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PSLE Foundation Mathematics

<p>The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$.</p>
<p>Give all arguments as principal arguments in radians.</p>
<ol class="abc-list"><li data-list-item-id="q003-part-0" data-list-item-path="q003-part-0"><div class="list-item-paragraph">Find the modulus and principal argument of $wz$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q003-part-1" data-list-item-path="q003-part-1"><div class="list-item-paragraph">Express $w$ and $z$ in Cartesian form. Hence find the exact real and imaginary parts of $wz$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q003-part-2" data-list-item-path="q003-part-2"><div class="list-item-paragraph">Deduce that \[\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}.\]<span class="inline-marks">[2]</span></div></li></ol>

The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$.

Give all arguments as principal arguments in radians.

Find the modulus and principal argument of $wz$.

Express $w$ and $z$ in Cartesian form. Hence find the exact real and imaginary parts of $wz$.

Deduce that \[\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}.\]

Modulus and Argument of Complex Numbers

Using Complex Numbers to Deduce Trigonometric Values

Arithmetic with Complex Numbers in Cartesian Form

Modulus, Argument, and Operations on Complex Numbers

A product giving an exact cosine value

Complex Numbers

<p><strong>Do not use a caulculator in answering this question.</strong> </p>
<p>A complex number $z$ is given by $z = \frac{\sqrt{3} + \mathrm{i}}{\sqrt{3} - \mathrm{i}}$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Find $\vert{}z\vert{}$ and $\arg(z)$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The point $A$ represents the complex number $z$ on an Argand diagram. The point $A$ is rotated anticlockwise through $\frac{1}{2}\pi$ about the origin $O$ to the point $B$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Plot the points $A$ and $B$ on an Argand diagram. State in terms of $z$ the complex number that is represented by $B$, explaining your answer.<span class="inline-marks">[3]</span></div></li></ol>

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

A complex number $z$ is given by $z = \frac{\sqrt{3} + \mathrm{i}}{\sqrt{3} - \mathrm{i}}$.

Find $\vert{}z\vert{}$ and $\arg(z)$.

The point $A$ represents the complex number $z$ on an Argand diagram. The point $A$ is rotated anticlockwise through $\frac{1}{2}\pi$ about the origin $O$ to the point $B$.

Plot the points $A$ and $B$ on an Argand diagram. State in terms of $z$ the complex number that is represented by $B$, explaining your answer.

Geometrical Transformations of Complex Numbers

Representing Complex Numbers on an Argand Diagram

N2025 P1 Q3

<p>Use complex number methods to find the exact value of $\cos\dfrac{7\pi}{12}$ and $\sin\dfrac{7\pi}{12}$.</p>

Use complex number methods to find the exact value of $\cos\dfrac{7\pi}{12}$ and $\sin\dfrac{7\pi}{12}$.

Properties of Trigonometric and Exponential Forms

Exact trigonometric ratios via the exponential form

<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.**

Finding Roots of Equations with Complex Coefficients

<p>The complex number $w$ has modulus $\sqrt2$ and argument $-3\pi/4$, and the complex number $z$ has modulus $2$ and argument $-\pi/3$. Given that&nbsp;$w=-1-i$ and $z=1-\sqrt3\,i$.</p>
<ol class="abc-list"><li data-list-item-id="item-1784107715811-wpdirzdf3" data-list-item-path="item-1784107715811-wpdirzdf3"><div class="list-item-paragraph">Find the exact values of the modulus and argument of $wz$.</div></li><li data-list-item-id="item-1784107728124-98v5ysayo" data-list-item-path="item-1784107728124-98v5ysayo"><div class="list-item-paragraph">By first expressing $w$ and $z$ in the form $x+y\mathbf{i}$, find the exact real and imaginary part of $wz$.</div></li><li data-list-item-id="item-1784107803405-q1ru6n5ma" data-list-item-path="item-1784107803405-q1ru6n5ma"><div class="list-item-paragraph">Hence show that&nbsp;$\sin \frac{\pi }{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$ .</div></li></ol>

The complex number $w$ has modulus $\sqrt2$ and argument $-3\pi/4$, and the complex number $z$ has modulus $2$ and argument $-\pi/3$. Given that $w=-1-i$ and $z=1-\sqrt3\,i$.

