Do not use a calculator in answering this question. Find the complex number $z$ which satisfies th

Question

Do not use a calculator in answering this question. Find the complex number $z$ which satisfies the equation $\frac{4|z|}{15-z^*} = 5i$.[3] The complex number $w$ is such that $(w-i)^3 = -i$. Given that one possible value of $w$ is $2i$, find the two other possible values of $w$. Give your answers in cartesian form $a+bi$.[4] The points $W_1$, $W_2$ and $W_3$ on the Argand diagram represent the three roots of the equation $(w-i)^3 = -i$, and the point $A$ represents the complex number $ki$, where $k$ is a positive real number. Show that the points $W_1$, $W_2$ and $W_3$ lie on a circle with centre $A$ for some value of $k$, stating the value of $k$.[2]

Accepted Answer

(a)z=15-20i (b)(i)$w=\frac{\sqrt 3}{2}+\frac{1}{2}i$ or $w=-\frac{\sqrt 3}{2} +\frac{1}{2}i$ (b)(ii)k=1

2025 VJC P1 Q7

2025 VJC P1 Q7