ACJC Complex Numbers Tutorial Q13 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
2021 YIJC P2 Q3 Do not use a calculator in answering this question. (a) The complex number $z$ is given by $z=frac{{{left( 1-mathbf{i} right)}^{3}}}{sqrt{2}{{left( a+mathbf{i} right)}^{2}}}$, where $a<0$. (i) Given
2021 NYJC P1 Q9 The complex numbers $z$ and $w$ where $wne 0$ satisfy the relation $2z=left| w right|+1$. (i) It is given that $a$ is a real number and
2021 ASRJC P2 Q1 The complex numbers $z$ and $w$ have moduli $k$ and $3{{k}^{2}}$ respectively and arguments $alpha $ and $4alpha $ respectively, where $k$ is a positive constant
2021 ACJC P1 Q9 (a) (i) Show that the cubic polynomial ${{x}^{3}}+p{{x}^{2}}+{{p}^{2}}x+q$ can be reduced to ${{y}^{3}}+left( frac{2{{p}^{2}}}{3} right)y+alpha $ by the substitution $x=y-frac{p}{3}$, where $alpha $ is to be
2021 SAJC P1 Q7 (a) It is given that $z=1+sqrt{3},mathbf{i}$ is a root of the equation $3{{z}^{3}}+a{{z}^{2}}+bz-8=0$, where $a$ and $b$ are real numbers. Find the exact values of $a$
2021 ASRJC P1 Q1 Do not use a calculator in answering this question. It is given that $text{f}left( z right)={{z}^{4}}+2sqrt{2}{{z}^{3}}+{{z}^{2}}+8sqrt{2}z-12$. One of the roots of the equation $text{f}left( z right)=0$
2021 HCI P1 Q5 Do not use a calculator for this question. (a) The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2mathbf{i}$, ${{z}_{2}}=2{{text{e}}^{frac{pi }{6},mathbf{i}}}$ and ${{z}_{3}}=frac{1}{7}left( cos
2017 DHS P1 Q8 (c) The complex number $z$ is given by $z=1+{{text{e}}^{mathbf{i},alpha }}$. (i) Show that $z$ can be expressed as $2cos left( frac{1}{2}alpha right){{e}^{mathbf{i},left( frac{1}{2}alpha right)}}$. [2] (ii)
NJC Complex Numbers Tutorial Q6 (i) Prove that for any complex variable $z$ and complex number $alpha $, $left( z-alpha right)left( z-{{alpha }^{*}} right)={{z}^{2}}-left( 2operatorname{Re}left( alpha right) right)z+{{left| alpha right|}^{2}}$.