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Date May Specimen paper Marks available 2 Reference code SPM.2.AHL.TZ0.6
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Write down and Hence Question number 6 Adapted from N/A

Question

A city has two cable companies, X and Y. Each year 20 % of the customers using company X move to company Y and 10 % of the customers using company Y move to company X. All additional losses and gains of customers by the companies may be ignored.

Initially company X and company Y both have 1200 customers.

Write down a transition matrix T representing the movements between the two companies in a particular year.

[2]
a.

Find the eigenvalues and corresponding eigenvectors of T.

[4]
b.

Hence write down matrices P and D such that T = PDP−1.

[2]
c.

Find an expression for the number of customers company X has after nn years, where nN.

[5]
d.

Hence write down the number of customers that company X can expect to have in the long term.

[1]
e.

Markscheme

(0.80.10.20.9)      M1A1

[2 marks]

a.

|0.8λ0.10.20.9λ|=0      M1

λ=1 and 0.7      A1

eigenvectors (12) and (11)     (M1)A1

Note: Accept any scalar multiple of the eigenvectors.

[4 marks]

b.

EITHER

P(1121)  D(1000.7)       A1A1

OR

P = (1112)  D = (0.7001)      A1A1

[2 marks]

c.

P−1 = 13(1121)       A1

13(1121)(1000.7n)(1121)(12001200)       M1A1

attempt to multiply matrices         M1

so in company A, after n years, 400(2+0.7n)         A1

[5 marks]

d.

400 × 2 = 800        A1

[1 mark]

e.

Examiners report

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a.
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b.
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c.
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d.
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e.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.19—Transition matrices – Markov chains
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Topic 4—Statistics and probability

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