Further Mathematics 7367: Complex Numbers Topical Past Paper Questions

The Argand diagram below shows a circle C (a) Write down the equation of the locus of C in the form $$ \left|z-w\right|=a $$ where w is a complex number whose real and imaginary parts are integers, and a is an integer. (b) It is given that $ z_{1} $ is a complex number representi…

Given that $ z = \cos\theta + i\sin\theta $, use de Moivre's theorem to show that $$ z^{n}-z^{-n}\equiv2\mathrm{i}\sin n0 $$ [2 marks] (b) The series $S$ is defined as $$ S=\sin\theta+\sin3\theta+\ldots+\sin(2n-1)\theta $$ Use part (a) to express $S$ in the form $$ S=\frac{1}{2\m…

Find the imaginary part of $$ \frac{5+\mathrm{i}}{1-\mathrm{i}} $$ Circle your answer.