O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2021 · Q129 Marks
The diagram shows part of the curve y = (9-x)(x-3) and the line y = k−3, where k > 3. The line through the maximum point of the curve, parallel to the y-axis, meets the x-axis at A. The curve meets the x-axis at B, and the line y=k-3 meets the curve at the point C(k, k-3). Find the area of the shaded region.
- The diagram shows part of the curve y = (9-x)(x-3) and the line y = k−3, where k > 3. The line through the maximum point of the curve, parallel to the y-axis, meets the x-axis at A. The curve meets the x-axis at B, and the line y=k-3 meets the curve at the point C(k, k-3). Find the area of the shaded region.[9]
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Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [B1] x-coordinate of A = 6 soi
- [B1] x-coordinate of B = 9 soi
- [M1] k-3 = (9-k)(k-3)
- [A1] k = 8 [therefore C(8, 5)]
- [B1] (8-6)×5 or 10 oe soi
- [M2] ∫(12x-27-x²) dx
- [M1] for 2 correct terms (for integration)
- [M1] [6x² - 27x - x³/3] (DEP on at least M1 for integration)
- [M1] their10 + F(their 9) – F(their 8)
- [A1] 38/3 or 12 2/3 or 12.7 or 12.66[66...] rot to 3/4 or more figs nfww
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