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Date May 2022 Marks available 3 Reference code 22M.2.AHL.TZ2.5
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number 5 Adapted from N/A

Question

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.

The model is of the form

Xn+1Zn+1=MXnZn

where Xn is the probability of the gene being in its normal state after dividing for the nth time, and Zn is the probability of it being in another state after dividing for the nth time, where n.

Matrix M is found to be 0.94  b0.06  0.98.

The gene is in its normal state when n=0. Calculate the probability of it being in its normal state

Write down the value of b.

[1]
a.i.

What does b represent in this context?

[1]
a.ii.

Find the eigenvalues of M.

[3]
b.

Find the eigenvectors of M.

[3]
c.

when n=5.

[2]
d.i.

in the long term.

[2]
d.ii.

Markscheme

0.02         A1

 

[1 mark]

a.i.

the probability of mutating from ‘not normal state’ to ‘normal state’         A1


Note: The A1 can only be awarded if it is clear that transformation is from the mutated state.

 

[1 mark]

a.ii.

det0.94-λ  0.020.06  0.98-λ=0         (M1)


Note: Award M1 for an attempt to find eigenvalues. Any indication that detM-λI=0 has been used is sufficient for the (M1).


0.94-λ0.98-λ-0.0012=0  OR  λ2-1.92λ+0.92=0         (A1)

λ=1, 0.92  2325         A1

 

[3 marks]

b.

0.94  0.020.06  0.98xy=xy  OR  0.94  0.020.06  0.98xy=0.92xy         (M1)


Note: Award M1 can be awarded for attempting to find either eigenvector.


0.02y-0.06x=0  OR  0.02y+0.02x=0

13  and  1-1         A1A1


Note: Accept any multiple of the given eigenvectors.

 

[3 marks]

c.

0.94  0.020.06  0.98510  OR  0.744  0.08520.256  0.91510         (M1)


Note: Condone omission of the initial state vector for the M1.


0.744   0.744311            A1

 

[2 marks]

d.i.

0.250.75         (A1)


Note: Award A1 for 0.250.75  OR  0.25  0.250.75  0.75 seen.


0.25          A1

 

[2 marks]

d.ii.

Examiners report

There was some difficulty in interpreting the meaning of the values in the transition matrix, but most candidates did well with the rest of the question. In part (d) there was frequently evidence of a correct method, but a failure to identify the correct probabilities. It was surprising to see a significant number of candidates diagonalizing the matrix in part (d) and this often led to errors. Clearly this was not necessary.

a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.
[N/A]
d.i.
[N/A]
d.ii.

Syllabus sections

Topic 1—Number and algebra » AHL 1.15—Eigenvalues and eigenvectors
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Topic 1—Number and algebra

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