June 2024 Paper 2 Q1

Notes

(a)(i) If “\(+\,c\)” is included with either derivative penalise it only once on the first occurrence.

M1: Award for \(x^3 \rightarrow x^2\) or \(x^2 \rightarrow x\) or \(5x \rightarrow 5\) or \(-10 \rightarrow 0\)
Indices may be unprocessed e.g. \(x^3 \rightarrow x^{3-1}\) or \(x^2 \rightarrow x^{2-1}\) or \(5x \rightarrow 5x^0\)

A1: Correct simplified expression with indices processed \(12x^2 - 14x + 5\).
Do not allow \(x^1\) for \(x\) or \(5x^0\) for 5.
Apply isw if necessary once a correct answer is seen.
The “\(\dfrac{\mathrm{d}y}{\mathrm{d}x} =\)” is not required.

(ii)

A1ft: Correct simplified second derivative \(24x - 14\) or follow through their first derivative.
Must be simplified so do not allow e.g. \(x^1\) for \(x\) or \(x^0\) for 1 as above.
Apply isw if necessary once a correct answer is seen.
The “\(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} =\)” is not required.

Notes

M1: Sets their second derivative of the form \(ax + b,\ a, b \neq 0\) equal to 0 and proceeds to a value for \(x\). Condone slips in rearranging as long as a value for \(x\) is obtained.
This may be implied by their value of \(x\) or may be implied by their working e.g.
\(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} =\right) 24x - 14 \rightarrow 24x = 14 \Rightarrow x = \ldots\)
Condone one slip in copying their second derivative.
Also condone if they “cancel” e.g. \(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} =\right) 24x - 14 \rightarrow 12x - 7 = 0 \Rightarrow x = \ldots\)

A1: Correct value from correct work and a correct second derivative but allow recovery if they “cancel” their second derivative to obtain e.g. \(12x - 7\).
Allow exact equivalents e.g. \(\dfrac{14}{24}\) but not rounded decimals e.g. 0.583
Allow recurring decimal if clearly indicated e.g. \(0.58\dot{3}\)
Correct answer only from a correct second derivative (or correctly cancelled second derivative) scores both marks.
Isw after a correct answer is seen.