14 The expected value of a painting, £\(P\), is given by the formula
\(P = 2500 \times 1.2^n\)
where \(n\) is the number of years after it was bought and \(0 \leqslant n \leqslant 4\).
(a) Write down the value of the painting when it was bought. [1]
(b) Write down the annual percentage increase in the expected value of the painting. [1]
(c) The table shows the expected value of the painting \(n\) years after it was bought.
Years after the painting is bought (\(n\))
1
2
3
4
Expected value of the painting (£)
3000
3600
4320
5184
On the grid below, draw a suitable graph to show the expected value of the painting \(n\) years after it was bought, where \(0 \leqslant n \leqslant 4\). [3]
(d) An art collector correctly works out \(2500 \times 1.2^{10}\) as 15479.
They say,
The expected value of the painting 10 years after it was bought is £15479.
What assumption has the art collector made. [1]
Mark scheme (a)
Answer
Marks
Part marks and guidance
2500
1
Mark scheme (b)
Answer
Marks
Part marks and guidance
20
1
Mark scheme (c)
Answer
Marks
Part marks and guidance
Correct increasing curve with 2500 indicated as \(y\) – intercept
3
B1 for 3 of 4 correct plots at 1, 2, 3, 4
Accuracy use overlay as a guide 4320 and 5184 should be in the correct small square including the boundaries
B1 for plot or intercept at 2500
B1 for increasing curve from \(x = 1\) to \(x = 4\) through their 4 points
Do not accept ruled linear graph Ignore if joined to (0, 0)
Maximum 2 marks if curve incorrect
Mark scheme (d)
Answer
Marks
Part marks and guidance
That the annual percentage increase stays the same oe
1
Accept % increase/interest/%change remains constant each year If % value is given then accept 20% or their(b)
See appendix 3
Appendix 3: Question 14d
Response
Mark
A
The formula works beyond 4 years
1
B
The formula stays the same and nothing changes
1
C
It remains at \(2500 \times 1.2^n\) over the years
1
D
The painting increases by the same [%] rate/proportion each year
1
E
The painting will continue to increase in value
0
F
The value does not decrease at any point
0
G
The value of the painting remains the same each year
0
H
The price is going to raise each year
0
I
The increase in the expected value remains constant
0
J
The painting consistently rises by the same amount each year
