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O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2012 · Q1210 Marks

Answer only one of the following alternatives. EITHER The point A(0, 10) lies on the curve for which dy/dx = e⁻ˣ/⁴. The point B, with x-coordinate –4, also lies on the curve. (i) Find, in terms of e, the y-coordinate of B. (ii) The tangents to the curve at the points A and B intersect at the point C. Find, in terms of e, the x-coordinate of the point C.

  1. (i)Find, in terms of e, the y-coordinate of B.[5]
  2. (ii)Find, in terms of e, the x-coordinate of the point C.[5]

Mark scheme

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

  1. [M1] Integrate to find y
  2. [A1] y = -4e⁻ˣ/⁴ (+ c)
  3. [DM1] Substitute (0, 10)
  4. [A1] c = 14
  5. [A1] y = 14 - 4e⁻ˣ/⁴
  6. [A1] 14 - 4e
  7. [B1] Tangent at A is y – 10 = x
  8. [B1] Gradient tangent at B is e
  9. [B1] Tangent at B is y + 4e - 14 = ex + 4e
  10. [M1] Solve equations of tangents
  11. [A1] x = 4 / (1 - e) o.e.

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About this question

Topic
Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 10 marks.
Source
O-Level Additional Mathematics, Oct/Nov 2012 Paper 2 Variant 2, question 12.

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