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PSLE

PSLE Foundation Mathematics

<p>A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-26" data-list-item-path="p1421-item-26"><div class="list-item-paragraph">Find $z$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-28" data-list-item-path="p1421-item-28"><div class="list-item-paragraph">Find $z-z^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-30" data-list-item-path="p1421-item-30"><div class="list-item-paragraph">Use the given sum and product to find $\dfrac1z+\dfrac1{z^*}$, without evaluating the two reciprocals separately.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)>0$.

Applying Complex Conjugate Identities

Solving Equations Involving Complex Conjugates

Recovering a number from its sum and product

Complex Numbers

<p>Find the roots of ${{z}^{2}}+z+2=0$.</p>
<p>Hence, solve</p>
<ol class="i-list"><li data-list-item-id="item-1762581706526-h8s5i1hwk" data-list-item-path="item-1762581706526-h8s5i1hwk"><div class="list-item-paragraph">${{(3w-1)}^{2}}+(1-3w)+2=0$</div></li><li data-list-item-id="item-1762581707074-mz7bmbcgx" data-list-item-path="item-1762581707074-mz7bmbcgx"><div class="list-item-paragraph">$(w^*)^2-w^*+2=0$, where&nbsp;$w^*$ is the complex conjugate of $w$.</div></li><li data-list-item-id="item-1762581707898-luhk6ylp0" data-list-item-path="item-1762581707898-luhk6ylp0"><div class="list-item-paragraph">$\frac{1}{{{w}^{2}}}-\frac{1}{w}+2=0$.</div></li></ol>

Solving Complex Equations by Substitution

Complex Roots of Quadratics with Real Coefficients

<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.**

Using Complex Numbers to Deduce Trigonometric Values

Arithmetic with Complex Numbers in Cartesian Form

Modulus, Argument, and Operations on Complex Numbers

Solving Complex Equations

2024 RI P1 Q8

<p>Given that ${{\left| \frac{2\mathbf{i}-{{z}^{*}}}{z}-1 \right|}^{2}}+z=-\mathbf{i}$, find $z$ in the form $x+\mathbf{i}y$.</p>
<p>State the exact values of $\left| z \right|$ and $\arg \left( z \right)$.</p><p style="text-align: right;">[5]</p>

Given that ${{\left| \frac{2\mathbf{i}-{{z}^{*}}}{z}-1 \right|}^{2}}+z=-\mathbf{i}$, find $z$ in the form $x+\mathbf{i}y$.

State the exact values of $\left| z \right|$ and $\arg \left( z \right)$.

Modulus and Argument of Complex Numbers

RI Complex Numbers Tutorial Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that that for any complex number $z=r{{\mathrm{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $,</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{z}{z-r}=\frac{1}{2}-\frac{1}{2}\mathbf{i}\cot \left( \frac{\theta }{2} \right)$.</div><p style="text-align: right;">[2]</p></li></ol>

Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that

Show that that for any complex number $z=r{{\mathrm{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $,

Properties of Trigonometric and Exponential Forms

2014 SAJC P2 Q4 [Modified]

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that $\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Given that $\left| z \right|=1$, find $\left| \frac{z-3\,\mathbf{i}}{1+3\,\mathbf{i}\,z} \right|$.</span></div><p style="text-align: right;">[3]</p></li></ol>

Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that $\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.

Given that $\left| z \right|=1$, find $\left| \frac{z-3\,\mathbf{i}}{1+3\,\mathbf{i}\,z} \right|$.

Math Question Bank