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O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2013 · Q1112 Marks

A curve has equation y = 3x + 1/(x-4)³.

  1. (i)Find dy/dx and d²y/dx².[4]
  2. (ii)Show that the coordinates of the stationary points of the curve are (5, 16) and (3, 8).[2]
  3. (iii)Determine the nature of each of these stationary points.[2]
  4. (iv)Find ∫(3x + 1/(x-4)³) dx.[2]
  5. (v)Hence find the area of the region enclosed by the curve, the line x = 5, the x-axis and the line x = 6.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

  1. [B1 + B1] (i) dy/dx = 3 - 3(x - 4)⁻⁴ o.e. isw
  2. [M1] (i) d²y/dx² = (their 12)(x – 4)^(their -5) o.e.
  3. [A1] (i) d²y/dx² = 12(x – 4)⁻⁵ o.e. isw
  4. [M1] (ii) Verifies dy/dx = 0 when x = 3 and x = 5
  5. [A1] (ii) Shows that x = 3 ⇒ y = 8 and x = 5 ⇒ y = 16
  6. [M1] (iii) x = 5, d²y/dx² (=12) > 0 ⇒ min or x = 3, d²y/dx² (=-12) < 0 ⇒ max
  7. [A1] (iii) Both correct cao
  8. [B1 + B1] (iv) 3x²/2 - (x-4)⁻²/2 (+c) o.e. isw
  9. [M1] (v) their [(3(6)²/2 - 1/(2(6-4)²)) - (3(5)²/2 - 1/(2(5-4)²))]
  10. [A1] (v) 16.875 to 3 or more sf or 135/8 or 16 7/8 cao

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About this question

Topic
Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 12 marks.
Source
O-Level Additional Mathematics, May/June 2013 Paper 2 Variant 2, question 11.

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