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NSW HSC 2003 : MATHEMATICS (GENERAL)

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2003 H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N General Mathematics General Instructions Reading time 5 minutes 1 Working time 2 hours 2 Write using black or blue pen Calculators may be used A formulae sheet is provided at the back of this paper Total marks 100 Section I Pages 2 11 22 marks Attempt Questions 1 22 Allow about 30 minutes for this section Section II Pages 12 23 78 marks Attempt Questions 23 28 Allow about 2 hours for this section 372 Section I 22 marks Attempt Questions 1 22 Allow about 30 minutes for this section Use the multiple-choice answer sheet. Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely. Sample: 2+4= (A) 2 A (B) 6 (C) 8 B C (D) 9 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. correct A B C 2 D 1 A number of men and women were surveyed at a railway station. They were asked whether or not they were travelling to work. The table shows the results. Men Women Going to work 64 60 Not going to work 24 42 How many men were surveyed? (A) 64 (B) 88 (C) 124 (D) 190 2 Simplify 3y3 12y2 . (A) 4y (B) 4 y (C) 1 4y y (D) 3 4 Dora works for $9.60 per hour for eight hours each day on Thursday and Friday. On Saturday she works for six hours at time-and-a-half. How much does Dora earn in total for Thursday, Friday and Saturday? (A) $192.00 (B) $211.20 (C) $240.00 (D) $316.80 4 If d = h , what is the value of d, correct to one decimal place, when h = 28? 5 (A) 1.1 (B) 2.4 (C) 2.8 (D) 5.6 3 5 Jim bought a new car at the beginning of 2001 for $40 000. At the end of 2001 the value of the car had depreciated by 30%. In 2002 the value of the car depreciated by 25% of the value it had at the end of 2001. What was the value of the car at the end of 2002? (A) $18 000 (B) $19 600 (C) $21 000 (D) $22 000 6 From 5 boys and 7 girls, two children will be chosen at random to work together on a project. Which of the following probability trees could be used to determine the probability of choosing a boy and a girl? (A) (B) 5 12 5 12 7 12 4 12 7 12 7 12 girl girl boy 7 11 girl 5 11 boy girl boy girl girl (C) boy 6 11 boy 7 12 5 12 4 11 girl 5 12 5 12 girl 6 12 boy boy 7 12 boy (D) 5 11 5 12 7 12 boy boy 6 11 girl 4 11 5 12 boy 7 12 girl 7 11 girl 4 boy girl 7 At the same time, Alex and Bryan start riding towards each other along a road. The graph shows their distances (in kilometres) from town after t minutes. Distance (km) 20 Alex 16 12 Bryan 8 4 0 10 20 30 Time (minutes) 40 t How many kilometres has Alex travelled when they meet? (A) 4 (B) 8 (C) 12 (D) 20 8 Which scatterplot shows a low (weak) positive correlation? (A) (B) (C) (D) 5 9 A swimming pool has a length of 6 m and a width of 5 m. The depth of the pool is 1 m at one end and 3.5 m at the other end, as shown in the diagram. 6m 5m 3.5 m 1m NOT TO SCALE 6.5 m What is the volume of this pool in cubic metres? (A) 67.5 (B) (C) 109.375 (D) 10 105 113.75 Kathmandu is 30 west of Perth. Using the longitude difference, what is the time in Kathmandu when it is noon in Perth? (A) 10:00 am (B) (C) 12:30 pm (D) 11 11:30 am 2:00 pm The council wants to put new grass on a park which is in the shape of an ellipse. If grass costs $7.50 per square metre, what is the total cost to the nearest dollar? NOT TO SCALE 80 m 120 m (A) $7 540 (B) $30 159 (C) $56 549 (D) $226 195 6 Use the following information to answer Questions 12 and 13. Joy asked the students in her class how many brothers they had. The answers were recorded in a frequency table as follows. Number of brothers 0 1 2 3 4 12 Frequency 5 10 3 1 1 What is the mean number of brothers? (A) 1.15 (B) 2 (C) 2.3 (D) 4 13 One of the students is chosen at random. What is the probability that this student has at least two brothers? (A) 0.10 (B) 0.15 (C) 0.25 (D) 0.75 14 x NOT TO SCALE 10 60 7 Which equation should be used to obtain the value of x in this triangle? (A) 7 x = sin 60 sin 10 (B) x 2 = 10 2 + 72 2 10 7 cos 60 (C) cos 60 = (D) x 2 = 10 2 72 x 2 + 10 2 72 2 10 7 7 15 Kylie and Danny work in a music store. The weekly wage $W of an employee at