Hint
Use the same x-value in both equations. Substitute the line into the curve.
Worked solution
(a)
\(\left(x - 15\right) \left(x - 11\right) = \left(x - 15\right) \left(x - 11\right)\); solve and substitute each root into the line.
Answer: \(\left\{\left( 11, \ 34\right), \left( 15, \ 42\right)\right\}\)
(b)
Between the two roots, (x−u)(x−v)<0, so the curve lies below the line.
Answer: \(2 x + \left(x - 15\right) \left(x - 11\right) + 12 < 2 x + 12\) on \(\left(11, 15\right)\)
(c)
\(\int\limits_{11}^{15} \left(- \left(x - 15\right) \left(x - 11\right)\right)\, dx\)=\(\displaystyle \frac{32}{3}\)
Answer: \(\displaystyle \frac{32}{3}\)
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) · 3 marks
- 1 mark: Substitute the line into the curve and solve the resulting quadratic.
- 1 mark: Find the matching y-coordinate for each root.
- 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: An x-root alone is not a complete intersection point.
(b) · 2 marks
- 1 mark: Determine the sign of curve minus line between the intersection roots.
- 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 add areas with an unexamined sign.
(c) · 2 marks
- 1 mark: Integrate upper minus lower between the two intersection abscissae.
- 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: An x-root alone is not a complete intersection point.
Total: 7 marks.