1.02y Partial fractions: decompose rational functions

1. A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, \(N\), in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study.
Given that

• there were 1000 bacteria in this population at the start of the study
• it took exactly 5 hours from the start of the study for this population to double
  1. find a complete equation for the model.
  2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study.
Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of \(T\).

11. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$

1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \quad x > 3$$
2. Prove that \(\mathrm { f } ( x )\) is a decreasing function.

1. (a) Express \(\frac { 3 } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.

When chemical \(A\) and chemical \(B\) are mixed, oxygen is produced.
A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. The total volume of oxygen produced, \(V \mathrm {~m} ^ { 3 } , t\) hours after the chemicals were mixed, is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 3 V } { ( 2 t - 1 ) ( t + 1 ) } \quad V \geqslant 0 \quad t \geqslant k$$ where \(k\) is a constant.
Given that exactly 2 hours after the chemicals were mixed, a total volume of \(3 \mathrm {~m} ^ { 3 }\) of oxygen had been produced,
(b) solve the differential equation to show that $$V = \frac { 3 ( 2 t - 1 ) } { ( t + 1 ) }$$ The scientist noticed that

• there was a time delay between the chemicals being mixed and oxygen being produced
• there was a limit to the total volume of oxygen produced

Deduce from the model
(c) (i) the time delay giving your answer in minutes,
(ii) the limit giving your answer in \(\mathrm { m } ^ { 3 }\)

1. (a) Express \(\frac { 1 } { V ( 25 - V ) }\) in partial fractions.

The volume, \(V\) microlitres, of a plant cell \(t\) hours after the plant is watered is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1 } { 10 } V ( 25 - V )$$ The plant cell has an initial volume of 20 microlitres.
(b) Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.
(c) Show that $$V = \frac { A } { \mathrm { e } ^ { - k t } + B }$$ where \(A , B\) and \(k\) are constants to be found. The model predicts that there is an upper limit, \(L\) microlitres, on the volume of the plant cell.
(d) Find the value of \(L\), giving a reason for your answer.

1. (a) Express \(\frac { 1 } { P ( 11 - 2 P ) }\) in partial fractions.

A population of meerkats is being studied.
The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began,
(b) determine the time taken, in years, for this population of meerkats to double,
(c) show that $$P = \frac { A } { B + C \mathrm { e } ^ { - \frac { 1 } { 2 } t } }$$ where \(A , B\) and \(C\) are integers to be found.

2 Express \(\frac { 13 - x } { ( x - 3 ) ( x + 2 ) }\) in partial fractions.

2 Express \(\frac { 5 x + 1 } { x ^ { 2 } - x - 12 }\) in partial fractions.

17

1. Express \(\frac { \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
2. Express \(y\) in terms of \(x\) given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \text { and } y = 16 \text { when } x = 3 .$$ \section*{END OF QUESTION PAPER}

10

1. Express \(\frac { 1 } { ( 4 x + 1 ) ( x + 1 ) }\) in partial fractions.
2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation \(\frac { d y } { d x } = \frac { y } { ( 4 x + 1 ) ( x + 1 ) }\),
   for \(x > - \frac { 1 } { 4 }\).
   Show by integration that \(\mathrm { y } = \mathrm { A } \left( \frac { 4 \mathrm { x } + 1 } { \mathrm { x } + 1 } \right) ^ { \mathrm { B } }\) where \(A\) and \(B\) are constants to be determined.

5 Express \(\frac { 12 x ^ { 3 } } { ( 2 x + 1 ) \left( 2 x ^ { 2 } + 1 \right) }\) using partial fractions.

5. $$f ( x ) = 3 - \frac { x - 1 } { x - 3 } + \frac { x + 11 } { 2 x ^ { 2 } - 5 x - 3 } , \quad x \in \mathbb { R } , \quad x < - 1$$

1. Show that $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 } , \quad x \in \mathbb { R } , \quad x < 1$$

1. Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
2. Find an expression for the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
3. Solve the equation \(\mathrm { f } ( x ) = 2\).

