Do not use a calculator in answering this question. Express \(z\), where \(z=-1+\sqrt3\mathrm{i}\), in the form \(re^{\mathrm{i}\theta}\), where \(r>0\) and \(-\pi<\theta\leq\pi\). Find the smallest positive integer value of \(n\) such that \(\frac{z^n}{\mathrm{i}z^*}\) is purely imaginary. Find the complex numbers \(v\) and \(w\) which satisfy the following simultaneous equations: \[2v+|w|=1\] \[3v-\mathrm{i}w=-3+4\mathrm{i}\] Give your answers in the form \(a+b\mathrm{i}\), where \(a\) and \(b\) are real numbers.

Do not use a calculator in answering this question.

Express \(z\), where \(z=-1+\sqrt3\mathrm{i}\), in the form \(re^{\mathrm{i}\theta}\), where \(r>0\) and \(-\pi<\theta\leq\pi\).

Find the smallest positive integer value of \(n\) such that \(\frac{z^n}{\mathrm{i}z^*}\) is purely imaginary.

Find the complex numbers \(v\) and \(w\) which satisfy the following simultaneous equations: \[2v+|w|=1\] \[3v-\mathrm{i}w=-3+4\mathrm{i}\] Give your answers in the form \(a+b\mathrm{i}\), where \(a\) and \(b\) are real numbers.

N2023 P1 Q8

N2023 P1 Q8