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NSW HSC 2005 : MATHEMATICS (GENERAL)

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2005 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 24 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. 2+4= Sample: (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 1 B C What is the mean of the set of scores? 3, 4, 5, 6, 6, 8, 8, 8, 15 (A) 6 (B) 7 (C) 8 (D) 9 2 What is the value of a b , if a = 240 and b = 56? 4 (A) 4 (B) 46 (C) 226 (D) 736 2 D 3 Four radio stations reported the probability of rain as shown in the table. Radio station Probability of rain 2AT 0.53 2BW 17% 2CZ 13 25 2DL 0.6 Which radio station reported the highest probability of rain? (A) 2AT (B) 2BW (C) 2CZ (D) 2DL 4 The diagram is a scale drawing of a butterfly. Wingspan 1 cm SCALE What is the actual wingspan of the butterfly? (A) 2.5 cm (B) 3 cm (C) 15 cm (D) 18.75 cm 3 5 Frankie Toby 80 20 m NOT TO SCALE 30 Sarah Which formula should be used to calculate the distance between Toby and Frankie? (A) (B) c2 = a2 + b2 (C) A= (D) 6 a b = sin A sin B c 2 = a 2 + b 2 2 ab cos C 1 ab sin C 2 Janet s gross income last year was $60 000. She had allowable tax deductions of $5000. Janet paid 1.5% of her taxable income for the Medicare levy. How much was Janet s Medicare levy? (A) $750 (B) $825 (C) $900 (D) $975 4 7 Simplify 2m2 3mp2 . (A) 5m2 p2 (B) 5m3p2 (C) 6m2 p2 (D) 6m3p2 8 If tan = 85, what is the value of , correct to the nearest minute? (A) 11 25' (B) 11 26' (C) 89 19' (D) 89 20' A set of data is represented by the cumulative frequency histogram and ogive. Cumulative frequency 9 800 600 400 200 0 25 35 45 55 65 Score 75 85 What is the best approximation for the interquartile range for this set of data? (A) 25 (B) 30 (C) 35 (D) 40 5 10 The table is used to calculate monthly loan repayments. Monthly loan repayments (in dollars) per $1000 borrowed Interest rate % pa 5 years 10 years 15 years 20 years 5% 18.87 10.61 7.91 6.60 6% 19.33 11.10 8.44 7.16 7% 19.80 11.61 8.99 7.75 8% 20.28 12.13 9.56 8.36 9% 20.76 12.67 10.14 9.00 Samantha has borrowed $70 000 at 8% per annum for 15 years. What is her monthly loan repayment? (A) $143.40 (B) $669.20 (C) $8 030.40 (D) $10 038.00 11 The diagram shows a spinner. 2 4 7 7 9 1 The arrow is spun and will stop in one of the six sections. What is the probability that the arrow will stop in a section containing a number greater than 4? (A) 2 5 (B) 2 3 (C) 1 3 (D) 1 2 6 12 The shaded region represents a block of land bounded on one side by a road. road 23 m 15 m 40 m NOT TO SCALE 19 m What is the approximate area of the block of land, using Simpson s rule? (A) 680 m2 (B) 760 m2 (C) 840 m2 (D) 1360 m2 13 Last year, Helen bought 150 shares at $2.00 per share. They are now worth $2.50 per share. Helen receives a dividend of $0.10 per share. What is the dividend yield? (A) 4% (B) 20% (C) $15 (D) $75 7 14 Using the formula d = 5t3 2, Marcia tried to find the value of t when d = 137. Here is her solution. She has made one mistake. d = 5t 3 2 137 = 5t 3 2 . . . . . . . . . . Line A 135 = 5t 3 . . . . . . . . . . Line B 27 = t 3 . . . . . . . . . . Line C t=3 . . . . . . . . . . Line D Which line does NOT follow correctly from the previous line? (A) Line A (B) Line B (C) Line C (D) Line D 15 A car bought for $50 000 is depreciated using the declining balance method. Which graph best represents the salvage value of the car over time? (A) Salvage value (B) $50 000 Salvage value $50 000 Time (C) Time Salvage value (D) $50 000 Salvage value $50 000 Time Time 8 16 On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table. Male viewers Female viewers Contestant 1 1372 3915 Contestant 2 2054 3269 One male viewer was selected at random from all of the male viewers. What is the probability that he voted for Contestant 1? (A) (B) 1372 5287 (C) 1372 3426 (D) 17 1372 10 610 1372 2054 The total cost, $C, of a school excursion is given by C = 2n + 5, where n is the number of students. If three extra students go on the excursion, by how much does the total cost increase? (A) $6 (B) $11 (C) $15 (D) $16 9 18 A model yacht has two triangular sails. These triangles are similar to each other. Some dimensions of the sails, in centimetres, are shown on the diagram. 