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O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2012 · Q1111 Marks

Answer only one of the following two alternatives. EITHER The diagram shows part of the curve y = sin(½x). The tangent to the curve at the point P (3π/2, √2/2) cuts the x-axis at the point Q. [Figure 11.1]

  1. (i)Find the coordinates of Q.[4]
  2. (ii)Find the area of the shaded region bounded by the curve, the tangent and the x-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. [M1] (i) dy/dx = ½ cos(½x)
  2. [A1] (i) Gradient tangent = -√2/4 or -0.35...
  3. [M1] (i) y - √2/2 = -√2/4 (x - 3π/2)
  4. [A1] (i) y = 0, x = 3π/2 + 2 or 6.71
  5. [B1] (ii) ∫sin(½x)dx = -2cos(½x)
  6. [B1] (ii) Identify 2π
  7. [M1] (ii) Use limits of 1.5π and (2π) on -2cos(½x) (2-√2 or 0.586)
  8. [M1] (ii) Attempt at area of triangle ½ × (base × height)
  9. [M1] (ii) (= √2/2 = 0.707)
  10. [M1] (ii) Idea of area of triangle subtract area under curve.
  11. [M1] (ii) Plan completely correct
  12. [A1] (ii) 3√2/2 - 2 or 0.121

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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 11.

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