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CARESA EDUCATION SERVICES TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION PRACTICE PAPER 1 GENERAL MATHEMATICS 2 UNIT Reading time – 5 minutes Working time –2 ½ hours DIRECTIONS TO CANDIDATES Section 1 22 marks Attempt Questions 1 – 22 Allow about 30 minutes for this section Section II 78 marks Attempt questions 23 – 28 Allow about 2 hours for this section * * * * A formula sheet is provided at the end of the paper. Board approved calculators may be used. Answer each question of section II on a separate page. You may ask for extra paper if you need it. Section I 1. Lyn was rolling a normal die. She rolled a 5 followed by a 3. How would you describe the probability of her rolling an odd number on the next roll of the die? (a) Likely (b) certain (c) Unlikely (d) even 2. Sylvie had a spherical candle mould whose volume was exactly one litre. (1000 cm3) What was the radius of the mould in cm? (a) 2356.2 (b) 238.7 (c) 13.3 (d) 6.2 3. Jack marked an "x" on the ground and used a sextant to determine that the angle of elevation from his "x" to the top of a tall gum tree was 35o. He then used a tape measure to determine that the "x" was 42m from the base of the tree. What was the height of the tree in metres? (a) 24.1 (b) 29.4 (c) 34.4 (d) 60.0 4. Raj scored 60, 70, 90, 80 and 70 out of 100 in his first five tests. He needs to maintain a mean of 75 or above to remain in the advanced class. What must he score in his sixth test to have a mean of 75? (a) 74 (b) 75 (c) 80 (d) 82 5. A web designer earns $500 per week plus $150 commission for every website she designs. What is the minimum number of websites she must design in 4 weeks to earn at least $5000? (a) 8 (b) 20 (c) 34 (d) 80 6. Luc tossed a coin 1000 times. What is the number of times he could expect to toss heads? (a) approximately 500 (b) exactly 500 (c) significantly more than 500 (d) significantly less than 500 7. What equation represents the relationship between x and y in the following table? x y -4 -1 -1 0 2 1 5 2 (a) y = 3x - 1 (b) y = 3x + 1 (c) x = 3y - 1 (d) x = 3y + 1 8. Two hundred students sat for an English exam. The results were normally distributed with a mean of 60 and a standard deviation of 10. How many students scored a mark of 50 or more? (a) 84 (b) 151 (c) 168 (d) 185 9. Ted bought a triangular block of land with sides of length 40, 50 and 60 metres respectively. What is the area of his land in m2? (a) 1000 sin 45o25' (b) 1200 sin 45o25' (c) 1500 sin 45o25' (d) 60 000 sin 45o25' 10. Sharyn drove her car at 70 km/h. What was her speed in m/s? (a) 1.2 (b) 19.4 (c) 252.0 (d) 1166.7 11. Simplify 2(5-3x)-3(2x-5) (a) -5 (b) 25 (c) 25 + 12x (d) 25 - 12x 12. Chris takes out a loan of $12500 to buy a car. Interest is charged at a flat rate of 11% p.a.and Chris repays the loan with monthly repayments over 4 years. What is the total amount that Chris will repay? (a) $375 (b) $5500 (c) $12500 (d) $18000 13. The following table shows the income tax payable by Australian residents for the 2006-2007 financial year. Taxable Income ($) 1 - 6000 6001 - 30 000 30 001 - 75 000 75 001 - 150 000 150 001 and over Tax payable on taxable income. nil 15c for each $1 over $6000 $3600 + 30c for each $1 over $30 000 $17 100 + 40c for each $1 over $75 000 $47 000 + 45c for each $1 over $150 000 Lois earned $45 600 during the 2006 - 2007 financial year. How much income tax did she have to pay? (a) $4680 (b) $8280 (c) $13 680 (d) $15 600 14. Timbuktu is situated at longitude 17oN and latitude 3oW. Sydney is situated at longitude 33oS and latitude 151oE. Raj phoned his friend in Timbuktu