A-LevelMathematicsIntegration9709/12 · May/June 2015 · Q1010 Marks

The equation of a curve is y = 4 / (2x – 1).

(i)Find, showing all necessary working, the volume obtained when the region bounded by the curve, the x-axis and the lines x = 1 and x = 2 is rotated through 360° about the x-axis.[4]
(ii)Given that the line 2y = x + c is a normal to the curve, find the possible values of the constant c.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[B1] (i) ∫ 16/(2x-1)² dx = -16/(2x-1) Correct without the ÷2
[B1] (i) ÷2 For the ÷2 even if first B1 is lost
[M1] (i) Vol = π [-8/(2x-1)] with limits 1 and 2 Use of limits in a changed expression.
[A1] (i) → 16π/3 co
[M1] (ii) m = ½m of tangent = -2 Use of m₁m₂ = -1
[B1] (ii) dy/dx = -4(2x-1)⁻² × 2 Correct without the ×2
[B1] (ii) For the ×2 even if first B1 is lost
[DM1] (ii) Equating their dy/dx to -2
[A1] (ii) → x = 3/2 or -1/2 co
[A1] (ii) → c = 5/2 or -7/2 co

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, May/June 2015 Paper 1 Variant 2, question 10.

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