Hint
Rearrange \(\displaystyle C(n)\ge 60\) before factorising. Use completed-square form for the largest possible score.
Worked solution
(a)
\(\displaystyle -n^{2}+14n+15\ge 60\) is equivalent to \(\displaystyle n^{2}-14n+45\le 0\). This factors as \(\displaystyle (n-5)(n-9)\le 0\), so \(\displaystyle 5\le n\le 9\).
Answer: \(5\le n\le9\).
(b)
Intersect [5,9] with the integer domain \(\displaystyle 0\le n\le 15\). The five allowable settings are 5, 6, 7, 8 and 9.
Answer: \(n=5,6,7,8,9\).
(c)
\(\displaystyle C(n)=-(n-7)^{2}+64\). The maximum is 64 at \(\displaystyle n=7\), which is a permitted integer setting.
Answer: Maximum score \(64\), at \(n=7\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Rearrange with the correct inequality direction.
- 1 mark: Factor or find roots 5 and 9.
- 1 mark: Give the inclusive interval [5,9].
Alternative methods: Completing the square to \(\displaystyle (n-7)^{2}\le 4\) is accepted.
Common error: Failing to reverse the inequality when multiplying by −1.
(b) · 2 marks
- 1 mark: Apply the integer restriction.
- 1 mark: List exactly the five settings, including both boundary settings.
Alternative methods: Integer set notation with the same five values is accepted.
Common error: Including noninteger settings or omitting 5 and 9.
(c) · 2 marks
- 1 mark: Use a valid maximum method and check the setting is permitted.
- 1 mark: State score 64 at \(\displaystyle n=7\).
Alternative methods: Symmetry, differentiation with the domain check, or a complete integer comparison is accepted.
Common error: Reporting the setting as the score.
Total: 7 marks.