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

Answer only one of the following alternatives. OR The diagram shows part of the curve y = 8 + e⁻ˣ/³ crossing the y-axis at P. The normal to the curve at P meets the x-axis at Q. (i) Find the coordinates of Q. (ii) The line through Q, parallel to the y-axis, meets the curve at R and OQRS is a rectangle. Find ∫[0 to -3] (8 + e⁻ˣ/³) dx and hence find the area of the shaded region.

  1. (i)Find the coordinates of Q.[4]
  2. (ii)Find ∫[0 to -3] (8 + e⁻ˣ/³) dx and hence find the area of the shaded region.[6]

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Mark scheme

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

  1. [M1] dy/dx = (-1/3)e⁻ˣ/³
  2. [A1] at (0, 9) dy/dx = -1/3
  3. [M1] Grad normal = 3
  4. [A1] Point Q is (-3, 0)
  5. [M1] Area rectangle 24 + 3e (32.1)
  6. [M1] Integrate: 8x and e⁻ˣ/³ seen
  7. [A1] [8x - 3e⁻ˣ/³]
  8. [M1] Correct use of limits their -3 and 0
  9. [A1] 21+3e (29.1)
  10. [A1] Shaded area =3

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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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