A-LevelMathematicsIntegration9709/12 · Oct/Nov 2012 · Q99 Marks
The diagram shows part of the curve y = 9/(2x + 3), crossing the y-axis at the point B (0, 3). The point A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C. [Figure 9]
- (i)Find the equation of the tangent to the curve at A.[4]
- (ii)Determine, showing all necessary working, whether C is nearer to B or to O.[1]
- (iii)Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through 360° about the x-axis.[4]
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 (Oct/Nov 2012). The mark or mark type is shown in brackets.
- [B1] dy/dx = -9/(2x+3)² × 2
- [M1] → m=-⅔
- [M1] → y-1 = -⅔(x-3)
- [A1] y = -⅔x + 3
- [B1] Meets the y-axis when x = 0, y = 3 (implies C(0,3)).
- [B1] This is nearer to B than to O.
- [B1] Integral of π ∫ y² dx = π ∫ 81/(2x+3)² dx
- [B1] = -81/(2(2x+3)) (correct integration)
- [M1] Uses limits 0 to 3
- [A1] → 9π
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