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O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2012 · Q1211 Marks

Answer only one of the following two alternatives. OR The diagram shows part of the curve y = xe⁻ˣ and the tangent to the curve at the point R (2, 2/e²). [Figure 12.1]

  1. (i)Given that y = xe⁻ˣ, find dy/dx and hence show that ∫xe⁻ˣ dx = −xe⁻ˣ − e⁻ˣ + c.[4]
  2. (ii)Find the area of the shaded region bounded by the curve, the tangent and the y-axis.[7]

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Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

  1. [A1] (i) Uses product rule
  2. [M1] (i) (1-x)e⁻ˣ
  3. [A1AG] (i) ∫(1-x)e⁻ˣdx = xe⁻ˣ
  4. [M1] (i) ∫xe⁻ˣdx = -xe⁻ˣ + ∫e⁻ˣdx = -xe⁻ˣ - e⁻ˣ
  5. [B1] (ii) gradient tangent = -1/e² or = -0.135
  6. [M1] (ii) y - 2/e² = -1/e² (x-2)
  7. [A1] (ii) Uses line cuts y-axis at 4/e² or 0.541
  8. [M1] (ii) Area trapezium (= 6/e² or 0.812)
  9. [M1] (ii) Uses limits of 2 and 0 on -xe⁻ˣ - e⁻ˣ (=1 - 3/e² or 0.594)
  10. [M1] (ii) Evaluate area of trapezium subtract area under curve
  11. [A1] (ii) 9/e² - 1 or 0.218

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About this question

Topic
Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 11 marks.
Source
O-Level Additional Mathematics, May/June 2012 Paper 2 Variant 2, question 12.

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