Complex Number

2022 DHS P1 Q4 For this question, you may use the result ${{left( cos theta +mathbf{i}sin theta right)}^{n}}=(cos ntheta +mathbf{i}sin ntheta )$. Find the exact value(s) of $theta $, where

2022 YIJC P1 Q1 Do not use a calculator in answering this question.It is given that $2-mathbf{i}$ is a root of the equation $2{{z}^{3}}+a{{z}^{2}}-2z+b=0$. (a) Find the values of the

2019 TJC P1 Q8 Do not use a calculator in answering this question. The complex numbers $z$ and $w$ are given by $z=frac{{{left( 1+mathbf{i} right)}^{4}}}{{{left( 1-mathbf{i} right)}^{2}}}$ and $w=frac{8}{{{left( sqrt{3}+mathbf{i}

2020 ACJC CA2 Q3 The complex number $z$ is given by $z=frac{{{left( k+kamathbf{i} right)}^{2}}}{{{left( 1-amathbf{i} right)}^{2}}}$, where $a$ and $k$ are positive real numbers. (i) Given that $left| z right|=3$,

Home 2018 ACJC CA2 Q2 Given that ${{left( frac{{{left( 2-2mathbf{i} right)}^{2}}}{-sqrt{3}+kmathbf{i}} right)}^{*}}=4{{text{e}}^{mathbf{i}theta }}$, where $k$ is a positive real number and $-pi <theta le pi $, (i) Show that $k=1$,

Home 2020 ASRJC J2 MYCT Q5 (b) Express $2{{text{e}}^{text{i}left( frac{pi }{4} right)}}-2{{text{e}}^{text{i}left( frac{pi }{3} right)}}$ in the form $r{{text{e}}^{text{i}left( kpi right)}}$, where $r$ is a positive real constant in exact

Home 2022 NYJC J2 CT2 P1 Q3 The complex number $z$ is given by $z=frac{{{left( cos left( frac{1}{12}pi right)-text{i}sin left( frac{1}{12}pi right) right)}^{3}}}{{{left( a+text{i} right)}^{2}}}$ where $a$ is a positive

Home 2022 TJC J2 MYE P1 Q6 Do not use a calculator in answering this question. The complex number $z$ is given by $z=1-sqrt{3}text{i}$. (i) Express $z$ in the form

(a) Do not use a calculator in answering this question. The complex number $w$ is given by $w=frac{{{left( cos frac{pi }{3}+text{i}sin frac{pi }{3} right)}^{4}}}{{{left( cos frac{pi }{6}-text{i}sin frac{pi }{6} right)}^{5}}}$.

Home 2022 CJC J2 MYE P1 Q7  (a) The complex number $z$ is given by $z=r{{text{e}}^{text{i}theta }}$ where $r>0$ and $-frac{2pi }{3}le theta le frac{2pi }{3}$. (i) Given that $w=left(