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PSLE Foundation Mathematics

<p>It is given that $z=1+2\mathrm{i}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Without using a calculator, find the values of ${{z}^{2}}$ and $\frac{1}{{{z}^{3}}}$ in cartesian form $x+\mathrm{i}y$, showing your working.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The real numbers $p$ and $q$ are such that $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$ is real. Find, in terms of $p$, the value of $q$ and the value of $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$.</div><p style="text-align: right;">[3]</p></li></ol>

Conditions for Real and Purely Imaginary Values

Arithmetic with Complex Numbers in Cartesian Form

Solving Complex Equations

N2014 P1 Q5

Complex Numbers

<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>

Applying Complex Conjugate Identities

Finding Square Roots of Complex Numbers Algebraically

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>

2023 DHS P1 Q4

<ol class="abc-list"><li data-list-item-id="item-1760360484890-wm5l1q7ci" data-list-item-path="item-1760360484890-wm5l1q7ci"><div class="list-item-paragraph">Find the complex numbers $w$ and $z$ which satisfy the simultaneous equations</div><div class="list-item-paragraph" style="text-align: center">$wz=\mathbf{i}$ and $w-2\mathbf{i}\,z=2$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760360485828-mzt38ff6e" data-list-item-path="item-1760360485828-mzt38ff6e"><div class="list-item-paragraph">The complex number $v$ satisfies $vv^*=3$ and $\arg(v/v^*)=-\pi/2$, where&nbsp;$v^*$ is the complex conjugate of $v$. Arguments are principal values in $(-\pi,\pi]$.</div><ol class="i-list"><li data-list-item-id="subitem-1760360506085-3087nvu2u" data-list-item-path="item-1760360485828-mzt38ff6e/subitem-1760360506085-3087nvu2u"><div class="list-item-paragraph">Find the modulus of $v$ and all possible values of its argument.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760360507052-a167borh8" data-list-item-path="item-1760360485828-mzt38ff6e/subitem-1760360507052-a167borh8"><div class="list-item-paragraph">Find the smallest positive integer $n$ such that&nbsp;$v^n$ is real for either possible value of $v$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Modulus, Argument, and Operations on Complex Numbers

Properties of Trigonometric and Exponential Forms

Solving Simultaneous Complex Equations

Simultaneous Equations Involving Complex Numbers and Conjugates

<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

N2015 P1 Q9

<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 $\alpha$.

2009 CJC 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

<p>The complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are such that&nbsp;${{z}_{1}}=3-\mathbf{i}\sqrt{3}$, ${{z}_{2}}=\frac{1}{2}{{\mathrm{e}}^{\mathbf{i}\frac{2\pi }{5}}}$ and ${{z}_{3}}={{z}_{1}}\times {{z}_{2}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1759136999626-5bb1vq0l7" data-list-item-path="item-1759136999626-5bb1vq0l7"><div class="list-item-paragraph">Find exactly the modulus and argument of ${{z}_{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759137000594-c1senjq3s" data-list-item-path="item-1759137000594-c1senjq3s"><div class="list-item-paragraph">Sketch an Argand diagram showing $z_1$, $z_2$ and $z_3$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759137001171-elu7gadlo" data-list-item-path="item-1759137001171-elu7gadlo"><div class="list-item-paragraph">Find the smallest positive integer value of $n$ for which ${{z}_{3}}^{n}$ is purely imaginary. State the modulus and argument of ${{z}_{3}}^{n}$ in this case, giving the modulus in the form $k\sqrt{3}$, where&nbsp;$k$ is an integer.<span class="inline-marks">[4]</span></div></li></ol>

The complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are such that ${{z}_{1}}=3-\mathbf{i}\sqrt{3}$, ${{z}_{2}}=\frac{1}{2}{{\mathrm{e}}^{\mathbf{i}\frac{2\pi }{5}}}$ and ${{z}_{3}}={{z}_{1}}\times {{z}_{2}}$.

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