Cartesian, Trigonometric and Exponential Forms

<ol class="abc-list"><li data-list-item-id="item-1770890447720-68w8dh04a" data-list-item-path="item-1770890447720-68w8dh04a"><div class="list-item-paragraph">The complex number $w$ has modulus $3$ and argument $\frac{3\pi }{4}$ and $w^*$ denotes the conjugate of $w$.  <br>Find the modulus and argument of $p$, where&nbsp;$p=\frac{1-\mathbf{i}}{{{w}^{*}}}$.</div></li></ol>

2013 PJC P1 Q10

<p>Given that&nbsp;$\left| z \right|=5$, $\arg z=\frac{\pi }{3}$ and $\left| w \right|=7$, $\arg w=-\frac{\pi }{2}$. Find the modulus and argument of </p>
<ol class="i-list"><li><div class="list-item-paragraph">${{z}^{2}}w$	</div></li><li><div class="list-item-paragraph">$\frac{{{z}^{*}}}{{{w}^{4}}}$	</div></li><li><div class="list-item-paragraph">${{w}^{-3}}{{({{z}^{*}})}^{2}}$	</div></li><li><div class="list-item-paragraph">$2\text{i}{{z}^{5}}$	</div></li><li><div class="list-item-paragraph">$\frac{{{w}^{3}}}{{{\left( z\mathbf{i} \right)}^{4}}}$</div></li></ol>

Given that $\left| z \right|=5$, $\arg z=\frac{\pi }{3}$ and $\left| w \right|=7$, $\arg w=-\frac{\pi }{2}$. Find the modulus and argument of 

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>It is given that&nbsp;$z = \sqrt{3} - \mathrm{i}$ and $w = 2\mathrm{i}z$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the modulus and argument of $z$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the modulus and argument of $w$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>$O$, $A$, $B$ and $C$ represent the complex numbers $0$, $z$, $z + w$ and $w$ respectively.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Illustrate these four points on a single diagram.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Identify the shape of the quadrilateral $OABC$ and state its area.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $z = \sqrt{3} - \mathrm{i}$ and $w = 2\mathrm{i}z$.

Deduce the modulus and argument of $w$.

$O$, $A$, $B$ and $C$ represent the complex numbers $0$, $z$, $z + w$ and $w$ respectively.

Illustrate these four points on a single diagram.

Identify the shape of the quadrilateral $OABC$ and state its area.

Geometric Problems on the Argand Diagram

2025 YIJC P2 Q2

<p>Given that $z={{e}^{-\frac{\pi }{6}\mathrm{i}}}$, find the smallest positive integer $n$ for ${{z}^{*}}{{z}^{n}}$ to be purely imaginary.</p>

Given that $z={{e}^{-\frac{\pi }{6}\mathrm{i}}}$, find the smallest positive integer $n$ for ${{z}^{*}}{{z}^{n}}$ to be purely imaginary.

Using Argument to Determine The Nature of a Complex Number Q3

<p>Given that $z={\mathrm{e}}^{-\frac{\pi}{3}\mathbf{i}}$, find the smallest positive integer $n$ for $z^{*}z^{2n}$ to be a negative real number.</p>

Given that $z={\mathrm{e}}^{-\frac{\pi}{3}\mathbf{i}}$, find the smallest positive integer $n$ for $z^{*}z^{2n}$ to be a negative real number.

