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<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.

Applying Complex Conjugate Identities

Properties of Complex Conjugates

Recognising real and imaginary expressions

Complex Numbers

<p>Let&nbsp;$z = x + y\mathbf{i}$ and $k$ is a real constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{({{z}^{*}})}^{*}}=z$</div></li><li><div class="list-item-paragraph">$z+{{z}^{*}}=2\mathrm{Re}(z)$</div></li><li><div class="list-item-paragraph">$z-z^*=\left(2\mathrm{Im}(z)\right)i$</div></li><li><div class="list-item-paragraph">$z{{z}^{*}}={{x}^{2}}+{{y}^{2}}$</div></li><li><div class="list-item-paragraph">${{({{z}{1}}\pm {{z}{2}})}^{}}={{z}{1}}^{}\pm {{z}{2}}^{*}$</div></li><li><div class="list-item-paragraph">${{(kz)}^{}}=k{{z}^{}}$</div></li><li><div class="list-item-paragraph">${{({{z}_{1}}{{z}_{2}})}^{*}}={{z}_{1}}^{*}{{z}_{2}}^{*}$</div></li><li><div class="list-item-paragraph">${{\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)}^{*}}=\frac{{{z}_{1}}^{*}}{{{z}_{2}}^{*}}$</div></li></ol>

For complex numbers, conjugation respects addition, multiplication, division and integer powers.

Properties of Conjugates

<p>Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{{u}_{r}}}$ in terms of ${r}$, where&nbsp;${{{u}_{r}}}$ is the ${r}$th term of the series.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that the series is geometric and convergent. Hence, state the sum to infinity.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By expressing the sum to infinity in terms of ${\sin \theta }$ and ${\cos \theta }$, show that</div><div class="list-item-paragraph" style="text-align: center">${\cos \theta +\frac{1}{2}\cos 2\theta +\frac{1}{4}\cos 3\theta +…=\frac{4\cos \theta -2}{5-4\cos \theta }}$.<span class="inline-marks">[5]</span></div></li></ol>

Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$

Proving the progression

Geometric Progression

Properties of Trigonometric and Exponential Forms

Sum to infinity

Application of Sum to Infinity

2023 TGM P1 Q8

Sequence and Series

<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;${\left| {{\mathrm{e}}^{\mathrm{i}\theta }} \right|=1}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Consider the geometric series ${1+\frac{1}{3}{{\mathrm{e}}^{\mathrm{i}\theta }}+\frac{1}{9}{{\mathrm{e}}^{2\mathrm{i}\theta }}+…}$</div><div class="list-item-paragraph">Write down the common ratio, ${z}$, of the series and show that&nbsp;${\left| z \right|=\frac{1}{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, show that&nbsp;${\sin \theta +\frac{1}{3}\sin 2\theta +\frac{1}{9}\sin 3\theta +…=\frac{9\sin \theta }{10-6\cos \theta }}$.<span class="inline-marks">[8]</span></div></li></ol>

<p>On an Argand diagram, the points $P$ and $Q$ represent the complex numbers $p$ and $q$ respectively where $p=\cos \theta +\mathbf{i}\sin \theta $, $0&lt;\theta &lt;\frac{\pi }{4}$ and $q=2{{\mathrm{e}}^{\mathbf{i}\,\,\phi }}$, $\frac{\pi }{4}&lt;\,\phi &lt;\frac{\pi }{2}$ and $\theta +\,\phi &gt;\frac{\pi }{2}$. Find the moduli and arguments of the complex numbers $a$, $b$,$c$, $d$ and $\mathrm{e}$, where $a=\mathbf{i}\,p$, $b={{q}^{2}}$, $c=\frac{1}{q}$ and $d=p+{{p}^{*}}$ and $e=pq$.</p>
<p>On your Argand diagram, sketch and label the points $P$, $Q$, $A$, $B$, $C$,$D$ and $E$ representing the complex numbers $p$, $q$, $a$, $b$, $c$, $d$ and $e$ respectively.</p>

On an Argand diagram, the points $P$ and $Q$ represent the complex numbers $p$ and $q$ respectively where $p=\cos \theta +\mathbf{i}\sin \theta $, $0<\theta <\frac{\pi }{4}$ and $q=2{{\mathrm{e}}^{\mathbf{i}\,\,\phi }}$, $\frac{\pi }{4}<\,\phi <\frac{\pi }{2}$ and $\theta +\,\phi >\frac{\pi }{2}$. Find the moduli and arguments of the complex numbers $a$, $b$,$c$, $d$ and $\mathrm{e}$, where $a=\mathbf{i}\,p$, $b={{q}^{2}}$, $c=\frac{1}{q}$ and $d=p+{{p}^{*}}$ and $e=pq$.

