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

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

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

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

Topic Past Papers – Further Integration Advanced Higher Topic Past Papers – Further Integration Unit 2 Outcome 2 2001 A3 A5 A10. a) b) k = -0.231 c) 44.5% d) Yes B2 2002 A5. 2ln2 – 1 = 0.3863 A9. a) y = 2x b) 2003 A3 A10 a) b) Proof c) 6 – 16e-1 = 0.1139 (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

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