Complex Number

2016 JJC P2 Q4 The complex numbers $a$ and $b$ are given by $a=-left( 1+sqrt{3}mathbf{i} right)$ and $b=frac{1}{2}left( 1-mathbf{i} right)$. (i) Without using a calculator, find the value of ${{a}^{2}}b$

2017 HCI P2 Q2 The complex numbers $z$ and $w$ satisfy the following equations $2z+3w=20$,$w-z{{w}^{*}}=6+22mathbf{i}$. (i) Find $z$ and $w$ in the form $a+bmathbf{i}$, where $a$ and $b$ are real,

2022 YIJC P1 Q6 (a) Show that $1+{{text{e}}^{-text{i}alpha }}=2cos frac{alpha }{2}{{text{e}}^{-text{i}frac{alpha }{2}}}$, where $-pi <alpha le pi $. [2] (b) Hence or otherwise, show that ${{left( 1+{{text{e}}^{-text{i}alpha }} right)}^{3}}-{{left( 1+{{text{e}}^{text{i}alpha

2022 HCI P2 Q3 The complex number $z$ is given by $z=2(cos beta +text{i}sin beta )$ where $0<beta <frac{pi }{2}$. (i) Show that $frac{z}{4-{{z}^{2}}}=(koperatorname{cosec}beta )text{i}$, where $k$ is a positive

2022 TMJC P2 Q2 [Modified] It is given that $z=-sqrt{6}-text{i}sqrt{2}$ and $w=3left( cos frac{5pi }{7}-text{i}sin frac{5pi }{7} right)$. Without the use of a calculator, find the modulus and argument of

2020 ACJC P2 Q4 (a) It is given that $text{f}left( x right)={{x}^{6}}-a{{x}^{4}}-{{x}^{2}}-b$, where $a$ and $b$ are real numbers. (i) Show that $text{f}left( x right)=text{f}left( -x right)$. [1] The diagram

2020 DHS P2 Q4 (a) (i) Using ${{text{e}}^{mathbf{i}theta }}=cos theta +mathbf{i}sin theta $ and the trigonometry formulas, show that $left( {{text{e}}^{mathbf{i}{{theta }_{1}}}} right)left( {{text{e}}^{mathbf{i}{{theta }_{2}}}} right)={{text{e}}^{mathbf{i}left( {{theta }_{1}}+{{theta }_{2}} right)}}$.

2019 NYJC P1 Q9 (a) The complex numbers $z$ and $w$ satisfy the simultaneous equations $left| z right|-{{w}^{*}}=-3-sqrt{2}mathbf{i}$ and ${{w}^{*}}+w+5z=1+20mathbf{i}$, where ${{w}^{*}}$ is the complex conjugate of $w$. Find the