Hint
Use integer powers and consistent complex-exponential definitions. Pair the \(\displaystyle z^{n}\) and \(\displaystyle z^{−n}\) terms in the exponential expansion.
Worked solution
(a)
\(\displaystyle 2^{1-9}=\frac{1}{256}\)
Answer: \(\displaystyle \frac{1}{256}\).
Answer: \(\displaystyle \frac{1}{256}\)
(b)
\(\displaystyle \frac{9}{2^{8}}= \frac{9}{256}\)
Answer: \(\displaystyle \frac{9}{256}\).
Answer: \(\displaystyle \frac{9}{256}\)
(c)
\(\displaystyle (- 1)^{\frac{9- 1}{2}}2^{1- 9}= \frac{1}{256}\)
Answer: \(\displaystyle \frac{1}{256}\).
Answer: \(\displaystyle \frac{1}{256}\)
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: Pair the \(\displaystyle z^{n}\) and \(\displaystyle z^{−n}\) terms in the exponential expansion.
- 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 drop powers of i in a sine expansion.
(b) · 2 marks
- 1 mark: Use the binomial coefficient for one negative-power selection.
- 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 coefficient of the highest multiple angle is not generally one.
(c) · 2 marks
- 1 mark: Retain the factor \(\displaystyle i^{−n}\) in the sine expansion.
- 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 drop powers of i in a sine expansion.
Total: 6 marks.