2 = 41( ) = 8897 (A1)

Transcription

1 . Find the sum of the arithmetic series (Total 4 marks) R (n )0 = 47 0(n ) = 400 n = 4 (A) 4 S 4 = ((7) + 40(0)) = 4(7 + 00) = 8897 (A) OR 4 S 4 = (7 + 47) 4 = (44) = 8897 (A) (C4) [4] 4. Find the sum of the infinite geometric series (Total 4 marks) u R. S = (A) r = (A) 5 = (A) (C4) 5 [4]

2 5. The Acme insurance company sells two savings plans, Plan A and Plan B. For Plan A, an investor starts with an initial deposit of $000 and increases this by $80 each month, so that in the second month, the deposit is $080, the next month it is $60 and so on. For Plan B, the investor again starts with $000 and each month deposits 6% more than the previous month. (a) Write down the amount of money invested under Plan B in the second and third months. () Give your answers to parts (b) and (c) correct to the nearest dollar. (b) (c) Find the amount of the th deposit for each Plan. Find the total amount of money invested during the first months (i) under Plan A; () under Plan B. R. (a) Plan A: 000, 080, Plan B: 000, 000(.06), 000(.06) nd month: $060, rd month: $.60 (A)(A) () (Total 0 marks) (b) For Plan A, T = a + d = (80) = $880 (A) For Plan B, T = 000(.06) = $898 (to the nearest dollar) (A) 4 (c) (i) For Plan A, S = [000 + (80)] = 6(880) = $780 (to the nearest dollar) (A) 000(.06 ) For Plan B, S =.06 = $6870 (to the nearest dollar) (A) 4 [0] 6. Each day a runner trains for a 0 km race. On the first day she runs 000 m, and then increases the distance by 50 m on each subsequent day. (a) (b) On which day does she run a distance of 0 km in training? What is the total distance she will have run in training by the end of that day? Give your answer exactly. (Total 4 marks)

3 R. (a) a = 000, a n = (n )50 = n = + = She runs 0 km on the 7th day. (A) 7 (b) S 7 = ( ) She has run a total of 0.5 km (A) [4] 7. Portable telephones are first sold in the country Cellmania in 990. During 990, the number of units sold is 60. In 99, the number of units sold is 40 and in 99, the number of units sold is 60. In 99 it was noticed that the annual sales formed a geometric sequence with first term 60, the nd and rd terms being 40 and 60 respectively. (a) What is the common ratio of this sequence? () Assume that this trend in sales continues. (b) How many units will be sold during 00? (c) In what year does the number of units sold first exceed 5000? Between 990 and 99, the total number of units sold is 760. (d) What is the total number of units sold between 990 and 00? () During this period, the total population of Cellmania remains approximately (e) R. (a) r = Use this information to suggest a reason why the geometric growth in sales would not continue. () (Total marks) =.5 (A) (b) 00 is the th year. u = 60(.5) = 0759 (Accept 0760 or 0800.) (A) (c) 5000 = 60(.5) n 5000 = (.5) n log = (n )log.5 60

4 n = 5000 log 60 log.5 n = th year 999 = 8.49 (A) (A) OR Using a gdc with u = 60, u k+ = uk, u 9 = 400, u 0 = 650 (M) 999 (G) 4 d).5 S = 60.5 = 6958 (Accept 6960 or 6000.) (A) (e) Nearly everyone would have bought a portable telephone so there would be fewer people left wanting to buy one. OR (R) Sales would saturate. (R) [] 8. The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square. A Q B P R (a) (i) Show that PQ = cm. D S C Find the area of PQRS. The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. 4

5 A Q B W X P R Z Y D S C (b) (i) Write down the area of the third square, WXYZ. Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence. The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. (c) (i) Find the area of the th square. Calculate the sum of the areas of all the squares. (Total 0 marks) R. (a) (i) PQ = = AP AQ = 4 = cm (A)(AG) Area of PQRS = ( )( ) = 8 cm (A) (b) (i) Side of third square = = 4 = cm Area of third square = 4 cm (A) st nd 6 8 nd rd Geometric progression, r = 6 8 (A) (c) (i) u = u r 0 = = 04 = ( = = 0.056, sf) (A) 64 5

6 S = u = r 6 = (A) 4 [0]. (a) Consider the geometric sequence, 6,, 4,. (i) Write down the common ratio. Find the 5 th term. Consider the sequence x, x +, x + 8,. (b) When x = 5, the sequence is geometric. (i) Write down the first three terms. Find the common ratio. () (c) (d) Find the other value of x for which the sequence is geometric. For this value of x, find (i) the common ratio; the sum of the infinite sequence. (Total marks) R. (a) (i) r = A N u 5 = () 4 (A) = 495 (accept 4900) A N (b) (i), 6, 8 A N r = A N (c) Setting up equation (or a sketch) M x x8 x x x + x + = x + x 4 x = 5 x = 5 or x = 5 (or correct sketch with relevant information) A (A) x = 5 A N Notes: If trial and error is used, work must be documented with several trials shown. 6

7 (d) (i) r = Award full marks for a correct answer with this approach. If the work is not documented, award N for a correct answer. A N For attempting to use infinite sum formula for a GP 8 S = S = 6 A N [] 7