the store is given by W = 0.75 n + 50, where n is the number of CDs the employee sells. If Kylie sells two more CDs than Danny in one week, how much more will she earn? (A) $0.75 (B) $1.50 (C) $50.75 (D) $51.50 16 Pauline calculates the present value (N) of an annuity. The interest rate is 4% per annum, compounded monthly. In five years the future value will be $100 000. Which calculation will result in the correct answer? (A) (B) N= (C) N= (D) 17 N= N= 100 000 (1 + 0.04)5 100 000 (1 + 0.04 12)5 100 000 (1 + 0.04)60 100 000 (1 + 0.04 12)60 If an electrical current varies inversely with resistance, what is the effect on the current when the resistance is doubled? (A) The current is doubled. (B) The current is exactly the same. (C) The current is halved. (D) The current is squared. 8 18 George measures the breadth and length of a rectangle to the nearest centimetre. His answers are 10 cm and 15 cm. Between what lower and upper values must the actual area of the rectangle lie? (A) 10 15 cm2 and 11 16 cm2 (upper) (B) 10 15 cm2 (lower) and 10.5 15.5 cm2 (upper) (C) 9.5 14.5 cm2 (lower) and 10 15 cm2 (upper) (D) 9.5 14.5 cm2 19 (lower) (lower) and 10.5 15.5 cm2 (upper) The roof of the Sydney Opera House is covered with 1.056 million tiles. If each tile covers 175 cm2, what area is covered by the tiles? (A) 184.8 m2 (B) 18 480 m2 (C) 184 800 m2 (D) 1 848 000 m2 20 Iliana buys several items at the supermarket. The docket for her purchases is shown. What is the amount of GST included in the total? (A) $1.15 (B) $1.27 (C) $1.55 (D) $1.71 9 21 The graph below shows the numbers of the two major types of cameras, analog and digital, sold in Australia in the years 1999 2002. Camera market size by type (in thousands) 1400 1200 1000 997 800 600 1070 1088 1000 400 200 0 35 1999 98 2000 Analog 200 2001 350 2002 Digital In 2001, what percentage of the cameras sold were digital cameras? (To the nearest per cent.) (A) 16% (B) 19% (C) 23% (D) 84% 10 Charlie surveyed 12 school friends to find out their preferences for chocolate. They were asked to indicate their liking for milk chocolate on the following scale. Dislike Like strongly moderately a little a little moderately strongly 0 1 2 3 4 5 They were also asked to do this for dark chocolate. Charlie displayed the results in a spreadsheet and graph as shown below. A B C D E F 1 Chocolate Survey Results 2 3 4 Milk chocolate Dark chocolate 5 4 2 6 3 3 7 4 3 8 2 4 9 5 1 10 4 1 11 2 0 12 1 3 13 5 4 14 4 4 15 3 1 16 5 Chocolate Survey 5 2 Dark chocolate 22 4 3 2 1 0 1 2 3 Milk chocolate 4 5 17 Charlie assumes that these 12 students are representative of the 600 students at the school. What is Charlie s estimate of the number of students in the school who like milk chocolate but dislike dark chocolate? (A) 50 (B) 200 (C) 250 (D) 450 11 Section II 78 marks Attempt Questions 23 28 Allow about 2 hours for this section Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. All necessary working should be shown in every question. Marks Question 23 (13 marks) Use a SEPARATE writing booklet. (a) Keryn is designing a new watering system for the shrubs in her garden. She knows that each shrub needs 1.2 litres of water per day. To minimise evaporation, Keryn designs a system to drip water into a tube that takes the water to the roots. (i) 1 (ii) Keryn pays 94.22 cents per kilolitre for water. Calculate the total cost of watering 13 shrubs for one week. 1 (iii) Keryn knows that 1 mL = 15 drops. Find the number of drops that one shrub needs daily. 1 (iv) (b) What is the number of litres of water required daily for 13 shrubs? How many drops per minute are required for one shrub if the system is in use for 10 hours per day? 1 In her garden, Keryn has a birdbath in the shape of a hemisphere (half a sphere). The internal diameter is 45 cm. 2 NOT TO SCALE What is the internal surface area of this birdbath? (Give your answer to the nearest square centimetre.) Question 23 continues on page 13 12 Marks Question 23 (continued) (c) A river has a cross-section as shown below, with measurements in metres. 