb{N}}$

Show clearly that ${z}$ is a real number.

<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

<ol class="abc-list"><li><div class="list-item-paragraph">Find the set of values of $\alpha $ for which the expression $\frac{2-\mathbf{i}\sin \,2\alpha }{1+2\,\mathbf{i}\,\sin \,2\alpha }$ is purely real.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">If&nbsp;$z$ and $w$ are different non-zero complex numbers and $\left| w \right|=1$, find $\left| \frac{3\left( w-z \right)}{1-{{z}^{*}}w} \right|$.<span class="inline-marks">[4]</span></div></li></ol>

Solving Complex Equations

2023 DHS P1 Q4

<ol class="abc-list"><li data-list-item-id="item-1760094713308-68eugrclx" data-list-item-path="item-1760094713308-68eugrclx"><div class="list-item-paragraph">The complex number $w$ is such that&nbsp;$w=a+\mathrm{i}b$, where&nbsp;$a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by $w^*$. Given that&nbsp;$\frac{{{w}^{2}}}{w^*}$ is purely imaginary, find the possible values of $w$ in terms of $a$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="n2015-p1-q9-b" data-list-item-path="n2015-p1-q9-b"><div class="list-item-paragraph">The complex number $z$ is such that&nbsp;$z^5=-32\mathrm i$.</div><ol class="i-list"><li data-list-item-id="n2015-p1-q9-bi" data-list-item-path="n2015-p1-q9-b/n2015-p1-q9-bi"><div class="list-item-paragraph">Find the modulus and argument of each of the possible values of $z$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="n2015-p1-q9-bii" data-list-item-path="n2015-p1-q9-b/n2015-p1-q9-bii"><div class="list-item-paragraph">Two of these values are $z_1$ and $z_2$, where&nbsp;$\frac12\pi&lt;\arg z_1&lt;\pi$ and $-\pi&lt;\arg z_2&lt;-\frac12\pi$. Find the exact value of $\arg(z_1-z_2)$ in terms of $\pi$ and show that&nbsp;$|z_1-z_2|=4\sin(\frac15\pi)$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

The complex number $z$ is such that $z^5=-32\mathrm i$.

Finding Roots of Complex Numbers

Properties of Trigonometric and Exponential Forms

N2015 P1 Q9

<ol class="i-list"><li><div class="list-item-paragraph">The complex number $w$ can be expressed as&nbsp;$\cos \,\theta +\mathrm{i sin}\,\theta .$</div><ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$w+\frac{1}{w}$ is a real number.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;$\frac{w-1}{w+1}$ can be expressed as&nbsp;$k\tan \frac{1}{2}\theta ,$ where&nbsp;$k$ is a complex number to be found.<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">The complex number $z$ has modulus $1$. Find the modulus of the complex number $\frac{z-3\mathrm{i}}{1+3\mathrm{i}z}.$<span class="inline-marks">[5]</span></div></li></ol>

N2019 P1 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">The complex number $z$ is given by $z=1+\cos \alpha +\mathrm{i}\sin \alpha $, where $0<\alpha <\frac{\pi }{2}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that $z$ can be expressed as $\left( 2\cos \frac{\alpha }{2} \right)\left( \cos \frac{\alpha }{2}+\mathrm{i}\sin \frac{\alpha }{2} \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $z{{z}^{*}}$, where ${{z}^{*}}$ denotes the conjugate of $z$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $\alpha =\frac{\pi }{3}$, without using a calculator, find the values of $\left| {{z}^{6}} \right|$ and $\arg \left( {{z}^{6}} \right)$. Deduce the value of ${{z}^{6}}$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">The complex numbers $u$ and $v$ are such that $\frac{\mathrm{i}\left( u+v \right)}{u-v}$ is real. Show that $\left| u \right|=\left| v \right|$.</div><p style="text-align: right;">[3]</p></li></ol>

2016 RI Promo Q8

<p>For positive integer $n$, a complex number $z$ is such that ${{\left| z \right|}^{n}}=\frac{1}{\sqrt{2}}$. By considering the conjugate of $1+2{{z}^{2n}}$, show that $\frac{2{{z}^{n}}}{1+2{{z}^{2n}}}$ is a real number.</p><p style="text-align: right;">[4]</p>

For positive integer $n$, a complex number $z$ is such that ${{\left| z \right|}^{n}}=\frac{1}{\sqrt{2}}$. By considering the conjugate of $1+2{{z}^{2n}}$, show that $\frac{2{{z}^{n}}}{1+2{{z}^{2n}}}$ is a real number.

2020 ASRJC P1 Q8 (b)

<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>

Modulus and Argument of Complex Numbers

Geometric Progression

2019 RI P1 Q7

Sequence and Series

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The complex numbers $u$ and $v$ are such that $\frac{u-2v}{u+2v}\mathrm{i}$ is a real number. Show that $\left| u \right|=2\left| v \right|$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that $z=\cos \left( \frac{3\pi }{16} \right)+\cos \left( \frac{7\pi }{16} \right)+\mathrm{i}\left[ \sin \left( \frac{3\pi }{16} \right)+\sin \left( \frac{7\pi }{16} \right) \right]$. Without using a calculator, express $z$ in the form of $r{{e}^{i\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</span></div><div class="list-item-paragraph">Hence find the smallest positive integer $n$ such that ${{z}^{n}}=k\left( -1-\mathrm{i} \right)$ for some positive real number $k$.</div><p style="text-align: right;">[5]</p></li></ol>

The complex numbers $u$ and $v$ are such that $\frac{u-2v}{u+2v}\mathrm{i}$ is a real number. Show that $\left| u \right|=2\left| v \right|$.

It is given that $z=\cos \left( \frac{3\pi }{16} \right)+\cos \left( \frac{7\pi }{16} \right)+\mathrm{i}\left[ \sin \left( \frac{3\pi }{16} \right)+\sin \left( \frac{7\pi }{16} \right) \right]$. Without using a calculator, express $z$ in the form of $r{{e}^{i\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.

2018 HCI JC2 MYE P1 Q5

<p>The complex number $w$ can be expressed as $2\left( \cos \theta +\mathrm{i}\sin \theta \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $w-\frac{4}{w}$ is a purely imaginary number.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $\frac{w+2}{w-2}$ can be expressed as $k\cot \frac{1}{2}\theta $, where $k$ is a complex number to be found.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the modulus of the complex number $\frac{w+6\mathrm{i}}{2-3\mathrm{i}w}$.</span></div><p style="text-align: right;">[4]</p></li></ol>

The complex number $w$ can be expressed as $2\left( \cos \theta +\mathrm{i}\sin \theta \right)$.

Show that $w-\frac{4}{w}$ is a purely imaginary number.

Show that $\frac{w+2}{w-2}$ can be expressed as $k\cot \frac{1}{2}\theta $, where $k$ is a complex number to be found.

Find the modulus of the complex number $\frac{w+6\mathrm{i}}{2-3\mathrm{i}w}$.

Math Question Bank