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PSLE

PSLE Foundation Mathematics

<p>The complex number $z$ is given by \[z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.\]</p>
<p>Give all arguments as principal arguments in radians.</p>
<ol class="abc-list"><li data-list-item-id="q001-part-0" data-list-item-path="q001-part-0"><div class="list-item-paragraph">Express $z$ in the form $x+iy$, and find $|z|$ and $\arg z$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q001-part-1" data-list-item-path="q001-part-1"><div class="list-item-paragraph">The point $A$ represents $z$ on an Argand diagram. It is rotated clockwise through $\frac{\pi}{2}$ about the origin to the point $B$. Sketch $A$ and $B$, and state the complex number represented by $B$, both in terms of $z$ and in Cartesian form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $z$ is given by \[z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.\]

Give all arguments as principal arguments in radians.

Express $z$ in the form $x+iy$, and find $|z|$ and $\arg z$.

The point $A$ represents $z$ on an Argand diagram. It is rotated clockwise through $\frac{\pi}{2}$ about the origin to the point $B$. Sketch $A$ and $B$, and state the complex number represented by $B$, both in terms of $z$ and in Cartesian form.

Modulus and Argument of Complex Numbers

Geometrical Transformations of Complex Numbers

Arithmetic with Complex Numbers in Cartesian Form

Representing Complex Numbers on an Argand Diagram

A quotient followed by a clockwise rotation

Complex Numbers

<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 $-\frac{\pi}{3}$.

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

Using Complex Numbers to Deduce Trigonometric Values

Modulus, Argument, and Operations on Complex Numbers

A product giving an exact cosine value

<p>The complex number $w$ is given by $1 + 2\mathrm{i}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find $w^3$ in the form $x + \mathrm{i}y$, showing your working.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Given that&nbsp;$w$ is a root of the equation $az^3 + 5z^2 + 17z + b = 0$, find the values of the real numbers $a$ and $b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using these values of $a$ and $b$, find all the roots of this equation in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number $w$ is given by $1 + 2\mathrm{i}$.

Equating Real and Imaginary Parts

Applying the Conjugate Root Theorem

N2013 P1 Q4

<p>A graphic calculator is not to be used in answering this question.</p>
<ol class="i-list"><li data-list-item-id="item-1770891051005-db9u4khhg" data-list-item-path="item-1770891051005-db9u4khhg"><div class="list-item-paragraph">It is given that&nbsp;${{z}_{1}}=1+\sqrt{3}\mathbf{i}$. Find the value of ${{z}_{1}}^{3}$, showing clearly how you obtain your answer.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1770891051641-3s5k7iqvi" data-list-item-path="item-1770891051641-3s5k7iqvi"><div class="list-item-paragraph">Given that&nbsp;$1+\sqrt{3}\mathbf{i}$ is a root of the equation $2{{z}^{3}}+a{{z}^{2}}+bz+4=0$, find the values of the real numbers $a$ and $b$.	<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1770891052242-oerr5ud5g" data-list-item-path="item-1770891052242-oerr5ud5g"><div class="list-item-paragraph"><p>For these values of $a$ and $b$, solve the equation in part <strong>(ii)</strong>, and show all the roots on an Argand diagram.</p><span class="inline-marks">[4]</span></div></li></ol>

A graphic calculator is not to be used in answering this question.

N2008 P1 Q8

<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

<p>Find $\frac{1+2\mathbf{i}}{3-4\mathbf{i}}$.</p>

Find $\frac{1+2\mathbf{i}}{3-4\mathbf{i}}$.

Worked Example 1

<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>Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$. Deduce that&nbsp;$\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.</p>

Evaluate the product $\frac{1+\mathbf{i}}{1+\sqrt{3}\mathbf{i}}$. Deduce that $\tan \left( \frac{\pi }{12} \right)=2-\sqrt{3}$.

Proving Trigonometric Ratios by Complex Numbers Q2

<p>Evaluate the product $(1+i)(1+\sqrt{3}i)$. Deduce that $\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.</p>

Evaluate the product $(1+i)(1+\sqrt{3}i)$. Deduce that $\tan \left( \frac{7\pi }{12} \right)=-(2+\sqrt{3})$.

Proving Trigonometric Ratios by Complex Numbers Q1

<p>Without the use of the calculator, find the exact value of ${{\left( 1-\mathrm{i} \right)}^{5}}$.</p>

Without the use of the calculator, find the exact value of ${{\left( 1-\mathrm{i} \right)}^{5}}$.

Forms of Complex Numbers Q6

<p>Without the use of the calculator, find the exact value of ${{\left( 1+\mathbf{i}\sqrt{3} \right)}^{3}}$.</p>

Without the use of the calculator, find the exact value of ${{\left( 1+\mathbf{i}\sqrt{3} \right)}^{3}}$.

Forms of Complex Numbers Q5

<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 $w-\dfrac1w=\dfrac1{10}+\dfrac{21}{10}i$.

Applying Complex Conjugate Identities

Finding Roots of Equations with Complex Coefficients

<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>Using calculator to evaluate $\left( \frac{(3 - i)(i + 5)}{5 + i} \right)^{*}$.</p>

Using calculator to evaluate $\left( \frac{(3 - i)(i + 5)}{5 + i} \right)^{*}$.

Complex Conjugates

<p>Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.</p>

Using calculator to evaluate $\left( \frac{7-3\mathbf{i}}{3+2\mathbf{i}} \right)^{*}$.

<p>Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.</p>

Using calculator to evaluate $\left( \frac{1+2\mathbf{i}}{3-9\mathbf{i}} \right)^{*}$.

<p>Find $\frac{2+9\mathbf{i}}{5-2\mathbf{i}}$.</p>

Find $\frac{2+9\mathbf{i}}{5-2\mathbf{i}}$.

<p>Find $\frac{1+2\mathbf{i}}{3-4\mathbf{i}}$.</p>

<p style="margin-bottom: 1.5rem">Let&nbsp;$z=x+y\text{i}$. The conjugate of ${{z}^{*}}$, is defined by ${{z}^{*}}=x-yi$.<br>Example:</p>
<p>If&nbsp;$z=6+7\mathbf{i}$, then&nbsp;${{z}^{*}}=6-7\mathbf{i}$<br>If&nbsp;$z=-4-\mathbf{i}$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$<br>If&nbsp;$z=-\mathbf{i}$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$<br>If&nbsp;$z=4$, then&nbsp;${{z}^{*}}=\text{}\hspace{0.5em}$<br></p>

For $z=x+yi$, the conjugate is $z^*=x-yi$. Thus $z$ and $z^*$ have equal real parts and opposite imaginary parts.

Multiply numerator and denominator by the conjugate of the nonzero divisor $z_2=x_2+y_2i$.

\[\frac{z_1}{z_2}=\frac{z_1z_2^*}{z_2z_2^*}=\frac{(x_1x_2+y_1y_2)+(x_2y_1-x_1y_2)i}{x_2^2+y_2^2},\qquad z_2\ne0.\]

Division

Conjugates and Division

<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>

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

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