3 4 3 5 3 5 2 3 4 NOT TO SCALE Calculate the area of this cross-section using Simpson s rule. (d) Peta is designing an eight-cylinder racing engine. Each cylinder has a bore (diameter) of 10.0 cm and a stroke (height) of 7.8 cm, as shown below. 10 cm NOT TO SCALE 7.8 cm (i) Calculate the volume of each cylinder, correct to the nearest cubic centimetre. 2 (ii) The capacity of the engine is the sum of the capacities of the eight cylinders. Does Peta s engine meet the racing requirement that the capacity should be under 5 litres? Justify your answer with a mathematical calculation. 3 End of Question 23 13 Marks Question 24 (13 marks) Use a SEPARATE writing booklet. (a) Minh invests $24 000 at an interest rate of 4.75% per annum, compounded monthly. What is the value of the investment after 3 years? (b) 2 Vicki earns a taxable income of $58 624 from her job with an insurance company. She pays $14 410.80 tax on this income. (i) Vicki has a second job which pays $900 gross income per month. 1 What is Vicki s total annual taxable income from both jobs, assuming that she has no allowable tax deductions? (ii) Use the tax table below to calculate the total tax payable on her income from both jobs. Taxable income 2 Tax payable $0 $6 000 NIL $6 001 $22 000 18 cents for each $1 over $6 000 $22 001 $55 000 $2 880 plus 30 cents for each $1 over $22 000 $55 001 $66 000 $12 780 plus 45 cents for each $1 over $55 000 $66 001 and over $17 730 plus 48 cents for each $1 over $66 000 (iii) Show that Vicki s monthly net income from her second job is $486.44. 3 (iv) Vicki plans to take a holiday in two years time which she estimates will cost $12 000. At the end of each month, Vicki invests the net income from her second job in an account which pays 4% per annum, compounded monthly. 2 Will she have enough in this account, immediately after the twenty-fourth payment, to pay for her holiday? Justify your answer with calculations. Question 24 continues on page 15 14 Marks Question 24 (continued) (c) Zo plans to borrow money to buy a car and considers the following repayment guide: Fortnightly car loan repayment guide Amount borrowed ($) 10 000 10 500 11 000 11 500 12 000 12 500 13 000 13 500 14 000 14 500 15 000 15 500 16 000 1 year Length of loan 2 years 3 years ($) 410 430 451 471 492 512 532 553 573 594 614 635 655 ($) 217 228 239 249 260 271 282 293 303 314 325 336 347 ($) 153 161 168 176 183 191 199 206 214 221 229 237 244 Zo wishes to borrow $15 500 and pay back the loan in fortnightly instalments over two years. What is the flat rate of interest per annum on this loan? End of Question 24 15 3 Marks Question 25 (13 marks) Use a SEPARATE writing booklet. A census was conducted of the 33 171 households in Sunnytown. Each household was asked to indicate the number of cars registered to that household. The results are summarised in the following table. Number of cars Frequency 0 2 735 1 12 305 2 13 918 3 3 980 4 233 Total 33 171 (ii) (1) Determine the mode number of cars in a household. 1 (2) Explain what is meant by the mode number of cars in a household. (i) 1 Sunnytown Council issued a free parking sticker for each car registered to a household in Sunnytown. 2 How many parking stickers were issued? (iii) The council represented the results of the census in a sector graph. 1 What is the angle in the sector representing the households with no cars? Give your answer to the nearest degree. (iv) Visitors to Sunnytown Airport have to pay for parking. The following step graph shows the cost of parking for t hours. 30 Parking fee ($) (a) 25 20 15 10 5 013 6 12 Length of time in carpark (hours) t What is the cost for a car that is parked one evening from 6 pm to 8:30 pm? Question 25 continues on page 17 16 1 Marks Question 25 (continued) (b) Equal large numbers of primary and secondary school students in a city were surveyed about their method of travel to school. The results are summarised in the relative frequency column graphs below. Method of travel to school Primary school Secondary school 40 Percentage 50 40 30 20 30 20 Other Train Bicycle Car Bus Other Train Bicycle 0 Car 0 Bus 10 Walk 10 Walk Percentage 50 (i) 2 (ii) Suggest a possible reason for ONE of these differences. 