at 9.00 a.m. Tuesday, Sydney time. What would be the time in Timbuktu? (a) 10.44 p.m. Monday (b) 1.16 p.m. Tuesday (c) 5.20 a.m. Tuesday (d) 12.20 p.m. Tuesday 15. The Moon is 384 392 km from the Earth. What is this distance expressed in scientific notation to 3 significant figures? (a) 384 000 km (b) 384 x 103 km (c) 3.843 x 105 km (d) 3.84 x 105 km 16. Ted constructed a concrete pipe. The pipe had an external diameter of 40 cm and an internal diameter of 32 cm. The pipe was 2.0 m long. What is the formula for the volume of concrete in the pipe? (a) 200 π (40 - 32)2 (b) 200 π (402- 322) (c) 200 π (20- 16)2 (d) 200 π (202- 162) 17. Chris bought a block of land that was bound by two parallel, straight fences, 200m apart, a straight road and a river. The fences were perpendicular to the road. He measured the distance between the road and the river at 50m intervals and used Simpson's Rule in two steps to calculate the area. A table of his measurements is shown below. Distance from first fence (m) Distance between road and river (m) What is the area of his land in m2? (a) 4000 (b) 4833 (c) 8833 (d) 9000 18. Sharyn used the capture-recapture technique to estimate the number of possums in a region of bushland. She caught, tagged and released 20 possums. The following night she caught 40 possums and found that 5 of them had been tagged. What is the best estimate for the number of possums in the region? (a) 40 (b) 160 (c) 320 (d) 800 19. Lyn and Sylvie and three of their friends went to dinner at their favourite restaurant. They sat in random order at a circular table. What is the probability that Lyn and Sylvie sat next to each other? (a) 1/5 (b) 2/5 (c) 1/4 (d) 1/2 20. The volume of a given mass of gas varies inversely as the pressure. When the pressure acting on a gas is two atmospheres its volume is 600 cm3. What will be the volume of the same gas if the pressure is increased to 3 atmospheres? (a) 300 cm3 0 30 50 40 100 50 150 50 200 40 (b) 400 cm3 (c) 900 cm3 (d) 1200 cm3 21. Jack measured the diameter of his birthday cake and found it was 30 cm. He cut himself a slice of cake in the form of a sector of a circle. He used a protractor to show that the angle of the sector was 20o. What was the area to the nearest cm3 of the top of Jack's slice of cake? (a) 13 (b) 39 (c) 50 (d) 157 22. Luc paid $200 per month into an annuity. Interest was calculated at 4% p.a. paid monthly. What would be the value of Luc's annuity after 10 years? (to nearest dollar) (a) 2401 (b) 19754 (c) 29450 (d) 548 313 Section II 23. (a) Jack bought a computer for $3100. He paid 10% deposit and paid the balance at $144.15 per month for 2 years. (i) What was the total price Jack paid for his computer? (ii) How much interest did Jack pay? (iii) Calculate the annual rate of simple interest charged for buying on terms. (b) Lyn paid $250 per month into a superannuation fund that earned 4% p.a. interest, compounded monthly. (i) What was her fund worth at maturity, 20 years later? (ii) Sylvie wanted to invest a lump sum now that would be equal to Lyn's superannuation payment in 20 years time. If the interest rate is 4% p.a. compounded monthly, how much should Sylvie invest? (c) Ted earned $870 per week. However Ted's pay cheque each week was less than this because income tax had been deducted. A table showing the tax scales for annual incomes is shown below. Taxable Income ($) 1 - 6000 6001 - 30 000 30 001 - 75 000 75 001 - 150 000 150 001 and over Tax payable on taxable income. nil 15c for each $1 over $6000 $3600 + 30c for each $1 over $30 000 $17 100 + 40c for each $1 over $75 000 $47 000 + 45c for each $1 over $150 000 Calculate the value of Ted's weekly pay cheque. 