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<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The equation $iz^3-1=0$ has 3 roots $z_1$, $z_2$ and $z_3$, which are represented by points $A$, $B$ and $C$ respectively on an Argand diagram. It is given that&nbsp;$z_1$ is purely imaginary and the real component of $z_2$ is positive.</p>
<ol class="abc-list"><li data-list-item-id="hci-q07-item01" data-list-item-path="hci-q07-item01"><div class="list-item-paragraph">Find the exact values of $z_1$, $z_2$ and $z_3$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q07-item02" data-list-item-path="hci-q07-item02"><div class="list-item-paragraph">Determine the modulus and the arguments of $z_1$, $z_2$ and $z_3$, where&nbsp;$-\pi&lt;\arg(z)\leq\pi$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q07-item03" data-list-item-path="hci-q07-item03"><div class="list-item-paragraph">Sketch $A$, $B$ and $C$ on an Argand diagram and state the geometrical shape of $ABC$.<span class="inline-marks">[3]</span></div></li></ol>

<strong>Do not use a calculator in answering this question.</strong>

The equation $iz^3-1=0$ has 3 roots $z_1$, $z_2$ and $z_3$, which are represented by points $A$, $B$ and $C$ respectively on an Argand diagram. It is given that $z_1$ is purely imaginary and the real component of $z_2$ is positive.

Find the exact values of $z_1$, $z_2$ and $z_3$.

Determine the modulus and the arguments of $z_1$, $z_2$ and $z_3$, where $-\pi<\arg(z)\leq\pi$.

Sketch $A$, $B$ and $C$ on an Argand diagram and state the geometrical shape of $ABC$.

Modulus and Argument of Complex Numbers

Geometric Problems on the Argand Diagram

Finding Roots of Complex Numbers

Representing Complex Numbers on an Argand Diagram

2026 HCI P1 Q7

Complex Numbers

<p>Consider the equation $z^3=c$, where&nbsp;$z\in\mathbb{C}$ and $c$ is a positive constant.</p>
<p>One of the solutions of the equation is $z=-2+2i\sqrt3$.</p>
<ol class="abc-list"><li data-list-item-id="p2-tza-q08-item01" data-list-item-path="p2-tza-q08-item01"><div class="list-item-paragraph">Find the value of $c$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item02" data-list-item-path="p2-tza-q08-item02"><div class="list-item-paragraph">Write down the other two solutions of $z^3=c$ giving your answers in the form $a+bi$, where&nbsp;$a,b\in\mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item03" data-list-item-path="p2-tza-q08-item03"><div class="list-item-paragraph">Find the solutions of $(iz)^3=c$ giving your answers in the form $a+bi$, where&nbsp;$a,b\in\mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p2-tza-q08-item04" data-list-item-path="p2-tza-q08-item04"><div class="list-item-paragraph">The solutions of $(iz)^3=c$ can be represented on an Argand diagram by three points $A$, $B$ and $C$.</div><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

Consider the equation $z^3=c$, where $z\in\mathbb{C}$ and $c$ is a positive constant.

One of the solutions of the equation is $z=-2+2i\sqrt3$.

Write down the other two solutions of $z^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$.

Find the solutions of $(iz)^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$.

De Moivre's Theorem

IB AA HL May 2026 P2 TZA Q8

<p>Solve the equations</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$z^4 = 1 - i$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$z^5 = 1 + \sqrt{3}i$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$(z + 2i)^5 = 1$</div></li></ol>

Obtaining roots of a complex number

<p>Find all the complex numbers $z$ such that</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$z^4 = -16$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$z^3 = -27i$</div></li></ol>

Find all the complex numbers $z$ such that

Solving Complex Equations

Obtaining the roots of a complex number

<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 $z^5=-32\mathrm i$.

Conditions for Real and Purely Imaginary Values

Applying Complex Conjugate Identities

Properties of Trigonometric and Exponential Forms

N2015 P1 Q9

<p>Write down the real root of</p>
<p style="text-align: center">${{w}^{3}}+8=0$,</p>
<p>Where&nbsp;$w$ is a complex number. </p>
<p>Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form. </p>
<p>Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$. </p>
<p>Determine the roots of $z$ such that&nbsp;${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.</p>

Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form. 

Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$. 

Determine the roots of $z$ such that ${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.

Solving Complex Equations by Substitution

2009 NJC P1 Q9 (b) [modified]

<p>The fixed nonzero complex number $a$ has modulus $R$ and argument $\alpha$, and $a^*$ denotes its complex conjugate. Given that&nbsp;$q=\dfrac{a^3}{a^*}$, express $q$ in trigonometric form.<span class="inline-marks">[3]</span></p>
<p>If one of the sixth roots of $q$ is purely imaginary, determine the least positive value of $\alpha$.<span class="inline-marks">[3]</span></p>

The fixed nonzero complex number $a$ has modulus $R$ and argument $\alpha$, and $a^*$ denotes its complex conjugate. Given that $q=\dfrac{a^3}{a^*}$, express $q$ in trigonometric form.

If one of the sixth roots of $q$ is purely imaginary, determine the least positive value of $\alpha$.

Modulus, Argument, and Operations on Complex Numbers

2009 CJC P1 Q8 (b)

<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.

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