A-LevelMathematicsDifferentiation9709/12 · May/June 2025 · Q45 Marks

A point P is moving along the curve with equation y = ax^3/2 – 12x in such a way that the x-coordinate of P is increasing at a constant rate of 5 units per second.

(a)Find the rate at which the y-coordinate of P is changing when x = 9. Give your answer in terms of the constant a.[3]
(b)Given that the curve has a minimum point when x = 1/4, find the value of a.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[1] *M1 For attempt at differentiation; at least one correct term needed. Condone poor notation throughout. dy/dx = (3/2)ax^(1/2) - 12
[1] DM1 For correct use of chain rule with 5, x=9 and their dy/dx. Condone missing brackets and allow errors in their working. dy/dt = dy/dx * dx/dt
[1] A1 OE simplified form. dy/dt = 45a/2 - 60 or 22.5a - 60 or 15/2(3a-8)
[1] M1 For setting their dy/dx = 0 with at least one term correct and substituting x = 0.25. Condone missing brackets. Allow a restart for dy/dx if 2 terms seen and at least one term correct. (3/2)a(1/4)^(1/2) - 12 = 0
[1] A1 a=16

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