O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2015 · Q910 Marks
[Figure 9.1] The diagram shows parts of the line y = 3x + 10 and the curve y = x³ – 5x² + 3x + 10. The line and the curve both pass through the point A on the y-axis. The curve has a maximum at the point B and a minimum at the point C. The line through C, parallel to the y-axis, intersects the line y = 3x + 10 at the point D.
- (i)Show that the line AD is a tangent to the curve at A.[2]
- (ii)Find the x-coordinate of B and of C.[3]
- (iii)Find the area of the shaded region ABCD, showing all your working.[5]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.
- [M1] for differentiation
- [A1] comparing both gradients
- [M1] equating gradient to zero and valid attempt to solve
- [A1,A1] Al for each
- [B1] area of the trapezium
- [M1] attempt to obtain the area enclosed by the curve and the coordinate axes, by integration
- [A1] integration all correct
- [DM1] correct application of limits (must be using their 3 from (ii) and 0)
- [A1] = 24.7 or 24.8
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