15 5 NOT TO SCALE 4 3 What is the total area of both sails? (A) 24 cm2 (B) 27 cm2 (C) 60 cm2 (D) 97 cm2 19 The location of Town A is 25 N 45 E. The location of Town B is 10 N 105 E. Which of the following is true? (Ignore time zones.) (A) Town A is four hours behind Town B. (B) Town A is four hours ahead of Town B. (C) Town A is one hour behind Town B. (D) Town A is one hour ahead of Town B. 20 Dave s school has computer security codes made up of four digits (eg 0773). Juanita s school has computer security codes made up of five digits (eg 30568). How many more codes are available at Juanita s school than at Dave s school? (A) 10 (B) 50 (C) 90 000 (D) 100 000 10 21 Yousef used the capture-recapture technique to estimate the number of kangaroos living in a particular area. He caught, tagged and released 50 kangaroos. Later, he caught 200 kangaroos at random from the same area. He found that 5 of these 200 kangaroos had been tagged. What is the correct estimate for the total number of kangaroos living in this area, using the capture-recapture technique? (A) 245 (B) 250 (C) 2000 (D) 10 000 22 Two groups of people were surveyed about their weekly wages. The results are shown in the box-and-whisker plots. People under 21 years People 21 years and older 200 250 300 350 Weekly wages ($) 400 450 Which of the following statements is true for the people surveyed? (A) The same percentage of people in each group earned more than $325 per week. (B) Approximately 75% of people under 21 years earned less than $350 per week. (C) Approximately 75% of people 21 years and older earned more than $350 per week. (D) Approximately 50% of people in each group earned between $325 and $350 per week. 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) There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table. Friend Number of tickets Danielle 45 Khalid 5 Nancy 10 Shani 14 Herman 26 Total (i) 100 Justine claims that each of her friends is equally likely to win first prize. 1 Give a reason why Justine s statement is NOT correct. (ii) What is the probability that first prize is NOT won by Khalid or Herman? Question 23 continues on page 13 12 2 Marks Question 23 (continued) (b) A clay brick is made in the shape of a rectangular prism with dimensions as shown. 9 cm NOT TO SCALE 8 cm 21 cm (i) 1 Calculate the volume of the clay brick. Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm. 9 cm NOT TO SCALE 8 cm 21 cm (ii) What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.) 3 (iii) What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.) 1 Question 23 continues on page 14 13 Marks Question 23 (continued) (c) Moheb owns five red and seven blue ties. He chooses a tie at random for himself and puts it on. He then chooses another tie at random, from the remaining ties, and gives it to his brother. (i) What is the probability that Moheb chooses a red tie for himself? 1 (ii) Copy the tree diagram into your writing booklet. 2 Complete your tree diagram by writing the correct probability on each branch. Moheb s tie Brother s tie Red Red Blue Red Blue Blue (iii) Calculate the probability that both of the ties are the same colour. End of Question 23 14 2 Marks Question 24 (13 marks) Use a SEPARATE writing booklet. (a) (i) 2 Draw a stem-and-leaf plot for the following set of scores. 21 45 29 27 19 35 23 58 34 27 (ii) 1 (iii) (b) What is the median of the set of scores? Comment on the skewness of the set of scores. 1 2A is used to calculate the dosage of Hackalot cough medicine 15 to be given to a child. The formula D = D is the dosage of Hackalot cough medicine in millilitres (mL). A is the age of the child in months. (i) If George is nine months old, what dosage of Hackalot cough medicine should he be given? 