Hint
Identify intercepts or curve intersections first. Use the first zero factor.
Worked solution
(a)
The intercept factors give the two endpoints; the leading coefficient is negative, so the curve is above the axis between them.
Answer: \(5\le x\le 8\)
(b)
Integrate \(\left(8 - x\right) \left(2 x - 10\right)\) from \(5\) to \(8\). A primitive is \(\displaystyle - \frac{2 x^{3}}{3} + 13 x^{2} - 80 x\), and upper minus lower gives \(9\).
Answer: \(9\)
(c)
The quadratic is nonnegative between its roots.
\(9\)
Answer: \(9\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Solve both intercepts and establish the sign of the curve between them.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not take an absolute value before checking internal sign changes.
(b) · 2 marks
- 1 mark: Integrate the nonnegative curve between its two intercepts.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: For two curves use upper minus lower.
(c) · 2 marks
- 1 mark: The quadratic is nonnegative between its roots.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not take an absolute value before checking internal sign changes.
Total: 6 marks.