On your Argand diagram, sketch and label the points $P$, $Q$, $A$, $B$, $C$,$D$ and $E$ representing the complex numbers $p$, $q$, $a$, $b$, $c$, $d$ and $e$ respectively.

Geometrical Transformations of Complex Numbers

Representing Complex Numbers on an Argand Diagram

ACJC Complex Numbers Tutorial Q13

<ol class="abc-list"><li data-list-item-id="item-1760257139205-71hk9ns32" data-list-item-path="item-1760257139205-71hk9ns32"><div class="list-item-paragraph">Write down the modulus and argument of the complex number $w$, where&nbsp;$w=1+\mathbf{i}\sqrt{3}$. Hence find the set of values of the positive integer $n$ for which ${{w}^{n}}$is real, and show that, for these values of $n$, the value of ${{w}^{n}}$ is either ${{2}^{n}}$or $-{{2}^{n}}$.</div><div class="list-item-paragraph">Express ${{w}^{100}}-{{\left( w^* \right)}^{100}}$ in the form $k\mathbf{i}$, where&nbsp;$k$ is real, giving the exact value of $k$ in non-trigonometrical form.</div></li></ol>

De Moivre's Theorem

Cartesian, Trigonometric and Exponential Forms

N1993 P2 Q12

<p><strong>Do not use a graphing calculator in answering this question.</strong></p>
<p>The complex number $w$ is a root to the equation</p>
<p style="text-align: center">${{z}^{4}}+a{{z}^{3}}+46{{z}^{2}}+bz+125=0$,</p>
<p>where $a$,$b\in \mathbb{R}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Prove that the product of the factors $z-w$ and $z-{{w}^{*}}$ is a quadratic polynomial with real coefficients.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $\left| w \right|=\sqrt{5}$ and $\arg (w)=-{{\tan }^{-1}}(2)$ where $-\frac{\pi }{2}<-{{\tan }^{-1}}(2)<0$, show that $w=1-2\mathrm{i}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find all the roots of the equation and the values $a$ and $b$.</div><p style="text-align: right;">[5]</p></li></ol>

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

The complex number $w$ is a root to the equation

${{z}^{4}}+a{{z}^{3}}+46{{z}^{2}}+bz+125=0$,

Modulus and Argument of Complex Numbers

Applying the Conjugate Root Theorem

2020 HCI P1 Q8

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Prove that for any complex variable $z$ and complex number $\alpha $,</span></div><div class="list-item-paragraph" style="text-align: center">$\left( z-\alpha \right)\left( z-{{\alpha }^{*}} \right)={{z}^{2}}-\left( 2\operatorname{Re}\left( \alpha \right) \right)z+{{\left| \alpha \right|}^{2}}$.</div></li><li><div class="list-item-content"><span class="list-item-text">Explain why the argument $\theta $ of any root $\alpha $ of the equation ${{z}^{5}}+32=0$ must be of the form $\frac{2k+1}{5}\pi $ for some integer $k$ and find the common modulus of all the roots of this equation.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use your results in parts (i) and (ii) to express ${{z}^{5}}+32$ as a product of one linear factor and two quadratic factors, all with real coefficients.</span></div><div class="list-item-paragraph">(You may express the coefficients in trigonometric form, where necessary.)</div></li></ol>

Prove that for any complex variable $z$ and complex number $\alpha $,

Explain why the argument $\theta $ of any root $\alpha $ of the equation ${{z}^{5}}+32=0$ must be of the form $\frac{2k+1}{5}\pi $ for some integer $k$ and find the common modulus of all the roots of this equation.

Use your results in parts (i) and (ii) to express ${{z}^{5}}+32$ as a product of one linear factor and two quadratic factors, all with real coefficients.

Math Question Bank