SSCE HSC Mathematics Extension 2 Modulus and argument: Which of the following statement

Question

Which of the following statements about complex numbers is true?

A. For all real numbers $x$, $y$, $	heta$ with $x \neq 0$,
   $$ \tan \theta = \frac{y}{x} \Rightarrow x + iy = re^{i\theta} $$
   for some real number $r$.

B. For all non-zero complex numbers $z_1$ and $z_2$, 
   $$ \text{Arg}(z_1) = \theta_1 \text{ and } \text{Arg}(z_2) = \theta_2 \Rightarrow \text{Arg}(z_1z_2) = \theta_1 + \theta_2, $$
   where Arg denotes the principal argument.

C. For all real numbers $r_1$, $r_2$, $	heta_1$, $	heta_2$ with $r_1, r_2 > 0$, 
   $$ r_1 e^{i \theta_1} = r_2 e^{i \theta_2} \Rightarrow r_1 = r_2 \text{ and } \theta_1 = \theta_2. $$

D. For all real numbers $x$, $y$, $r$ with $r > 0$ and $x \neq 0$, 
   $$ x + iy = re^{i\theta} \Rightarrow \theta = \text{arctan} \left( \frac{y}{x} \right). $$

Which of the following statements about complex numbers is true? A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2023 - Paper 1

Worked Solution & Example Answer:Which of the following statements about complex numbers is true? A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2023 - Paper 1

A. For all real numbers $x$, $y$, $ heta$ with $x \neq 0$

B. For all non-zero complex numbers $z_1$ and $z_2$

C. For all real numbers $r_1$, $r_2$, $ heta_1$, $ heta_2$ with $r_1, r_2 > 0$

D. For all real numbers $x$, $y$, $r$ with $r > 0$ and $x \neq 0$

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