1 (ii) The correct dosage of Hackalot cough medicine for Sam is 4 mL. 3 What is the difference in the ages of Sam and George, in months? (c) Make L the subject of the equation T = 2 L2 . Question 24 continues on page 16 15 2 Marks Question 24 (continued) The sector graph shows the proportion of people, as a percentage, living in each region of Sumcity. There are 24 000 people living in the Eastern Suburbs. Proportion of people living in each region of Sumcity Southern Suburbs (30%) Northern Suburbs (45%) Eastern Suburbs (10%) (15%) Western Suburbs (i) Show that the total number of people living in Sumcity is 160 000. 1 Jake used the information above to draw a column graph. Number of people (d) (ii) 84 000 72 000 60 000 48 000 36 000 24 000 12 000 0 Northern Suburbs Southern Eastern Suburbs Suburbs Region Western Suburbs The column graph height is incorrect for one region. Identify this region and justify your answer. End of Question 24 16 2 Marks Question 25 (13 marks) Use a SEPARATE writing booklet. (a) Reece is preparing his annual budget for 2006. His expected income is: $90 every week as a swimming coach Interest earned from an investment of $5000 at a rate of 4% per annum. His planned expenses are: $30 every week on transport $12 every week on lunches $48 every month on entertainment. Reece will save his remaining income. He uses the spreadsheet below for his budget. A B C D E F G H 1 REECE S ANNUAL BUDGET FOR 2006 2 3 INCOME EXPENSES 4 $Y $4,680 Transport 5 Wages Lunches $624 $X 6 Interest on investment Entertainment $Z 7 8 9 (i) Determine the values of X , Y and Z . (Assume there are exactly 52 weeks in a year.) 3 (ii) At the beginning of 2006, Reece starts saving. 2 Will Reece have saved enough money during 2006 for a deposit of $2100 on a car if he keeps to his budget? Justify your answer with suitable calculations. Question 25 continues on page 18 17 Marks Question 25 (continued) (b) A 12 cm NOT TO SCALE 13 cm C B 5 cm (i) 1 (ii) (c) Use Pythagoras theorem to show that ABC is a right-angled triangle. Calculate the size of ABC to the nearest degree. 2 Robyn plays a game in which she randomly chooses one of these five cards. She plays the game 60 times, replacing the card after each game. Win Win Win Win nothing Lose $4 $4 $4 $0 $8 (i) How many times would she expect to win $4? 1 (ii) What is the financial expectation of the game? 2 (iii) Another card is added to the game with Win nothing $0 written on it. Robyn claims that the financial expectation will not change. 2 Do you agree? Justify your answer with suitable calculations. End of Question 25 18 Marks Question 26 (13 marks) Use a SEPARATE writing booklet. (a) 2 A printing machine worth $150 000 is bought in December 2005. In December each year, beginning in 2006, the value of the printing machine is depreciated by 10% using the declining balance method of depreciation. In which year will the depreciated value first fall below $120 000? (b) Rod is saving for a holiday. He deposits $3600 into an account at the end of every year for four years. The account pays 5% per annum interest, compounding annually. The table shows future values of an annuity of $1. Future values of an annuity of $1 End of year 1 2 3 4 5 6 7 8 1% 1.0000 2.0100 3.0301 4.0604 5.1010 6.1520 7.2135 8.2857 2% 1.0000 2.0200 3.0604 4.1216 5.2040 6.3081 7.4343 8.5830 Interest rate 3% 1.0000 2.0300 3.0909 4.1836 5.3091 6.4684 7.6625 8.8923 4% 1.0000 2.0400 3.1216 4.2465 5.4163 6.6330 7.8983 9.2142 5% 1.0000 2.0500 3.1525 4.3101 5.5256 6.8019 8.1420 9.5491 (i) 2 (ii) (c) Use the table to find the value of Rod s investment at the end of four years. How much interest does Rod earn on his investment over the four years? 2 The weights of boxes of Brekky Bicks are normally distributed. The mean is 754 grams and the standard deviation is 2 grams. (i) What is the z-score of a box of Brekky Bicks with a weight of 754 g? 1 (ii) What is the weight of a box that has a z-score of 1? 