Using Argument to Determine The Nature of a Complex Number Q2

<p>The complex number $w$ has modulus $\sqrt{2}$ and argument $-\frac{3\pi}{4}$, and the complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$. Find the exact values of the modulus and argument of $wz$. By first expressing $w$ and $z$ in the form $x+y\mathrm{i}$, find the exact real and imaginary part of $wz$. Hence show that $\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$.</p>

The complex number $w$ has modulus $\sqrt{2}$ and argument $-\frac{3\pi}{4}$, and the complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$. Find the exact values of the modulus and argument of $wz$. By first expressing $w$ and $z$ in the form $x+y\mathrm{i}$, find the exact real and imaginary part of $wz$. Hence show that $\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}$.

Proving Trigonometric Ratios by Complex Numbers Q3

<p>Find the modulus and argument of ${{\left( \frac{3-i}{1-2i} \right)}^{22}}$ in exact form and hence write the complex number in the exponential form.</p>

Find the modulus and argument of ${{\left( \frac{3-i}{1-2i} \right)}^{22}}$ in exact form and hence write the complex number in the exponential form.

Properties from Exponential or Trigonometric form Q5

<p>The complex number $z$ has a modulus $r$ and principal argument $\theta$ where $0<\theta<\dfrac{\pi}{2}$.
</p>
<p>Find, in terms of $r$ and $\theta$, the modulus and argument of</p>
<ol class="i-list"><li><div class="list-item-paragraph">$z^{-1}$</div></li><li><div class="list-item-paragraph">$z^{2}$</div></li><li><div class="list-item-paragraph">$iz$</div></li><li><div class="list-item-paragraph">$z z^{*}$</div></li></ol>

The complex number $z$ has a modulus $r$ and principal argument $\theta$ where $0<\theta<\dfrac{\pi}{2}$.


Find, in terms of $r$ and $\theta$, the modulus and argument of

Properties from Exponential or Trigonometric form Q3

<p>Given that $| z |=3,\arg z=\dfrac{\pi }{6}$ and $| w |=2,\arg w=-\dfrac{5\pi }{6}$. Find the modulus and argument of</p>
<ol class="i-list"><li><div class="list-item-paragraph">$w^{5}z^{2}$</div></li><li><div class="list-item-paragraph">$\dfrac{w^{2}}{z^{5}}$</div></li><li><div class="list-item-paragraph">$z^{-2}w^{3}$</div></li><li><div class="list-item-paragraph">$\dfrac{(iw)^{7}}{2({z^{*}})^{3}}$</div></li><li><div class="list-item-paragraph">$\dfrac{z^{*}(1+i)^{2}}{w}$</div></li></ol>

Given that $| z |=3,\arg z=\dfrac{\pi }{6}$ and $| w |=2,\arg w=-\dfrac{5\pi }{6}$. Find the modulus and argument of

Properties from Exponential or Trigonometric form Q1

<p>Express the following complex numbers in the form $z = r e^{\mathbf{i}\theta}$ where $r \ge 0$, and $-\pi < \theta \le \pi$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$3\mathrm{i}e^{-\mathbf{i}\pi}$</div></li><li><div class="list-item-paragraph">$-2e^{-\mathbf{i}\frac{\pi}{3}}$</div></li><li><div class="list-item-paragraph">$e^{1-\mathbf{i}\frac{\pi}{3}}$</div></li><li><div class="list-item-paragraph">$\left( \sqrt{3}-\mathbf{i} \right)e^{-\mathbf{i}\frac{\pi}{4}}$</div></li><li><div class="list-item-paragraph">$-2\left( 1+\sqrt{3}\mathbf{i} \right)e^{\mathbf{i}\frac{2\pi}{5}}$</div></li></ol>

Express the following complex numbers in the form $z = r e^{\mathbf{i}\theta}$ where $r \ge 0$, and $-\pi < \theta \le \pi$.

