SSCE HSC Mathematics Extension 2 Complex numbers: 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 $x 
eq 0$, 
$$ 	an 	heta = \frac{y}{x} \Rightarrow x + iy = re^{i	heta} $$
for some real number $r$.

B. For all non-zero complex numbers $z_1$ and $z_2$, 
$$ 	ext{Arg}(z_1) = 	heta_1 	ext{ and } 	ext{Arg}(z_2) = 	heta_2 \Rightarrow 	ext{Arg}(z_1z_2) = 	heta_1 + 	heta_2 $$
where Arg denotes the principal argument.

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

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

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

For all real numbers $x, y, \theta$ with $x \neq 0$, \tan \theta = \frac{y}{x}$

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

For all real numbers $r_1, r_2, \theta_1, \theta_2$ with $r_1, r_2 > 0$

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

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