1 (iii) (c) Describe TWO differences in the method of travel between these primary school and secondary school students. There were 25 000 primary school students surveyed. How many of these students travelled to school by bus? 1 Results for an aptitude test are given as z-scores. In this test, Hardev gains a z-score of 1. (i) Interpret Hardev s score with reference to the mean and standard deviation of the test. 2 (ii) The scores for the test are normally distributed. What proportion of people sitting the test obtain a higher score than Hardev? 1 End of Question 25 17 Marks Question 26 (13 marks) Use a SEPARATE writing booklet. (a) At a World Cup rugby match, the stadium was filled to capacity for the entire game. At the end of the game, people left the stadium at a constant rate. The graph shows the number of people (N) in the stadium t minutes after the end of the game. N 60 000 0 30 t (minutes) The equation of the line is of the form N = a bt, where a and b are constants. (i) Write down the value of a, and give an explanation of its meaning. 2 (ii) (1) Calculate the value of b. 1 (2) What does the value of b represent in this situation? 1 (iii) Rearrange the formula N = a bt to make t the subject. 2 (iv) How long did it take 10 000 people to leave the stadium? 1 (v) Copy or trace the graph of N against t shown above into your answer booklet. 2 Suppose that 15 minutes after the end of the game, several of the exits had been closed, reducing the rate at which people left. On the same axes, carefully draw another graph of N against t that could represent this new situation. Your new graph should show N from t = 15 until all the people had left the stadium. Question 26 continues on page 19 18 Marks Question 26 (continued) (b) NOT TO SCALE T N River 50 m A B In the diagram above, the following measurements are given: TAB = 30 . B is 50 m due east of A. The bearing of T from B is 020 . Copy or trace ABT into your answer booklet. (i) Explain why ABT is 110 . 1 (ii) Calculate the distance BT (to the nearest metre). 3 End of Question 26 19 Marks Question 27 (13 marks) Use a SEPARATE writing booklet. (a) A celebrity mathematician, Karl, arrives in Sydney for one of his frequent visits. Karl is known to stay at one of three Sydney hotels. Hotel X is his favourite, and he stays there on 50% of his visits to Sydney. When he does not stay at Hotel X, he is equally likely to stay at Hotels Y or Z. (i) What is the probability that he will stay at Hotel Z ? (ii) 1 On his first morning in Sydney, Karl always flips a coin to decide if he will have a cold breakfast or a hot breakfast. If the coin comes up heads he has a cold breakfast. If the coin comes up tails he has a hot breakfast. (1) List all the possible combinations of hotel and breakfast choices. (2) Give a brief reason why these combinations are not all equally likely. 1 (3) Calculate the probability that Karl stays at Hotel Z and has a cold breakfast. (b) 2 1 Two unbiased dice are thrown. The dice each have six faces. The faces are numbered 1, 2, 3, 4, 5 and 6. (i) What is the probability that neither shows a 6? 1 (ii) Dale plays a game with these dice. There is no entry fee. 3 When the dice are thrown: Dale wins $20 if both dice show a 6. He wins $2 if there is only one 6. He loses $2 if neither shows a 6. What is his financial expectation from this game? Question 27 continues on page 21 20 Marks Question 27 (continued) (c) The diagram shows a radio mast AD with two of its supporting wires, BE and CE. The point B is half-way between A and C. D C NOT TO SCALE B A 65 15 m E (i) Calculate the height AB in metres, correct to one decimal place. 2 (ii) Calculate the distance CE in metres, correct to one decimal place. 2 End of Question 27 21 Marks Question 28 (13 marks) Use a SEPARATE writing booklet. (a) Sandra is on holiday in Mexico and she plans to buy some silver jewellery. Various silver pendants are on sale. The cost varies directly with the square of the length of the pendant. A pendant of length 30 mm costs 130 Mexican pesos. 