A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2013 · Q910 Marks

The diagram shows part of the curve y = 8/x + 2x and three points A, B and C on the curve with x-coordinates 1, 2 and 5 respectively.

(i)A point P moves along the curve in such a way that its x-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the y-coordinate of P is changing as P passes through A.[4]
(ii)Find the volume obtained when the shaded region is rotated through 360° about the x-axis.[6]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

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

[1] dy/dx = −8/x² + 2 (Attempt at differentiation. algebraic – unsimplified.)
[1] dy/dx at x=1 (point A) = −6
[1] dy/dt = (dy/dx) × (dx/dt) (Needs product of 0.04 and 'his' dy/dx)
[1] Rate = −0.24
[1] Use of integral of y² dx (e.g. ∫(8/x + 2x)² dx)
[1] Expansion of (8/x + 2x)² to 64/x² + 32 + 4x²
[1] Integration of 64/x² to -64/x
[1] Integration of 32 to 32x
[1] Integration of 4x² to 4x³/3
[1] Correct application of limits from 2 to 5 and final answer (271.2π or 852) (omission of π loses last mark)

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.