Hint
The argument is real unless stated otherwise. Substitute eᵘ=a and e⁻ᵘ=1/a.
Worked solution
(a)
\((8-1/8)/2=\frac{63}{16}\)
Answer: \(\frac{63}{16}\).
Answer: \(\frac{63}{16}\)
(b)
\((8+1/8)/2=\frac{65}{16}\)
Answer: \(\frac{65}{16}\).
Answer: \(\frac{65}{16}\)
(c)
\(\sinh u/\cosh u=\frac{63}{65}\)
Answer: \(\frac{63}{65}\).
Answer: \(\frac{63}{65}\)
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 eᵘ=a and e⁻ᵘ=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.