2 How much does a pendant of length 40 mm cost? (Answer correct to the nearest Mexican peso.) (b) In 2002, the population of Mexico was approximately 103 400 000. (i) 2 (ii) (c) The growth rate of Mexico s population is estimated to be 1.57% per annum. If y represents the estimated number of people in Mexico at a time x years after 2002, write a formula relating x and y in the form y = b(a x ). Use appropriate values for a and b in your formula. Using your formula, or otherwise, find an estimate for the size of Mexico s population two years after 2002. Express your answer to the nearest thousand. 2 (i) While Sandra is on holiday she visits countries where the Fahrenheit temperature scale is used. She knows that the correct way to convert from Celsius to Fahrenheit is: 1 Multiply the Celsius temperature by 1.8, then add 32. Find the value of A in the following table. Celsius 5 41 15 59 A (ii) Fahrenheit 77 Peter uses the following method to approximate the conversion from Celsius to Fahrenheit: Add 12 to the Celsius temperature, then double your result. Express Peter s rule as an algebraic equation. Use C for the Celsius temperature and F for the approximate Fahrenheit temperature. Question 28 continues on page 23 22 2 Marks Question 28 (continued) (d) While Sandra is on holidays she buys a digital camera for $800. Using the straight-line method of depreciation, the salvage value, S, of the camera will be zero in four years time, as shown on the graph below. S 800 0 2 4 n (time in years) Using the declining balance method of depreciation, at a rate of R% per annum, the salvage value is represented by the dotted curve on the graph below. S 800 NOT TO SCALE 0 2 4 n (time in years) Both methods give the same salvage value after two years. Find the value of R. End of paper 23 4 BLANK PAGE 24 Board of Studies NSW 2003 2003 HIGHER SCHOOL CERTIFIC ATE EXAMINATION General Mathematics FORMULAE SHEET Area of an annulus ( A = R2 r 2 Surface area ) Sphere R = radius of outer circle r = radius of inner circle Area of an ellipse A = ab A = 4 r 2 Closed cylinder A = 2 rh + 2 r 2 r = radius h = perpendicular height Volume A= V = r 2h V= 1 Ah 3 Sphere r2 360 1 V = r 2h 3 Pyramid Area of a sector Cone Cylinder a = length of semi-major axis b = length of semi-minor axis V= 43 r 3 = number of degrees in central angle r = radius h = perpendicular height A = area of base Arc length of a circle l= 2 r 360 = number of degrees in central angle Sine rule a b c = = sin A sin B sin C Simpson s rule for area approximation A ( h d + 4 dm + dl 3f ) h = distance between successive measurements Area of a triangle 1 A = ab sin C 2 d f = first measurement dm = middle measurement Cosine rule dl = last measurement c 2 = a 2 + b 2 2 ab cos C or cos C = 373 25 a2 + b2 c2 2 ab FORMULAE SHEET Simple interest Declining balance formula for depreciation I = Prn S = V0 (1 r )n P = initial quantity r = percentage interest rate per period, expressed as a decimal n = number of periods S = salvage value of asset after n periods r = percentage interest rate per period, expressed as a decimal Mean of a sample Compound interest A = P(1 + r ) A P n r = = = = x= n final balance initial quantity number of compounding periods percentage interest rate per compounding period, expressed as a decimal Future value ( A) of an annuity n (1 + r ) 1 A = M r x= x x n f = = = = x n fx f mean individual score number of scores frequency Formula for a z -score x x z= s M = contribution per period, paid at the end of the period s = standard deviation Present value ( N ) of an annuity Gradient of a straight line (1 + r )n 1 N = M n r (1 + r ) m= or vertical change in position horizontal change in position Gradient intercept form of a straight line A N= (1 + r )n y = mx + b m = gradient b = y-intercept Straight-line formula for depreciation S = V0 Dn Probability of an event S = salvage value of asset after n periods V0 = purchase price of the asset The probability of an event where outcomes are equally likely is given by: D = amount of depreciation apportioned per period n = number of periods P(event) = 26 number of favourable outcomes total number of outcomes

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Additional Info : New South Wales Higher School Certificate General Mathematics 2003
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