October 2020 Paper 3 Q6

OCR MEICurrent spec12 marksLogs & Exponentials Modelling

(a)

(i) Write down the derivative of \(\mathrm{e}^{kx}\), where \(k\) is a constant. [1]

(ii) A business has been running since 2009. They sell maths revision resources online.
Give a reason why an exponential growth model might be suitable for the annual profits for the business. [1]

Fig. 6 shows the relationship between the annual profits of the business in thousands of pounds (\(y\)) and the time in years after 2009 (\(x\)). The graph of \(\ln y\) plotted against \(x\) is approximately a straight line.

Fig. 6: graph of ln y (0 to 4.5) against x (0 to 10) on graph paper; ten plotted crosses at x = 0 to 9 lie close to a straight line from (0, 1.9) to about (9, 4.1) — Fig. 6

(b) Show that the straight line is consistent with a model of the form \(y = A\mathrm{e}^{kx}\), where \(A\) and \(k\) are constants. [2]

(d) Use the model to predict the profit in the year 2020. [3]

(e) How reliable do you expect the prediction in part (d) to be? Justify your answer. [1]

Mark scheme (a)

Scheme	Marks	AO
(i) \(k\mathrm{e}^{kx}\)	B1	1.2
	[1]
(ii) Reason referring to growth proportional to profit.	E1	3.3
	[1]

Notes

(ii) E1: E.g. As more people hear about the business, they will sell more so it is reasonable that the rate of growth is proportional to the profits

Scheme	Marks	AO
\(\ln y = \ln A + kx\)	M1	1.1
Equation is of form “\(y = mx + c\)” so a straight line - hence model is consistent with graph	E1	2.4
	[2]

Mark scheme (c)

Scheme	Marks	AO
\(\ln A = 1.9\)	M1	1.1a
\(A = 6.686\)	A1	2.2a
\(k = \dfrac{0.4}{1.6}\)	M1	1.1
\(k = 0.24\) to \(0.25\)	A1	2.2a
	[4]

Notes

M1: Intercept = 1.9 not enough for M1

A1: One or more d.p.
Allow \(\mathrm{e}^{1.9}\)

M1: Attempt to find gradient

A1: \(\frac{11}{45}\) gets M1A1

Mark scheme (d)

Scheme	Marks	AO
\(x = 11\)	B1	3.3
\(y = 6.686 \times \mathrm{e}^{0.25x}\) So \(y = 6.686 \times \mathrm{e}^{0.25 \times 11}\)	M1	3.4
104.586… £104 587	A1	3.2a
	[3]

Notes

M1: Use of model with their \(A\) and \(k\)

A1: Translation into pounds (may be to nearest thousand) (FT their \(k\) and \(a\))

Mark scheme (e)

Scheme	Marks	AO
Not reliable. Extrapolation may not be valid o.e. OR Fit of model is good so far and it’s only two more years o.e.	E1	3.5b
	[1]

Notes

E1: E.g.
• The relationship between sales and time may change
• There is a limit to the market for revision resources
• Changes in the economy may affect the business
‘Reliable’ plus ‘extrapolation’ does not score