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O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2014 · Q1210 Marks

The diagram below shows part of the curve y = 3x - 14 + 32/x² cutting the x-axis at the points P and Q. [Figure 12.1]

  1. (i)Show that x - 2 is a factor of 3x³ - 14x² + 32.[1]
  2. (ii)Hence factorise 3x³ - 14x² + 32 completely.[4]
  3. (iii)State the x-coordinates of P and Q.[1]
  4. (iv)Find ∫ (3x - 14 + 32/x²) dx and hence determine the area of the shaded region.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.

  1. [B1] (i) f(2)= 3(2³)-14(2²)+ 32 = 0. Or complete long division
  2. [M1] (ii) (x-2)(3x²-8x-16). 3x² and 16
  3. [A1] (ii) 8x and correct signs
  4. [M1] (ii) (x-2)(x-4)(3x+4). Factorise three term quadratic
  5. [A1] (ii)
  6. [B1] (iii) x = 2, 4
  7. [B1] (iv) ∫ (3x-14+32/x²) dx = 1.5x²-14x-32/x (+c). first 2 terms
  8. [B1] (iv) third term correct unsimplified
  9. [M1] (iv) Area = [1.5x²-14x-32/x] evaluated from 2 to 4. Limits of 2 and 4 and subtract
  10. [A1] (iv) = (-) 2

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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 2014 Paper 2 Variant 2, question 12.

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