Hint
The quotient formula requires \(\displaystyle z\ne 1\). Evaluate the complex geometric quotient before taking the real part.
Worked solution
(a)
\(\displaystyle S= \frac{z^{n}- 1}{z- 1}\) gives \(\Re(S)=-69\).
Answer: \(-69\).
Answer: \(-69\)
(b)
\(\Im(S)=17.3205080756888\)
Answer: \(17.3205080756888\).
Answer: \(\approx 17.3205080756888\)
(c)
\(|S|^2=\Re(S)^2+\Im(S)^2=5061\)
Answer: \(5061\).
Answer: \(\approx 5061\)
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: Evaluate the complex geometric quotient before taking the real part.
- 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: Do not interchange the real part of a quotient with the quotient of real parts.
(b) · 2 marks
- 1 mark: Take the imaginary part of the complete finite sum.
- 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: A complex modulus is not its real part.
(c) · 2 marks
- 1 mark: Add the squares of the real and imaginary parts.
- 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: Do not interchange the real part of a quotient with the quotient of real parts.
Total: 6 marks.