1.02y Partial fractions: decompose rational functions

4 The expression \(\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }\) can be written in the form \(2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }\), where \(A\) and \(B\) are constants.

1. Find the values of \(A\) and \(B\).
2. Hence find \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).

2

1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).

3

1. Given that \(\frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) }\) can be written in the form \(3 + \frac { A } { 3 x - 1 } + \frac { B } { x - 1 }\), where \(A\) and \(B\) are integers, find the values of \(A\) and \(B\).
2. Hence, or otherwise, find \(\int \frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) } \mathrm { d } x\).

2

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - 1 }\) up to the term in \(x ^ { 3 }\).
   2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { 1 + 3 x }\) up to the term in \(x ^ { 3 }\).
2. Express \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) in partial fractions.
3. 1. Find the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) up to the term in \(x ^ { 3 }\).
   2. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) is valid.

6

1. Express \(\frac { 2 } { x ^ { 2 } - 1 }\) in the form \(\frac { A } { x - 1 } + \frac { B } { x + 1 }\).
2. Hence find \(\int \frac { 2 } { x ^ { 2 } - 1 } \mathrm {~d} x\).
3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y } { 3 \left( x ^ { 2 } - 1 \right) }\), given that \(y = 1\) when \(x = 3\). Show that the solution can be written as \(y ^ { 3 } = \frac { 2 ( x - 1 ) } { x + 1 }\).

3

1. Find the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Express \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) in the form \(\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }\), where \(A\) and \(B\) are integers.
   2. Find the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) is valid.

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.

2

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).

9 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).

1. Find \(v\) in terms of \(t\).
2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
5. According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
   WEDNESDAY 21 MAY 2008
   Afternoon
   Time: Up to 1 hour
   Additional materials: Rough paper
   MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
   \includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
   This document consists of \(\mathbf { 6 }\) printed pages, \(\mathbf { 2 }\) blank pages and an insert. 1 Complete these Latin square puzzles.

6. 2	1	3
   3

7. \includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square

   1	2	3	4
   3	1	4	2
   2	4	1	3
   4	3	2	1

   could not be the solution to a Sudoku puzzle.
   Explain this briefly.
   3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.

   1	2	3	4

   4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.

   4	2	3	1
   		2	4
   		4	2
   2	4	1	3

   5 The number of \(9 \times 9\) Sudokus is given in line 121 .
   Without doing any calculations, explain why you would expect 9! to be a factor of this number.
   6 In the table below, \(M\) represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). \(s\) is the side length of the Sudoku grid and \(b\) is the side length of its blocks.

   Block side length, \(b\)	Sudoku, \(s \times s\)	\(M\)
   1	\(1 \times 1\)	-
   2	\(4 \times 4\)	12
   3	\(9 \times 9\)
   4	\(16 \times 16\)
   5

8. Complete the table.
9. Give a formula for \(M\) in terms of \(b\).
   7 A man is setting a Sudoku puzzle and starts with this solution.

   1	2	3	4	5	6	7	8	9
   4	5	6	8	9	7	3	1	2
   7	8	9	3	1	2	5	6	4
   2	3	1	5	6	4	8	9	7
   5	6	4	9	7	8	1	2	3
   8	9	7	1	2	3	6	4	5
   3	1	2	6	4	5	9	7	8
   6	4	5	7	8	9	2	3	1
   9	7	8	2	3	1	4	5	6

   He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
   [0pt] [2,2]

10. 1	2		4		6			9
    4			8	9			1
    	8						6
    2		1			4			7
    	6	4		7	8	1	2
    8	9			2			4
    	1		6	4		9	7
    6	4		7		9			1
    9		8	2		1	4		6

11. 1	2	3	4	5	6	7	8	9
    4	5	6	8	9	7	3	1	2
    7	8	9				5	6	4
    2	3	1	5	6	4	8	9	7
    5	6	4	9	7	8	1	2	3
    8	9	7				6	4	5
    3	1	2	6	4	5	9	7	8
    6	4	5	7	8	9	2	3	1
    9	7	8				4	5	6

12. \(\_\_\_\_\)
13. \(\_\_\_\_\)

2

1. Given that $$\frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { A } { 2 r - 1 } + \frac { B } { 2 r + 1 }$$ find the values of \(A\) and \(B\).
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
3. Find the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) differs from 0.5 by less than 0.001 .

