1. Topic Past Papers –Further Integration
Advanced Higher
Topic Past Papers –Further Integration
Unit 2 Outcome 2
2001
A3. Find the value of
(5)
A5. (a) Obtain partial fractions for
(b) Use the result of (a) to find
(2)
(4)
A10. A chemical plant food loses effectiveness at a rate proportional to the amount present in the soil.
The amount M grams of plant food effective after t days satisfies the differential equation
Find the general solution for M in terms of t where the initial amount of plant food
is M0 grams.
Find the value of k if, after 30 days, only half the initial amount of plant food is effective.
What percentage of the original amount of plant food is effective after 35 days?
The plant food has to be renewed when its effectiveness falls below 25%. Is the manufacturer
of the plant food justified in calling its product “sixty day super food”?
(3)
(3)
(2)
(2)
B2. Find the general solution of the following differential equation
(4)
2002
A5. Use integration by parts to evaluate
(5)
A9. Functions x(t) and y(t) satisfy
When t = 0, x = 1 and y = 2.
(a) Express in terms of x and y and hence obtain y as a function of x.
(b) Deduce that and obtain x as a function of t for t ≥ 0.
(5)
(5)
2003
A10.
Define
a) Use Integration by parts to obtain the value of
b) Similarly, show that
c) Evaluate
3 marks
4 marks
3 marks
Lanark Grammar Mathematics Department
Mrs Leck

2. Topic Past Papers – Further Integration
Advanced Higher
Topic Past Papers – Further Integration
Unit 2 Outcome 2
A11.
The volume V(t) of a cell at time t changes according to the law
Show that for some constant C.
Given that V(0)=5, show that
Obtain the limiting value of V(t) as
4 marks
3 marks
2 marks
2004 5.
Express in partial fractions.
Evaluate
2 marks
4 marks
15.
a) A mathematical biologist believes that the differential equation models a process.
Find the general solution of the differential equation.
Given that y = 2 when x = 1, find the particular solution, expressing y in terms of x.
b) The biologist subsequently decides that a better model is given by the equation
Given that y = 2 when x = 1, obtain y in terms of x.
5 marks
2 marks
4 marks
2005
13.
Express in partial fractions.
Obtain a formula for I(k) where expressing it in the form where
a and b depend on k.
Write down an expression for and obtain the value of
4 marks
4 marks
2 marks
15.
a) Given
b) If, in general, where g(x) > 0, show that
stating the value of k.
Hence or otherwise, find
c) Use integration by parts and the result of (b) to evaluate
2 marks
3 marks
4 marks
Lanark Grammar Mathematics Department
Mrs Leck

3. Topic Past Papers – Further Integration
Advanced Higher
Topic Past Papers – Further Integration
Unit 2 Outcome 2
2006
17.
(a) Show that
(b) By writing and using integration by parts, show that
(c) Show that
(d) Hence, using the above results, show that
1 mark
3 marks
3 marks
3 marks
2007 4.
Express in partial fractions.
Given that determine values for the integers m and n.
3 marks
3 marks
14.
A garden centre advertises young plants to be used as hedging.
After planting, the growth G metres (ie the increase in height) after t years is modelled by
the differential equation
where k is a constant and G = 0 when t = 0.
(a) Express G in terms of t and k.
(b) Given that a plant grows 0.6metres by the end of 5 years, find the value of k correct to
3 decimal places.
(c) On the plant labels it states that the expected growth after 10 years is approximately 1 metre.
Is this claim justified?
(d) Given that the initial height of the plants was 0.3m, what is the likely long term height
of the plants?
4 marks
2 marks
2 marks
2 marks
2008 4.
Express in partial fractions.
Hence evaluate
3 marks
3 marks 7.
Use integration by parts to obtain
5 marks
Lanark Grammar Mathematics Department
Mrs Leck

4. Topic Past Papers – Further Integration
Advanced Higher
Topic Past Papers – Further Integration
Unit 2 Outcome 2
2009 3.
Given that
and y = 0 when x = 1, find y in terms of x.
4 marks 9.
Use integration by parts to obtain the exact value of
5 marks
2010 3.
(b) Integrate x2lnx with respect to x.
4 marks
2011 9.
Given that y > 1 and x > 1, obtain the general solution of the differential equation
Expressing your answer in the form y = f(x).
5 marks
11.
(b) Find
4 marks
16.
Define for n  1.
Use integration by parts to show that
Find the values of A and B for which
and hence show that
(c) Hence obtain the exact value of
3 marks
5 marks
3 marks
Lanark Grammar Mathematics Department
Mrs Leck

5. Topic Past Papers – Further Integration
Advanced Higher
Topic Past Papers – Further Integration
Unit 2 Outcome 2
2001 A3 A5
A10. a) b) k =
c) 44.5% d) Yes B2
2002
A ln2 – 1 =
A9. a) y = 2x b)
2003 A3
A10 a) b) Proof c) 6 – 16e-1 =
(3)
(4)
(3)
A11 a) Proof b) Proof c)
2004 5
(2)
(4) 15
(5)
(2)
(4)
2005
2. x = 0 or x = 4
13.
(4)
(4)
(2) 15
(1)
(2)
(3)
(4)
2006
(2)
(3)
2007
(3)
(3)
(5)
(5)
14.a) b) c) d) Limit = 3.6 metres
(4)
(2)
(2)
Lanark Grammar Mathematics Department
Mrs Leck

6. Topic Past Papers – Further Integration
Advanced Higher
Topic Past Papers – Further Integration
Unit 2 Outcome 2
2008
4.a) b) 4 ln 3 = 4.39
5.a) b) 5y + 7x = 12
2009
2010
3. a) b)
13.
2011 3. 9.
11.
Lanark Grammar Mathematics Department
Mrs Leck