Photo AI
Which of the following statements about complex numbers is true? A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2023 - Paper 1
Question 7
Which of the following statements about complex numbers is true? A. For all real numbers $x$, $y$, $ heta$ with $x \neq 0$, $$ an \theta = \frac{y}{x} \Rightarrow... show full transcript
Worked Solution & Example Answer:Which of the following statements about complex numbers is true? A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2023 - Paper 1
Step 1
A. For all real numbers $x$, $y$, $\theta$ with $x \neq 0$, $$\tan \theta = \frac{y}{x} \Rightarrow x + iy = re^{i\theta},$$ for some real number $r$.
Answer
This statement is true. The relationship given defines the polar form of a complex number, indicating that any complex number can be expressed in this way. When , the indeed gives the correct ratio of , verifying the angle corresponds to that combination in the complex plane.
Step 2
B. For all non-zero complex numbers $z_1$ and $z_2$, $$\text{Arg}(z_1) = \theta_1 \text{ and } \text{Arg}(z_2) = \theta_2 \Rightarrow \text{Arg}(z_1 z_2) = \theta_1 + \theta_2,$$ where Arg denotes the principal argument.
Answer
This statement is also true. The argument of the product of two non-zero complex numbers is the sum of their arguments, which is a fundamental property of complex multiplication.
Step 3
C. For all real numbers $r_1$, $r_2$, $\theta_1$, $\theta_2$ with $r_1, r_2 > 0$, $$r_1 e^{i\theta_1} = r_2 e^{i\theta_2} \Rightarrow r_1 = r_2 \text{ and } \theta_1 = \theta_2.$$
Answer
This statement is true. In polar form, equality of two complex numbers requires both their magnitudes (radii) and arguments to be equal, confirming the distinct unambiguous representation.
Step 4
D. For all real numbers $x$, $y$, $r$, $0 < r$ and $x \neq 0$, $$x + iy = re^{i\theta} \Rightarrow \theta = \arctan \left(\frac{y}{x}\right).$$
Answer
This statement is true but requires clarification on the range of . While it provides a valid relationship, care must be taken with the quadrant where is calculated, as alone does not uniquely determine the angle in all cases.