Complex numbers questions | form six Pure Advanced Mathematics

Find Complex numbers examination questions, form six Pure Advanced Mathematics in acaproso.com

#	Question
1	Use Demoivre`s theorem to find the value of $left ( frac{1}{2} + frac{1}{2}i ight )^{10}$ Show that $left [ r(cos heta +i,sin heta) ight ]^{n}=r^{n}e^{in heta}$ and hence find in form of $re^{i heta}$ all complex numbers z, such that $z^{3}=frac{5+i}{2+3i}$ i) Solve the equation $x^{4}+1=0$ and leave the roots in radical form. ii) If $w=frac{z+2}{2}$ and $\|z\|=4$ , find the locus of the w. Mathematical Calculation
2	Express the complex number $left ( frac{1+i}{1-i} ight )+left ( frac{sqrt{3}}{1-i} ight )^{4}$ in the form a+ib. Mathematical Calculation
3	Show that $[r(cos heta + isin heta )]^{n}=r^{n}e^{in heta}$ Mathematical Calculation