Hint
The root count m is a positive integer. Take the mth root of the positive modulus.
Worked solution
(a)
\(\displaystyle |z|= (10^{3})^{\frac{1}{3}}= 10\)
Answer: \(10\).
Answer: \(10\)
(b)
\(\displaystyle \theta_{0}= \frac{\pi }{4\cdot 3}\)
Answer: \(0.261799387799149\).
Answer: \(\approx 0.261799387799149\)
(c)
\(\displaystyle \frac{2\pi }{3}\)
Answer: \(2.0943951023932\).
Answer: \(\approx 2.0943951023932\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Take the mth root of the positive modulus.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Divide the argument by m after including all 2kπ branches.
(b) · 2 marks
- 1 mark: Use \(\displaystyle k=0\) in the complete root formula.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The modulus of a root is the mth root of the original modulus.
(c) · 2 marks
- 1 mark: Compare consecutive k values.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Divide the argument by m after including all 2kπ branches.
Total: 6 marks.