Forms of Complex Numbers Q8

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The point $P$ represents the complex number $z=-2+b\,\mathbf{i}$, where $b>0$. The line $OP$ makes an angle of ${{45}^{{}^\circ }}$ with the horizontal.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the value of $b$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence find the modulus and argument of $\frac{1}{z}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show on an Argand diagram the points $P$, $Q$ and $R$ representing the complex numbers $z$, $\frac{1}{z}$ and $z+\frac{1}{z}$ respectively. Indicate on your diagram the value of $\angle PRQ$.</div><p style="text-align: right;">[3]</p></li></ol>

The point $P$ represents the complex number $z=-2+b\,\mathbf{i}$, where $b>0$. The line $OP$ makes an angle of ${{45}^{{}^\circ }}$ with the horizontal.

2021 ASRJC JC2 MYA P1 Q3

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="item-1760958279806-xq2aenzsd" data-list-item-path="item-1760958279806-xq2aenzsd"><div class="list-item-paragraph">The complex number $w$ is such that&nbsp;$w=a+ib$, where&nbsp;$a$ and $b$ are non-zero real numbers. The complex conjugate of $w$ is denoted by ${{w}^{*}}$. Given that</div><div class="list-item-paragraph" style="text-align: center">$w{{w}^{*}}=4-2i+2i{{w}^{*}}$,</div><div class="list-item-paragraph">find the two possible values of $w$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760958280459-pkr2zagji" data-list-item-path="item-1760958280459-pkr2zagji"><div class="list-item-paragraph">The complex number $z$ is given by $z=\frac{1-\sqrt{3}i}{-1+i}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760958315657-b2fvtu8co" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958315657-b2fvtu8co"><div class="list-item-paragraph">Find $\arg (z)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760958316353-3o0qjy3xy" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958316353-3o0qjy3xy"><div class="list-item-paragraph">Find $z$ in cartesian form $x+iy$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760958317155-wrr9fl1ge" data-list-item-path="item-1760958280459-pkr2zagji/subitem-1760958317155-wrr9fl1ge"><div class="list-item-paragraph">Hence find the value of $\tan \frac{\pi }{12}$ in the form $c+d\sqrt{3}$, where&nbsp;$c$ and $d$ are integers to be found.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Solving Complex Equations

Solving Equations Involving Complex Conjugates

2024 RI P1 Q8

<p>It is given that&nbsp;$z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where&nbsp;$b$ is real and positive.</p>
<ol class="i-list"><li data-list-item-id="item-1760957017876-s1hi3ko2r" data-list-item-path="item-1760957017876-s1hi3ko2r"><div class="list-item-paragraph">Given that&nbsp;$\left| \frac{z}{w} \right|=1$, show that&nbsp;$b=\sqrt{2}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760957018557-8rcw481un" data-list-item-path="item-1760957018557-8rcw481un"><div class="list-item-paragraph">Show that&nbsp;$\arg z+\arg(w^*)=13\pi/12$, where&nbsp;$w^*$ is the complex conjugate of $w$ and arguments are principal values.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760957019299-5idq5iawl" data-list-item-path="item-1760957019299-5idq5iawl"><div class="list-item-paragraph">Given that&nbsp;$\arg \left( p+z \right)-\arg \left( z \right)=-\frac{7\pi }{12}$, find the exact value of the real number $p$.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where $b$ is real and positive.

2021 CJC JC2 MYE P1 Q4 [Modified]

<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>

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

Applying Complex Conjugate Identities

<p style="text-align: center">${z={{\left( 2+3\mathbf{i} \right)}^{4n+2}}+{{\left( 3-2\mathbf{i} \right)}^{4n+2}}}$, ${n\in \mathbb{N}}$</p>
<p>Show clearly that&nbsp;${z=0}$ for all ${n\in \mathbb{N}}$.</p>

${z={{\left( 2+3\mathbf{i} \right)}^{4n+2}}+{{\left( 3-2\mathbf{i} \right)}^{4n+2}}}$, ${n\in \mathbb{N}}$

Show clearly that ${z=0}$ for all ${n\in \mathbb{N}}$.

Math Question Bank