Do not use a calculator in answering this question. The equation $iz^3-1=0$ has 3 roots $z_1$, \

Question

Do not use a calculator in answering this question. The equation $iz^3-1=0$ has 3 roots $z_1$, $z_2$ and $z_3$, which are represented by points $A$, $B$ and $C$ respectively on an Argand diagram. It is given that $z_1$ is purely imaginary and the real component of $z_2$ is positive. Find the exact values of $z_1$, $z_2$ and $z_3$.[4] Determine the modulus and the arguments of $z_1$, $z_2$ and $z_3$, where $-\pi<\arg(z)\leq\pi$.[4] Sketch $A$, $B$ and $C$ on an Argand diagram and state the geometrical shape of $ABC$.[3]

Accepted Answer

(a) $z_1=i$, $z_2=\frac{\sqrt3-i}{2}$, $z_3=\frac{-\sqrt3-i}{2}$ (b) all moduli $1$; arguments $\frac\pi2,-\frac\pi6,-\frac{5\pi}{6}$ (c) equilateral triangle

2026 HCI P1 Q7

2026 HCI P1 Q7

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