Hint
Integrate their vertical separation. The accumulated area increases strictly between the intersections.
Worked solution
(a)
x²=3x gives x(x−3)=0. The points are (0,0) and (3,9). On (0,3), 3x−x²=x(3−x)>0, so the line is above the parabola.
Answer: \((0,0)\) and \((3,9)\); y=3x is upper for 0<x<3.
(b)
\(A=\int_0^3(3x-x^2)\,dx=[3x^2/2-x^3/3]_0^3=27/2-9=9/2\) square units.
Answer: \(9/2\) square units.
(c)
The area from 0 to a is \(L(a)=3a^2/2-a^3/3\). At a=3/2 this is 27/8−9/8=9/4, half of 9/2. Also L′(a)=3a−a²=a(3−a)>0 on (0,3). The accumulated area is strictly increasing there, so it reaches half the total at only one cut.
Answer: Each area is 9/4; the split is unique since \(a(3-a)>0\) for 0<a<3.
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: Solve for both intersection inputs 0 and 3.
- 1 mark: Give both complete intersection points.
- 1 mark: Identify the line as upper with a valid sign or test-point argument.
Alternative methods: A correctly labelled graph is accepted.
Common error: Using only one intersection or assuming the parabola is upper from its formula.
(b) · 2 marks
- 1 mark: Set up and integrate upper minus lower over [0,3].
- 1 mark: Give 9/2 square units.
Alternative methods: Subtracting two correctly evaluated under-curve integrals is accepted.
Common error: Reversing the upper and lower functions.
(c) · 3 marks
- 1 mark: Use the correct accumulated-area expression.
- 1 mark: Evaluate 9/4 at a=3/2 and compare with the total area.
- 1 mark: Justify uniqueness by positive vertical separation or a positive derivative throughout the open interval.
Alternative methods: Symmetry about x=3/2 plus a valid strict-increase argument is accepted.
Common error: Verifying the proposed cut without addressing uniqueness.
Total: 8 marks.