O-LevelAdditional MathematicsDifferentiation and integration4037/21 · Oct/Nov 2011 · Q1211 Marks
Answer only one of the following two alternatives. EITHER [Figure: A diagram showing a curve y = ln(x+1) - lnx. A tangent line and a normal line are drawn at point P(1, ln 2). The tangent intersects the x-axis at A and the y-axis at B. The normal intersects the x-axis at C and the y-axis at D.] The diagram shows part of the curve y = ln(x+1) – lnx. The tangent to the curve at the point P (1, 1n 2) meets the x-axis at A and the y-axis at B. The normal to the curve at P meets the x-axis at C and the y-axis at D.
- (i)Find, in terms of 1n 2, the coordinates of A, B, C and D.[8]
- (ii)Given that Area of triangle BPD / Area of triangle APC = 1/k, express k in terms of 1n 2.[3]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.
- [1] dy/dx = 1/(x+1) - 1/x
- [1] gradient tangent = -1/2
- [1] y - ln 2 = -(1/2)(x - 1)
- [1] A(1 + 2 ln 2,0)
- [1] B(0, -1/2 + ln 2)
- [1] Uses m₁m₂ = -1 in equation of line (y – ln 2 = 2(x – 1))
- [1] C(1 - 1/2 ln 2,0)
- [1] D(0,-2 + ln 2)
- [1] Valid method for area of triangle
- [1] 1.25 or 1.25 × (ln 2)² or 0.601
- [1] k = (ln 2)²
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