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Date May 2021 Marks available 2 Reference code 21M.2.AHL.TZ2.4
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Hence or otherwise and Determine Question number 4 Adapted from N/A

Question

In a small village there are two doctors’ clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that 3.5% of Doctor Black’s patients moved to Doctor Green’s clinic and 5% of Doctor Green’s patients moved to Doctor Black’s clinic. All additional losses and gains of patients by the clinics may be ignored.

At the start of a particular year, it was noted that Doctor Black had 2100 patients on their register, compared to Doctor Green’s 3500 patients.

Write down a transition matrix T indicating the annual population movement between clinics.

[2]
a.

Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.

[2]
b.

Find a matrix P, with integer elements, such that T=PDP1, where D is a diagonal matrix.

[6]
c.

Hence, show that the long-term transition matrix T is given by T=10171017717717.

[6]
d.

Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.

[2]
e.

Markscheme

T=0.9650.050.0350.95        M1A1


Note: Award M1A1 for T=0.950.0350.050.965.
Award the A1 for a transposed T if used correctly in part (b) i.e. preceded by 1×2 matrix 2100    3500 rather than followed by a 2×1 matrix.


[2 marks]

a.

0.9650.050.0350.95221003500        (M1)

=22943306

so ratio is 2294:3306  =1147:1653, 0.693889        A1


[2 marks]

b.

to solve Ax=λx:

0.965-λ0.050.0350.95-λ=0        (M1)

0.965-λ0.95-λ-0.05×0.035=0

λ=0.915  OR  λ=1        (A1)

attempt to find eigenvectors for at least one eigenvalue        (M1)

when λ=0.915, x=1-1 (or any real multiple)        (A1)

when λ=1, x=107 (or any real multiple)        (A1)

therefore P=110-17 (accept integer valued multiples of their eigenvectors and columns in either order)        A1


[6 marks]

c.

P-1=110-17-1=1177-1011            (A1)


Note: This mark is independent, and may be seen anywhere in part (d).


D=0.915001            (A1)

Tn=PDnP-1=110-170.915n001n1177-1011            (M1)A1


Note:
Award (M1)A0 for finding P-1DnP correctly.


as n, Dn=0.915n001n0001                  R1

so Tn117110-1700017-1011                 A1

=10171017717717                   AG


Note: The AG line must be seen for the final A1 to be awarded.


[6 marks]

d.

METHOD ONE

1017101771771721003500=32942306            (M1)

so ratio is  3294:2306   1647:1153,  1.42844, 0.700060                A1

 

METHOD TWO

long term ratio is the eigenvector associated with the largest eigenvalue            (M1)

10:7                A1


[2 marks]

e.

Examiners report

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e.

Syllabus sections

Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations
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Topic 1—Number and algebra » AHL 1.15—Eigenvalues and eigenvectors
Topic 1—Number and algebra
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