24. (a) Raj mapped out a rectangle 20m long and 5m wide in his backyard. Within this rectangular area he constructed an elliptical pool so that the ellipse touched the sides of the rectangle at their 4 midpoints. The pool was 30cm deep. (i) Calculate the area of the ellipse. (ii) Calculate the volume of the pool in m3 (iii) Given that a litre of water occupies 1000 cm3, calculate the volume of water in litres that would be required to fill the pool. (iv) If the area within the rectangle around the pool is to be paved, what area of paving stones would be required? (b) Luc cut out a triangular sheet of paper such that it had sides of length 10cm, 15cm and 20cm. (i) Calculate the angle bounded by the 10cm and 15cm sides. (ii) Calculate the area of the triangle. (c) Ted was in a lighthouse and observed a launch 3 kilometres away at a bearing of 120oT and a yacht 6 kilometres away at a bearing of 240oT. What is the bearing of the launch from the yacht? 25. (a) Show that the point (1, 1) lies on the line 2x + y = 3. (b) Simplify: 8m2n ÷ 4n3 x 2m2n (c) When current flows through an electrical component of constant resistance it is found that the power is directly proportional to the square of the current. (i) Write an equation linking the power (P) with the current (C) and the resistance (R). (ii) If power is measured in watts, current in amps and resistance in ohms, calculate the current that would produce 800 watts of power when flowing through a 50 ohm resistance. (d) The radius of the base of a cone is given by r = √(3V/πh) where V = volume of cone h = height of cone Calculate the radius of the base of a cone that is 20cm high and has a volume of 200 cm3. (e) Sharyn is organising a social function for her classmates. The cost of the hall is $200 and refreshments cost $15 per person. Sharyn charges her classmates $20 per ticket for the function. (i) Write an equation connecting the total cost (C) with the cost of the hall (H), refreshments (R) and the number of people attending(n). (ii) How many people have to attend for Sharyn to break even? (iii) What is the profit or loss if 53 people attend the function? 26. (a) Thirty students enter a classroom in single file. Calculate the number of possibilities for the order in which the students can enter the classroom and express it in scientific notation correct to 3 significant figures. (b) Lois bought two tickets in a raffle where 500 tickets were sold. There are 10 prizes, each of a double movie pass. What is the probability that Lois will win at least one of the prizes? (c) A group of ten friends formed a Christmas club. They contributed some money each week and at Christmas had a raffle in which each of the friends was given a ticket to allocate the ten prizes. First prize was $100, second prize $90 and so on with tenth prize being $10. Raj said that he had just as much chance of winning the $100 as winning the $10. Comment on the accuracy of Raj's statement. (d) Ted's sock drawer contained 12 identical black socks and 8 identical brown socks. Ted went to his sock drawer in the dark and chose two socks. What is the probability that he chose (i) two black socks? (ii) two brown socks? (iii) a matching pair? (iv) one black and one brown sock? (e) Jack plays a game in which 16 cards are placed face down on a table. They consist of the jack, queen, king and ace of hearts, diamonds, clubs and spades. Jack pays $10 per game to choose a card. If he chooses an ace he gets $20, if he chooses a king he gets his $10 back and if he chooses a queen or jack he loses. Calculate Jack's financial expectation from playing 100 games. 