Mathematics Standard 2

Transcription

1 Western Mathematics Exams Mathematics Standard SOLUTIONS

2 Multiple Choice Worked Solutions No Working Answer The shape is a sector and we want its area, so from the formula sheet the C formula is. The radius, r = 4 with an angle of 60 o, so the calculation is Unordered selections of 3 from 0 = = 0 A 3 C 4 A goes to D() and E(7) B goes to A(0) and C to B(8) D D goes to E (9) and E goes to C(4) 5 3ab 3a(a 4b) a =3ab 6a +ab a =5ab 7a D

3 6 tan WTK = WTK = tan = 36 Bearing = = 34 7 Looking down the interest column, the interest amount increased in June. Int Rate May = $,4.95 $48, = (4 dec places) Int Rate June = $,34.43 $47,83.4 = (4 dec places) Increase in annual interest rate = = 0.5% ( dec place) C B 8 One variable decreases as the other increases, so the correlation will be negative. The data is grouped close to the line, so the value will be close to -. So 0.95 is the most likely. 9 Quarterly interest rate =0. 4 =0.03 After first year the investment = 450 (.03) 4 = $ After she added $ 80 value = $ $ 80 = $ 9.99 After second year the investment = $ 9.99 (.03) 4 = $ Area of the cloth = 30 = 900 cm. An area of Area of cloth = 900 =0.09 m 0000 Amount absorbed = = 0.45 L = 450 ml Under the declining balance method, each succeeding years price is a percentage of the previous year s price, which means the value goes down by gradually lesser amounts each year. Graph B indicates this. Graph C is a straight line depreciation and A and D are erratic amounts. A C B B 80% as a z score = =. A 3 Both C and D have the correct shape for an exponential with a negative in the index, but it should have a y intercept of, so only D is correct. D 3

4 4 A 5 Network B is not spanning as it does not include vertex L Networks A and C include all of the vertices, but also include a cycle or a loop. Only network D, is a spanning tree. D 4

5 09 Trial Higher School Certificate Examination Mathematics Standard Name Teacher Section I Multiple Choice Answer Sheet Allow about 5 minutes for this section Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely. Sample: + 4 = (A) (B) 6 (C) 8 (D) 9 A B C D If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer. A B C D If you change your mind and have crossed out what you consider to be the correct answer, then indicate the correct answer by writing the word correct and drawing an arrow as follows. A B C D. A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D

6 Solution Question 6 (a) The method will not provide a random sample and is likely to be biased based on the common background and interests of Barbara s friends. (b) The data is categorical and ordinal. Question 7 (i) Stamp duty = 3% of % of Marks Question 8 (a) (b) (c) P(Different) = P( same) = 9 66 = Question 9 50 km at 5 litres/00 km = 5 = 7.5 litres 50 km at 0 litres/00 km = 0.5 = 5 litres Total = = 3.5 litres Question 0 (ii) Convert to z-scores. For a diameter of 6.4 z = = For a diameter of 6.5 z = = Approximately 68% of scores have z-scores between and Percentage of z-scores less than - = =6% Approximately 95% of scores have z-scores between and Percentage of z scores more than = =.5% Total percentage rejected = = 8.5% 6

7 Question Absolute Error = half of the smallest unit (0.5 cm) = 0.5 cm Question Hourly earnings = = $68.50 Commission = = $640 Total Earnings =$ $640 = $ 3.50 Question 3 (a) Mean = 4.5 (b) Standard deviation = 5.4 Question 4 (a) Time difference = = 5 hours So Sydney is 5 hours ahead. (b) Sydney is 5 hours ahead of 6 am Wednesday 7th. 6 hours ahead of 6 am is pm Wed. 9 hours more ahead is 9 pm on Wednesday 7 th. (c) The time she leaves New York (Sydney time) is 9 pm Wednesday. Time for flight is 0 hours, so 3 hours to midnight and 7 hours after midnight so arrives at 5 pm Thursday 8th. Question 5 3 Starting at P, shortest to R is 5 and to Q is 4, shortest to S is through R 5+0 = 35 then to T is through S 35+6 = 5 and hence to U is through T = 59 So shortest time is 59 minutes going from P to R, to S, to T to U Or abbreviated to PRSTU. Question 6 The heater is 000 watts which is kw. It is run for 0 8 = 960 hours per year Power used = 960 = 90 kwh Cost at $0.30 per kwh = = $576 7

