June 2018 Paper 1 Q8

Notes

(a)(i)

M1: Attempts to differentiate (deals with the powers of \(v\) correctly).
Look for an expression for \(\dfrac{\mathrm{d}C}{\mathrm{d}v}\) in the form \(\dfrac{A}{v^2} + B\)

A1: \(\left(\dfrac{\mathrm{d}C}{\mathrm{d}v}\right) = -\dfrac{1500}{v^2} + \dfrac{2}{11}\)
A number of students are solving part (a) numerically or graphically. Allow these students to pick up the M1 A1 here from part (b) when they attempt the second derivative.

M1: Sets \(\dfrac{\mathrm{d}C}{\mathrm{d}v} = 0\) (which may be implied) and proceeds to an equation of the type \(v^n = k, k \gt 0\)
Allow here equations of the type \(\dfrac{1}{v^n} = k, k \gt 0\)

A1: \(v = \sqrt{8250}\) or \(5\sqrt{330}\) awrt 90.8 (km h⁻¹). Don't be concerned by incorrect / lack of units.
As this is a speed withhold this mark for answers such as \(v = \pm\sqrt{8250}\)

* Condone \(\dfrac{\mathrm{d}C}{\mathrm{d}v}\) appearing as \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) or perhaps not appearing at all. Just look for the rhs.

(a)(ii)

M1: For a correct method of finding \(C =\) from their solution to \(\dfrac{\mathrm{d}C}{\mathrm{d}v} = 0\).
Do not accept attempts using negative values of \(v\).
Award if you see \(v = \ldots, C = \ldots\) where the \(v\) used is their solution to (a)(i). You do not need to check this calculation.

A1ft: Minimum cost = awrt (£) 93. Condone the omission of units
Follow through on sensible values of \(v\). \(60 \lt v \lt 110\)

Notes

M1: Finds \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2}\) (following through on their \(\dfrac{\mathrm{d}C}{\mathrm{d}v}\) which must be of equivalent difficulty) and attempts to find its value / sign at their \(v\)
Allow a substitution of their answer to (a) (i) in their \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2}\)
Allow an explanation into the sign of \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2}\) from its terms (as \(v \gt 0\))

A1ft: \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2} = +0.004 \gt 0\) hence minimum (cost). Alternatively \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2} = +\dfrac{3000}{v^3} \gt 0\) as \(v \gt 0\)
Requires a correct calculation or expression, a correct statement and a correct conclusion.
Follow through on their \(v\) (\(v \gt 0\)) and their \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2}\)

* Condone \(\dfrac{\mathrm{d}^2C}{\mathrm{d}v^2}\) appearing as \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) or not appearing at all for the M1 but for the A1 the correct notation must be used (accept notation \(C''\)).

Notes

B1: Gives a limitation of the given model, for example

• It would be impossible to drive at this speed over the whole journey
• The traffic would mean that you cannot drive at a constant speed