8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0 , x = 2.5\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }\).
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\).
3. What is the long-term population of red squirrels predicted by this model? The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When \(t = 0 , y = 1\).
4. Express \(\frac { 1 } { 2 y - y ^ { 2 } }\) in partial fractions.
5. Hence show by integration that \(\ln \left( \frac { y } { 2 - y } \right) = 2 t\). Show that \(y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }\).
6. What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
   Paper B: Comprehension
   Monday
   23 JANUARY 2006
   Afternooon U Additional materials:
   Rough paper
   MEI Examination Formulae and Tables (MF2)
   4754(B) \section*{Up to 1 hour
   $$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}
   • Write your name, centre number and candidate number in the spaces at the top of this page.
   • Answer all the questions.
   • Write your answers in the spaces provided on the question paper.
   • You are permitted to use a graphical calculator in this paper.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • The insert contains the text for use with the questions.
   • You may find it helpful to make notes and do some calculations as you read the passage.
   • You are not required to hand in these notes with your question paper.
   • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
   • The total number of marks for this section is 18.

   For Examiner's Use
   Qu.	Mark
   1
   2
   3
   4
   5
   Total

   1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.

   Party	Votes (\%)
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3

7. Use the Trial-and-Improvement method, starting with values of \(10 \%\) and \(14 \%\), to find an acceptance percentage for 7 seats, and state the allocation of the seats.

   Acceptance percentage, \(\boldsymbol { a }\) \%		10\%	14\%
   Party	Votes (\%)	Seats	Seats	Seats	Seats	Seats
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Total seats

   Seat Allocation \(\quad \mathrm { P } \ldots\)... \(\mathrm { Q } \ldots\) R ... S ... T ... \(\mathrm { U } \ldots\).
8. Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.

   	Round
   Party	1	2	3	4	5	6	7	Residual
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Seat allocated to

   Seat Allocation \(\mathrm { P } \ldots\). Q … \(\mathrm { R } \ldots\). S … T … \(\mathrm { U } \ldots\). 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for \(a\) in Table 4. Treating Party E as Party 5, verify that \(\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }\).
   4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.

   Seats	Interval	Seats	Interval
   1	\(22.2 < a \leqslant 27.0\)	5
   2	\(16.6 < a \leqslant 22.2\)	6	\(10.6 < a \leqslant 11.1\)
   3		7
   4

9. Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.
10. Complete the table above. \begin{table}[h]

    	Round
    Party	1	2	3	4	5	6	Residual
    A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
    B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
    C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
    D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
    E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
    F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
    G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
    H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
    Seat allocated to	C	A	D	C	E	A

    \captionsetup{labelformat=empty} \caption{Table 5}
    \end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.
11. Relate this distinction to the use of inequality signs.
12. Show that the inequality on line 102 can be rearranged to give \(0 \leqslant V _ { k } - N _ { k } a < a\). [1]
13. Hence justify the use of the inequality signs in line 102.