8 Question 7 (a) (b) B = R + T A = A = (35) A = 0 A 360 A = A =90 Question 8 (a) (i) 6.6% pa %= 6.6 = 0.55% per month. From table interest factor for 0.55% for years (0 months) is Future value = = $76 97 (nearest dollar) (b) (ii) 9.0% pa %= 9.0 = 0.75% per month. Let I be the required interest factor Then 500 I = $ I = = From table, in the column for 0.75% the first interest factor to exceed 00 is for 3 months (00.936), So the minimum term is 0 years and 3 months 8

9 Question 9 (a) (i) It takes 4 years (from the graph). (b) (ii) The time to triple is about.5 years, compared to 4 years to double, so it is increasing at an increasing rate, or it is increasing exponentially. Question 30 Question 3 Monthly repayment =$93.33 from table. There are 5 = 60 repayments. Amount paid = $ = $

10 Question 3 (a) A minimum spanning tree (b) Using Prims Algorithm Select C as the first vertex (arbitrary as any other could be chosen) Highlight CF as lowest weight leaving C Highlight FE as lowest weight leaving the tree. Highlight CG as lowest weight leaving the tree. Highlight ED as lowest weight leaving the tree not creating a cycle. Every vertex is now included, so we have our minimum spanning tree Length of road to re-seal = = 9 km Question 33 Dosage for child - = age of child (in years) adult dose age of child (in years) + = = 0 0 = 6 ml Question 34 area of cross section = 3 = 8.3 m volume = = 45.4 m 3 capacity = 45.4 kl Number of 50 kl tanks = = tanks could be filled. Question 35 (a) (b) A high score is an outlier if it is greater than So is an outlier. 0

11 Question 36 Measurement =.7cm from Parkes to Dubbo (answers from.5 to.7) cm = 5km Allow for different So distance =.7 x 5km = 67.5 radius (6.5 for.5 cm) answers in A = r measuring, due to printing = 67.5 differences, = 434 km (7 if using.5 ) Question 37 (a) (i) Passes through (40, 480) and the origin, so Gradient = m = = C intercept = 0 Equation is C =N (b) (ii)gradient = 8 and C intercept is 00. Plot the intercept and use gradient to obtian a second point. (c) (iii) Break even point is (5, 300) which means that when 5 widgets are sold the income equals the expenses, so if less than 5 widgets are sold, a loss is made, but for more, a profit is made. Question 38 (a) Total residents = 00 (b) Total females = 49 P( no prob given female) = 5 49

12 Question 39 (a) BAC FEMALE = 0N 7.5H 5.5M = = = ( d p) Which is more than 0.08 (b) Question 40 (a) (i) From calculator Mean = 46% and SD = 9%. (b) (ii) The evening half hour class has an SD of 5% so has the greatest variability. Question 4 (a) (b) The circle around B passes through 5 edges, so the degree is 5 Incorrect It is not disconnected as every vertex can be reached from any other vertex. It has only one loop, not, which is at B with weight 6. Correct It is a directed network

13 Question 4 (a) (i) 6% p.a. interest = 0.5% per month. So r = Ten years is 0 months, so N = 0 From table the factor is So Monthly repayment = Monthly repayment = = $7.63 (b) (ii) Total repaid (ten years) = = $ Nine years is 08 months, so N = 08 From table the factor is So Monthly repayment = Monthly repayment = = $ Total repaid (nine years) = = $ Amount of interest saved =$ $ = $ Question 43 (a) (b) 3

14 Question 44 (a) Only 5 can flow into V so the maximum outflow is 5, but the pipe has a capacity of 30 so the excess flow is 30 5 = 5 Gigalitres. (b) The values of the various cuts are shown before reducing the flow to the actual amounts. So the maximum flow is the minimum cut of 40 Gigalitres per hour If the cuts were done after reducing the flows, all cuts would be 40. (c) The theoretical maximum flow from the diagram is 70 through the last 3 pipes. If UV were increased by 5 to 30 it would bring VY to maximum capacity To allow UV to receive the extra 5 and to bring UW to its capacity of 5, SU would need to be increased by 0 to 80 To allow TX to reach its capacity ST could be increased by 0 to 90 This achieves a maximum flow of 70 4

15 Question 45 (a) (b) AOB = 95 Area = ab sin O = sin 95 = 66 = m = 6. ha (c) BOC = 0 95 = 5 BC = cos 5 = 8797 BC = 536 m (d) 5

16 Question 46 (a) Plot point E on the graph. (b) A correlation approaching means that as one quantity increases, so does the other in a close linear relationship. (c) The line drawn on the graph. (d) Draw a right triangle Rise = = Answers could be Run = = 0 slightly different depending on the line of best fit. Y intercept = 0. Question 47 (a) (b) Amount owing on st July = = 300. So PV = 300 There are 3 days in July so n = 3 r = = (c) Interest = = $0.58 (or $0.30) 6