It is given that $z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where $b$ is

Question

It is given that $z=-\sqrt{3}+\mathbf{i}$ and $w=b-\mathbf{i}\sqrt{2}$, where $b$ is real and positive. Given that $\left| \frac{z}{w} \right|=1$, show that $b=\sqrt{2}$.[2] Show that $\arg z+\arg(w^*)=13\pi/12$, where $w^*$ is the complex conjugate of $w$ and arguments are principal values.[2] Given that $\arg \left( p+z \right)-\arg \left( z \right)=-\frac{7\pi }{12}$, find the exact value of the real number $p$.[3]

Accepted Answer

(i) $b=\sqrt2$. (ii) $\arg z+\arg(w^*)=13\pi/12$. (iii) $p=1+\sqrt3$.

2021 CJC JC2 MYE P1 Q4 [Modified]

2021 CJC JC2 MYE P1 Q4 [Modified]

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