27. (a) In a General Math’s class test the scores out of 50 were: 23, 29, 32, 43, 28, 35, 28, 42, 18, 48, 36, 33, 27, 38, 20 (i) Calculate the mean of these scores correct to one decimal place. (ii) Calculate the standard deviation of these scores correct to one decimal place. (iii) Sharyn scored the highest mark in the class. What was her z-score? (iv) Chris was absent for the test. If his score was recorded as zero, what effect will this have on the mean and standard deviation? (b) Chris scored 68 in his English test in which the mean was 76 and the z-score was 8. He scored 63 in his Math’s test in which the mean was 65 and the z-score was 5. He then said that he did better in Math’s than he did in English. Comment on the accuracy of his statement. (c) 200 students sat for an exam. The results were normally distributed with a mean of 65 and a z-score of 15. (i) What was the median mark for the exam? (ii) What was the most common score in the exam? (iii) How many students scored a mark of 50 or more? (d) The marks of five students are listed in the table below. Name Tom Dick Harry George Charlie English 60 85 72 68 69 Math’s 70 83 79 73 70 Science 75 78 83 71 70 (i) Name two students whose marks show a positive correlation. (ii) Name two students whose marks show a negative correlation. 28. (a) What is the name of the imaginary line that circles the Earth at 0o latitude? (b) Sydney is located at (33oS, 151oE) Valparaiso (Chile) is located at (33oS, 72oW) (i) What is the time in Valparaiso when it is noon on Friday, Sydney time? (ii) Harry calculated the distance between Sydney and Valparaiso by the following method: Angle between Sydney and Valparaiso = 151o + 72o = 223o 223 x 60 = 13380 nautical miles. What is wrong with Harry’s method of calculation? (c) Sharyn borrowed $300 000 to buy a house. She is paying it off monthly at an interest rate of 7.5% p.a. adjusted monthly. The payments for the first 3 months are shown in the table below. No. of months (N) 1 2 3 4 Principal (P) 300 000 299 775.00 299 548.59 Interest (I) 1 875.00 1873.59 1872.18 P+I 301 875.00 301 648.59 301 421.77 P+I–R R = repayment 299 775.00 299 548.59 299 320.77 Complete the next line of the table. (d) Luc has a 2.5m ladder that he leans against a wall. The point where the ladder touches the wall is 2.0m above the floor. (i) How far from the bottom of the ladder from the wall? (ii) What is the angle between the top of the ladder and the wall? ANSWERS Q.1. = D Q.2. = C Q.3. = B Q.4. = C Q.5. = B Q.6. = B Q.7. = C Q.8. = C Q.9. = C Q.10. = B Q.11. = D Q.12. = D Q.13. = B Q.14. = A Q.15. = D Q.16. = D Q.17. = C Q.18. = B Q.19. = D Q.20. = B Q.21. = B Q.22. = C Section B Q.23. (a) (i) Deposit = $310 Payments = $144.15 x 24 = $3459.60 Total price = $310 + $3459.60 = $3769.60 (ii) Interest = $3769.60 - $3100 = $669.60 (iii) I = Prn 669.60 = (3100 -310) x r x 2 r = 669.60/(2790 x 2) = 0.12 = 12% (1 r ) n 1 (b) (i) A M r 240 1.0033333 1 250 0.00333333 = $91 693.66 (ii) A = p(1 + r)n 91 693.66 = P(1.00333333)240 P = 91 693 66 / 1.003333333240 = $41 255.50 (c) $870 p.w. = 870 x 52 = $45 240 p.a. Tax on $45 240 = 3600 +(45 240 – 30 000) x 0.3 =$8172 $8172 ÷ 52 = $157.15 p.w. $870 - $157.15 = $712.85 24. (a) (i) For ellipse: major axis = 20m semimajor axis = 10m minor axis = 5m semiminor axis = 2.5m area of ellipse = ab = x 10 x 2.5 = 25 = 78.54m2 (ii) Volume = surface area x depth = 78.54 x 0.3 = 23.56 m3 (iii) 1m3 = 100 x 100 x 100 = 106 cm3 1m3 = 106 / 103 = 1000 litres Volume = 23.56 x 1000 litres = 23560 litres (iv) Area of rectangle = 20 x 5 = 100m2 Area of paving = rectangle – ellipse = 100 – 78.4 = 21.46 m2 (b) (i) Angle bounded by 10 & 15 cm sides is opposite 20 cm side Cos A = b2 + c2 – a2 / 2bc = (102 + 152 – 202) / (2 x 10 x 15) = -0.25 A = cos-1 -0.25 = 104o29’ (ii) Area = ½ ab sin C = ½ x 10 x 15 x sin 104o29’ = 72.62 cm (c) (i) Angle between yacht and launch = 240 – 120 = 120o Distance between yacht and launch found by cosine rule d2 = 32 + 62 – 2 x 3 x 6 cos 120o d2 = 63 d = 7.94m (ii) sin = 3 sin 120o / 7.94 Sin = 0.3272 = 19o6’ Bearing = 60o + 19o6’ = 79o6’ 25. (a) 2x + y = 3 substitute (1, 1) 2 + 1 = 3 satisfies So (1, 1) lies on the line 8m2n ÷ 4n3 x 2m2n = 8m2n x ¼ n3 x 2m2n = 4m4 / n (i) P = C2R (ii) 800 = C2 x 50 C2 = 800 / 50 = 16 C=4 (b) (c) (d) r = √(3V/h) = √(3 x 200 / x 20) =√(30 / ) = √9.549 = 3.09cm (i) C = 200 + 15n (ii) 20n = 200 + 15n 5n = 200 (iii) income = 53 x 20 = $1060 Expenditure = 200 + 53 x 15 = $995 Profit = 1060 – 995 = $65 n = 40 (e) 26. (a) Number of ways = 30! = 2.65 x 1032 (b) (i) Pevent number _ of _ favourable _ outcomes total _ number _ of _ outcomes = 2/500 = 1/250 (ii) Probability that Lois does not win a prize on first draw = 498/500 Probability that Lois does not win a prize on second draw = 497/499 Probability that Lois does not win a prize on third draw = 496/498 : : Probability that Lois does not win a prize on the tenth draw = 489/491 Probability that Lois does not win a prize = 498x497 x496x495x494x493x492x491x490x489 500x499x498x497 x496x495x494x493x492x491 490x 489 = = 0.960 500x 499 Probability that Lois wins at least one prize = 1-0.96 = 0.04 (c) Raj is correct. One person wins first prize One person wins last prize (d) (i) 12/20 x 11/19 = 33/95 (ii) 8/20 x 7/19 = 14/95 (iii) 33/95 + 14/95 = 47/95 (iv) 95/95 – 47/95 = 48/95 Probability of winning first = 1/10 Probability of winning last = 1/10 (e) Financial expectation per game = outcomes x probability of outcomes Card Ace King Queen Jack Probability 0.25 0.25 0.25 0.25 Return $20 $10 Zero zero Probability x return $5 $2.50 Zero zero Expected return per game = $5 + $2.50 = $7.50 Cost per game = $10 Expected loss per game = $2.50 Financial expectation per 100 games = $2.50 x 100 = $250 loss. 27. (a) (i) mean = 32 (from calculator) (ii) standard deviation = 8.27 = 8.3 (1 d.p.) (from calculator) (iii) Sharyn’s score = 48 (48 – 32) = 16 above mean z-score = 16/8.3 = 1.93 = 1.9 (1 d.p.) (iv) mean will be lower standard deviation will be higher English: 76 – 68 = 8 = 1 standard deviation below mean. Math’s: 65 – 63 = 2 2/5 = 0.4 standard deviation below mean. The statement is accurate since, despite the lower absolute mark, Chris performed better relative to his peers. For normal distribution: mean = mode = median (i) median = 65 (ii) most common score = 65 (iii) 32% score more than 1 s.d. from mean 16% score 1 s.d. or more below the mean. 84% score 50 or more 84% of 200 = 168 students (i) Tom & Harry (b) (c) (d) (ii) Tom & Dick or Dick & Harry 28. (a) (b) Equator (i) 223o x 4 minutes = 892 minutes = 14 hours 52 minutes Since Valparaiso is west of Sydney it is behind Sydney time Noon Fri. – 14 hours 52 minutes = 9.08 p.m. Thurs. (ii) Since Sydney and Valparaiso both lie on 33oS latitude they do not lie on the same great circle. The radius of the circle 33os is not the radius of the Earth and mence 1 minute is NOT equal to one nautical mile. (c) No. of months (N) 1 2 3 4 (d) P+I–R R = repayment 299 775.00 299 548.59 299 320.77 299 091.52 Principal (P) 300 000 299 775.00 299 548.59 299 320.77 Interest (I) 1 875.00 1873.59 1872.18 1870.75 P+I 301 875.00 301 648.59 301 421.77 301 191.52 (i) d2 = 2.52 – 22 = 6.25 – 4 = 2.25 d = 1.5 (ii) sin = 1.5/2.5 = 0.6 = 36o52’