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A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2018 · Q1114 Marks

EITHER The curve C is defined parametrically by x = 18t - t² and y = 8t³, where 0 < t < 4.

  1. (i)Show that at all points of C, d²y/dx² = 3(9+t) / (2t²(9-t)³).[4]
  2. (ii)Show that the mean value of d²y/dx² with respect to x over the interval 0 < x ≤ 56 is 3/70.[4]
  3. (iii)Find the area of the surface generated when C is rotated through 2π radians about the x-axis, showing full working.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

  1. [B1] AEF.
  2. [M1] Uses chain rule again to find second derivative.
  3. [dM1] Uses quotient (or product rule)
  4. [A1] AG.
  5. [B1] Uses correct formula for mean value
  6. [M1] Finding limits correctly
  7. [M1] Using expression
  8. [A1] Inserts correct values of t and obtains answer (AG.)
  9. [M1A1] Simplifies (dx/dt)² + (dy/dt)²
  10. [M1A1] Uses 2π ∫ y ds / dt dt.
  11. [M1] Integrates term by term.
  12. [A1] Accept exact answer or decimal rounding to 15300.

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About this question

Topic
Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 14 marks.
Source
A-Level Further Mathematics, Oct/Nov 2018 Paper 1 Variant 2, question 11.

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