Complex Number

NJC Complex Numbers Tutorial Q6 (i) Prove that for any complex variable $z$ and complex number $alpha $, $left( z-alpha right)left( z-{{alpha }^{*}} right)={{z}^{2}}-left( 2operatorname{Re}left( alpha right) right)z+{{left| alpha right|}^{2}}$.

2015 CJC P2 Q3 A calculator is not to be used in answering this question. The complex numbers $a$ and $b$ are given by $frac{1+mathbf{i}}{1-mathbf{i}}$ and $frac{surd 2}{1-mathbf{i}}$ respectively. Find

2014 SAJC P2 Q4 [Modified] (a) Find the complex number $z$ in the form $x+y,mathbf{i}$ where $x$, $yin mathbb{R}$ such that $frac{mathbf{i}z}{z-2{{z}^{*}}-2}=-1$. [3] (b) Show that that for any complex

2022 ACJC P1 Q5 Do not use a calculator in answering this question. Two complex numbers are ${{z}_{1}}=2left( cos frac{pi }{18}-mathbf{i}sin frac{pi }{18} right)$ and ${{z}_{2}}=2mathbf{i}$. (i) Show that $frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}}$

2022 ACJC P2 Q3 (i) Find the roots of the equation $mathbf{i}{{z}^{2}}-left( 5+mathbf{i} right)z+2-6mathbf{i}=0$, giving your answers in cartesian form $a+bmathbf{i}$, where $a$, $bin mathbb{R}$. [2] (ii) Hence find the

2022 VJC P1 Q5 Do not use a calculator in answering this question. The complex number $z$ satisfies the equation ${{z}^{2}}-left( 4+mathbf{i} right)z+2left( mathbf{i}-t right)=0$, where $t$ is a real

2022 HCI P1 Q8 Do not use a graphing calculator for this question. (i) The complex number $a+mathbf{i}$ is a root of the equation ${{z}^{2}}-2sqrt{2}z+b=0$ where $a$ and $b$ are

2022 DHS P2 Q2 Do not use a calculator in answering this question.The complex number $w$is given by $2-3mathbf{i}$. (a) Find ${{w}^{3}}$ in the form $x+mathbf{i}y$, showing your working. [2]

2022 ASRJC P2 Q1 (i) The equation $3{{z}^{3}}-7{{z}^{2}}+17z+m=0$, where $m$ is a real constant, has a root $z=1+2mathbf{i}$. Find the value of $m$.   Hence using an algebraic method, find