3 Express \(\frac { x + 6 } { x \left( x ^ { 2 } + 2 \right) }\) in partial fractions.

9

1. 1. By using a counter example, show that the answer obtained by Chloe cannot be correct.
      9
      1. (ii) Explain her mistake in Step 1.
         9
   2. Write \(\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }\) as partial fractions, with constant numerators.

16

1. Given that $$\frac { 1 } { 16 - 9 x ^ { 2 } } \equiv \frac { A } { 4 - 3 x } + \frac { B } { 4 + 3 x }$$ find the values of \(A\) and \(B\) 16
2. An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-29_469_812_404_616} The container has a small hole in the bottom. Water is poured into the container at a rate of 0.16 cubic metres per minute.
   At time \(t\) minutes after the container starts to be filled, the depth of water is \(d\) metres and water leaks out at a rate of \(0.36 d ^ { 2 }\) cubic metres per minute. At time \(t\) minutes after the container starts to be filled, the volume of water in the container is \(V\) cubic metres. 16 (b) (i) Show that $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 16 - 9 V ^ { 2 } } { 100 }$$ \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-30_2493_1721_214_150} \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-31_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-36_2498_1723_213_148}

5 Express $$\frac { 5 ( x - 3 ) } { ( 2 x - 11 ) ( 4 - 3 x ) }$$ in the form $$\frac { A } { ( 2 x - 11 ) } + \frac { B } { ( 4 - 3 x ) }$$ where \(A\) and \(B\) are integers.

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by \(32 \%\) each day.
10

1. The total number of plants damaged by insects, \(x\), is modelled by $$x = A \times B ^ { t }$$ where \(A\) and \(B\) are constants and \(t\) is the number of days after the gardener first noticed the damaged plants. 10
   1. (i) Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
      10
   2. (ii) Explain why this model is not realistic in the long term.
      10
   3. A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$ 10
   4. 1. Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where \(A\) and \(B\) are positive integers to be found.
         

         10 (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

3

1. Express \(\frac { 1 } { ( x + 2 ) ( x + 3 ) }\) in partial fractions.
2. Find \(\int \frac { 1 } { ( x + 2 ) ( x + 3 ) } \mathrm { d } x\) in the form \(\ln ( \mathrm { f } ( x ) ) + c\), where \(c\) is the constant of integration and \(\mathrm { f } ( x )\) is a function to be determined.

10

1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
2. Find the particular solution for which \(y = 0\) when \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\)

5
Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.

2 In this question you must show detailed reasoning.

1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).

9. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
   (Solutions based entirely on calculator technology are not acceptable.)

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$

1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.

1. (a) Express \(\frac { 1 } { x ( 2 x - 1 ) }\) in partial fractions.

The height above ground, \(h\) metres, of a carriage on a fairground ride is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 1 } { 50 } h ( 2 h - 1 ) \cos \left( \frac { t } { 10 } \right)$$ where \(t\) seconds is the time after the start of the ride.
Given that, at the start of the ride, the carriage is 2.5 m above ground,
(b) solve the differential equation to show that, according to the model, $$h = \frac { 5 } { 10 - 8 \mathrm { e } ^ { k \sin \left( \frac { t } { 10 } \right) } }$$ where \(k\) is a constant to be found.
(c) Hence find, according to the model, the time taken for the carriage to reach its maximum height above ground for the 3rd time.
Give your answer to the nearest second.
(Solutions relying entirely on calculator technology are not acceptable.)

8

1. Given that \(\frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \equiv \frac { A } { 2 x + 1 } + \frac { B } { x + 3 }\), find the values of the constants \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 2 } \frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \mathrm { d } x = \ln 15\).

13 By first factorising completely \(x ^ { 3 } + x ^ { 2 } - 5 x + 3\), find \(\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x\).

5 Express \(\frac { 7 - x } { ( x - 1 ) ( x + 2 ) }\) in partial fractions.