3

1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
   (2 marks)

1

1. Express \(\frac { 2 x + 3 } { 4 x ^ { 2 } - 1 }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { 2 x + 1 }\), where \(A\) and \(B\) are integers. (3 marks)
2. Express \(\frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 }\) in the form \(C x + \frac { D ( 2 x + 3 ) } { 4 x ^ { 2 } - 1 }\), where \(C\) and \(D\) are integers.
   (3 marks)
3. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers.
   (5 marks)

2 It is given that \(\mathrm { f } ( x ) = \frac { 7 x - 1 } { ( 1 + 3 x ) ( 3 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 3 - x } + \frac { B } { 1 + 3 x }\), where \(A\) and \(B\) are integers.
   (3 marks)
2. 1. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) in the form \(a + b x + c x ^ { 2 }\), where \(a\), \(b\) and \(c\) are rational numbers.
      (7 marks)
   2. State why the binomial expansion cannot be expected to give a good approximation to \(\mathrm { f } ( x )\) at \(x = 0.4\).
      (1 mark)

8

1. Express \(\frac { 1 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) in the form \(\frac { A } { 3 - 2 x } + \frac { B } { 1 - x } + \frac { C } { ( 1 - x ) ^ { 2 } }\).
   (4 marks)
2. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \sqrt { y } } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$$ where \(y = 0\) when \(x = 0\), expressing your answer in the form $$y ^ { p } = q \ln [ \mathrm { f } ( x ) ] + \frac { x } { 1 - x }$$ where \(p\) and \(q\) are constants.

1

1. 1. Express \(\frac { 5 x - 6 } { x ( x - 3 ) }\) in the form \(\frac { A } { x } + \frac { B } { x - 3 }\).
      (2 marks)
   2. Find \(\int \frac { 5 x - 6 } { x ( x - 3 ) } \mathrm { d } x\).
      (2 marks)
2. 1. Given that $$4 x ^ { 3 } + 5 x - 2 = ( 2 x + 1 ) \left( 2 x ^ { 2 } + p x + q \right) + r$$ find the values of the constants \(p , q\) and \(r\).
   2. Find \(\int \frac { 4 x ^ { 3 } + 5 x - 2 } { 2 x + 1 } \mathrm {~d} x\).

1

1. 1. Express \(\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }\) in the form \(\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }\), where \(A\) and \(B\) are integers.
      (3 marks)
   2. Hence show that \(\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2\), where \(p\) is rational.
      (4 marks)
2. 1. Given that \(\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\) can be written as \(C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }\), find the value of \(C\).
      (1 mark)
   2. Hence find the exact value of the area of the region bounded by the curve \(y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 0\). You may assume that \(y > 0\) when \(- 1 \leqslant x \leqslant 0\).

8

1. Express \(\frac { 16 x } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }\) in the form \(\frac { A } { 1 - 3 x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }\).
2. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 x \mathrm { e } ^ { 2 y } } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$$ where \(y = 0\) when \(x = 0\).
   Give your answer in the form \(\mathrm { f } ( y ) = \mathrm { g } ( x )\).
   [0pt] [7 marks]

1 It is given that \(\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }\) can be expressed as \(\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }\), where \(A\) and \(B\) are integers.

1. Find the values of \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5\), where \(k\) is a rational number.
   [0pt] [6 marks]

1. The function \(f\) is given by

$$f ( x ) = \frac { 3 ( x + 1 ) } { ( x + 2 ) ( x - 1 ) } , x \in \mathbb { R } , x \neq - 2 , x \neq 1$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, or otherwise, prove that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all values of \(x\) in the domain.

7. Given that $$\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) } \equiv \frac { A } { 1 - 2 x } + \frac { B } { 2 + x }$$

1. find the values of the constants \(A\) and \(B\).
2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of \(\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) }\), for \(| x | < \frac { 1 } { 2 }\).