1 (iii) Brekky Bicks boxes are labelled as having a weight of 750 g. 2 What percentage of boxes will have a weight less than 750 g? Question 26 continues on page 20 19 Marks Question 26 (continued) (d) Peta borrows $28 000 from a credit union at 6% per annum compounding monthly. She will repay the money over nine years. Peta uses the formula: (1 + r )n 1 28 000 = M n r (1 + r ) to calculate her monthly repayment, M . (i) Rewrite the formula, showing the correct substitutions for r and n . 2 (ii) Calculate Peta s monthly repayment. 1 End of Question 26 20 Marks Question 27 (13 marks) Use a SEPARATE writing booklet. The area graph shows sales figures for Shoey s shoe store. Number of shoes sold (a) 30 000 25 000 Boots School shoes Business shoes 20 000 15 000 10 000 5 000 0 ary ary rch April May June July ugust mber tober mber mber nu ebru Ma A epte Oc ove ece Ja F D N S Month (i) 1 (ii) For which month does the graph indicate that the same number of school shoes and business shoes was sold? 1 (iii) (b) Approximately how many school shoes were sold in January? Identify ONE trend in this graph, and suggest a valid reason for this trend. 2 This diagram represents Earth. O is at the centre, and A and B are points on the surface. 2 A O A: 35 N 20 E B: B 8 S 20 E Calculate the distance from A to B along the great circle through A and B. Give your answer in nautical miles. (Radius of Earth is 6400 km. 1.852 km = 1 nautical mile) Question 27 continues on page 22 21 Marks Question 27 (continued) (c) N B NOT TO SCALE A 36 km C The bearing of C from A is 250 and the distance of C from A is 36 km. (i) 1 (ii) (d) Explain why is 110 . If B is 15 km due north of A, calculate the distance of C from B , correct to the nearest kilometre. 3 Nine students were selected at random from a school, and their ages were recorded. Ages 12 11 16 14 16 15 14 15 14 (i) What is the sample standard deviation, correct to two decimal places? 2 (ii) Briefly explain what is meant by the term standard deviation. 1 End of Question 27 22 Marks Question 28 (13 marks) Use a SEPARATE writing booklet. (a) The Mitchell family has moved to a new house which has an empty swimming pool. The base of the pool is in the shape of a rectangle, with a semicircle on each end. y metres Base of pool x metres (i) Explain why the expression for the area of the base of the pool is 2xy + y2. (ii) 1 4 1.1 m The pool is 1.1 metres deep. The sides and base of the pool are covered in tiles. If x = 6 and y = 2.5, find the total area covered by tiles. (Give your answer correct to the nearest square metre.) (iii) Before filling the pool, the Mitchells need to install a new shower head, which saves 6 litres of water per minute. The shower is used 5 times every day, for 3 minutes each time. If the charge for water is $1.013 per kilolitre, how much money would be saved in one year by using this shower head? (Assume there are 365 days in a year.) Question 28 continues on page 24 23 2 Marks Question 28 (continued) Sue and Mikey are planning a fund-raising dance. They can hire a hall for $400 and a band for $300. Refreshments will cost them $12 per person. (i) Write a formula for the cost ($C ) of running the dance for x people. 1 The graph shows planned income and costs when the ticket price is $20. Planned income and costs for the dance 5000 me 4500 o nc I 4000 sts 3500 Dollars (b) Co 3000 2500 2000 1500 1000 500 0 0 100 Number of people (x) 200 (ii) Estimate the minimum number of people needed at the dance to cover the costs. 1 (iii) How much profit will be made if 150 people attend the dance? 1 (iv) Sue and Mikey plan to sell 200 tickets. They want to make a profit of $1500. 3 What should be the price of a ticket, assuming all 200 tickets will be sold? End of paper 24 Board of Studies NSW 2005 2005 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 2005
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