Hint
The argument is real unless stated otherwise. Substitute \(\displaystyle e^{u}=a\) and \(\displaystyle e^{− u}= \frac{1}{a}\).
Worked solution
(a)
\(\displaystyle \frac{8- \frac{1}{8}}{2}= \frac{63}{16}\)
Answer: \(\displaystyle \frac{63}{16}\).
Answer: \(\displaystyle \frac{63}{16}\)
(b)
\(\displaystyle \frac{8+ \frac{1}{8}}{2}= \frac{65}{16}\)
Answer: \(\displaystyle \frac{65}{16}\).
Answer: \(\displaystyle \frac{65}{16}\)
(c)
\(\displaystyle \frac{\sinh u}{\cosh u}= \frac{63}{65}\)
Answer: \(\displaystyle \frac{63}{65}\).
Answer: \(\displaystyle \frac{63}{65}\)
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: Substitute \(\displaystyle e^{u}=a\) and \(\displaystyle e^{− u}= \frac{1}{a}\).
- 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 fundamental identity has a minus sign.
(b) · 2 marks
- 1 mark: Add the exponential and its reciprocal before dividing by two.
- 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: Sech is a reciprocal, not an inverse function.
(c) · 2 marks
- 1 mark: Divide the two hyperbolic 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: The fundamental identity has a minus sign.
Total: 6 marks.