Recurrence Rel. Past Papers Unit 1 Outcome 4

Transcription

1 PSf Recurrence Rel. Past Papers Unit 1 utcome 4 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. A sequence is defined b the recurrence relation u n+1 = 1 4 u n + 8 with u 0 = 32. Evaluate u 2. A. 10 B. 12 C. 16 D. 32 Ke utcome Grade Facilit Disc. Calculator Content Source B 1.4 C NC A11 HSN 137 PSf hsn.uk.net Page 1

2 PSf 2. A sequence is defined b the recurrence relation u n+1 = 2 5 u n + 6 with u 0 = 10. What is the limit of the sequence? A. 10 B. 2 5 C D. 30 Ke utcome Grade Facilit Disc. Calculator Content Source A 1.4 C NC A13 HSN 088 PSf [END F MULTIPLE CHICE QUESTINS] hsn.uk.net Page 2

3 PSf Written Questions A sequence is defined b the recurrence relation u n = 0 9u n 1 + 2, u 1 = 3. (a) Calculate the value of u 2. 1 (b) What is the smallest value of n for which u n > 10? 1 (c) Find the limit of this sequence as n. 2 hsn.uk.net Page 3

4 PSf 5. A sequence is defined b the recurrence relation u n+1 = 0 3u n + 5 with first term u 1. (a) Eplain wh this sequence has a limit as n tends to infinit. 1 (b) Find the eact value of this limit Two sequences are generated b the recurrence relations u n+1 = au n + 10 and v n+1 = a 2 v n The two sequences approach the same limit as n. Determine the value of a and evaluate the limit. 5 Part Marks Level Calc. Content Answer U1 C4 4 C NC A13 a = 3 5, L = P1 Q5 1 A/B NC A12 1 ss: know how to find limit 2 pd: process 3 pd: process 4 ic: interpret coeff. of u n 5 pd: process 1 L = al + 10 or L = a 2 L + 16 or L = b 1 a 16 or L = 1 a 2 2 L = 10 1 a a or 16 1 a a 2 16a + 6 = 0 5 a = 3 5 and L = 25 hsn.uk.net Page 4

5 PSf hsn.uk.net Page 5

6 PSf 9. hsn.uk.net Page 6

7 PSf 10. hsn.uk.net Page 7

8 PSf 11. hsn.uk.net Page 8

9 PSf 12. hsn.uk.net Page 9

10 PSf 13. hsn.uk.net Page 10

11 PSf 14. hsn.uk.net Page 11

12 PSf 15. n the first da of March, a bank loans a man 2500 at a fied rate of interest of 1.5% per month. This interest is added on the last da of each month and is calculated on the amount due on the first da of the month. He agrees to make repaments on the first da of each subsequent month. Each repament is 300 ecept for the smaller final amount which will pa off the loan. (a) The amount that he owes at the start of each month is taken to be the amount still owing just after the monthl repament has been made. Let u n and u n+1 represent the amounts that he owes at the start of two successive months. Write down a recurrence relation involving u n+1 and u n. 2 (b) Find the date and the amount of the final pament. 4 Part Marks Level Calc. Content Answer U1 C4 (a) 2 C CN A10, A14 u n+1 = 1 015u n 300, u 0 = P2 Q3 (b) 4 C CR A11, A14 1 December, ic: interpret 1 5% 2 ic: state the recurrence relation 3 ss: use recurrence relation 4 pd: process 5 ic: start final date 6 pd: process final pament stated or implied b the start of (b) 2 u n+1 = 1 015u n 300 and initial value (e.g. u 0 = 2500) stated or implied b the start of (b) 3 u 1 i.e u 2 and u 3 i.e , for December pament hsn.uk.net Page 12

13 PSf 16. A man decides to plant a number of fast-growing trees as a boundar between his propert and the propert of his net door neighbour. He has been warned, however, b the local garden centre that, during an ear, the trees are epected to increase in height b 0 5 metres. In response to this warning he decides to trim 20% off the height of the trees at the start of an ear. (a) If he adopts the 20% pruning polic, to what height will he epect the trees to grow in the long run? 3 (b) His neighbour is concerned that the trees are growing at an alarming rate and wants assurances that the trees will grow no taller than 2 metres. What is the minimum percentage that the trees will need to be trimmed each ear so as to meet this condition. 3 Part Marks Level Calc. Content Answer U1 C4 (a) 3 C CN A13, A metres 2002 P2 Q4 (b) 3 C CN A12, A13 trim 25% 1 ic: interpret the deca factor 2 ss: strateg for limit 3 pd: process limit 4 ss: reverse strateg for limit 5 pd: process 6 ic: interpret scale factor stated or implied 2 e.g. l = 0 8l or l = < 0 8 < 1 so l = 2 5 metres 4 2 = 2m m = trim 25% hsn.uk.net Page 13

14 PSf 17. hsn.uk.net Page 14

15 PSf 18. [END F WRITTEN QUESTINS] hsn.uk.net Page 15