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NSW HSC 2008 : MATHEMATICS (GENERAL)

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2008 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 21 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 for Questions 1 22. 1 Which expression is equivalent to 12 k 3 4 k ? (A) 3 k 2 (B) 3k3 (C) 8k2 (D) 8 k 3 2 What is the surface area of the open box? 3 cm 5 cm 2 cm (A) 10 cm2 (B) 30 cm2 (C) 52 cm2 (D) 62 cm2 2 NOT TO SCALE 3 The stem-and-leaf plot represents the daily sales of soft drink from a vending machine. If the range of sales is 43, what is the value of N ? 2 N 5 5 3 4 7 7 9 4 0 5 8 5 2 6 0 7 (A) 4 (B) 5 (C) 24 (D) 25 4 Which graph best represents y = 3 x ? (A) y (B) y x (C) x y (D) x y x 3 5 What is the size of the smallest angle in this triangle? 7 6 NOT TO SCALE 8 (A) 29 (B) 47 (C) 58 (D) 76 6 Taxable income Tax payable $0 $12 000 Nil $12 001 $30 000 Nil plus 30 cents for each $1 over $12 000 $30 001 $45 000 $5400 plus 40 cents for each $1 over $30 000 $45 001 $60 000 $11 400 plus 50 cents for each $1 over $45 000 over $60 000 $18 900 plus 55 cents for each $1 over $60 000 Using the tax table, what is the tax payable on $43 561? (A) $5424.40 (B) $10 824.40 (C) $16 224.40 (D) $17 424.40 4 7 Luke s normal rate of pay is $15 per hour. Last week he was paid for 12 hours, at time-and-a-half. How many hours would Luke need to work this week, at double time, to earn the same amount? (A) 4 (B) 6 (C) 8 (D) 9 8 What is the median of the following set of scores? Score Frequency 12 13 14 6 16 2 18 12 Total 33 (A) 12 (B) 13 (C) 14 (D) 15 9 What is the value of x + 2y if x = 5.6 and y = 3.1, correct to 2 decimal places? 8y (A) 0.69 (B) 2.62 (C) 2.83 (D) 4.77 5 10 The marks for a Science test and a Mathematics test are presented in box-and-whisker plots. Science test Mathematics test Which measure must be the same for both tests? (A) Mean (B) Range (C) Median (D) Interquartile range 11 The diagram shows the floor of a shower. The drain in the floor is a circle with a diameter of 10 cm. What is the area of the shower floor, excluding the drain? NOT TO SCALE 1 m (A) 9686 cm2 (B) 9921 cm2 (C) 9969 cm2 (D) 10 000 cm2 6 12 A scatterplot is shown. R T Which of the following best describes the correlation between R and T ? (A) Positive (B) Negative (C) Positively skewed (D) Negatively skewed 13 The height of each student in a class was measured and it was found that the mean height was 160 cm. Two students were absent. When their heights were included in the data for the class, the mean height did not change. Which of the following heights are possible for the two absent students? (A) 155 cm and 162 cm (B) 152 cm and 167 cm (C) 149 cm and 171 cm (D) 143 cm and 178 cm 7 14 Danni is flying a kite that is attached to a string of length 80 metres. The string makes an angle of 55 with the horizontal. How high, to the nearest metre, is the kite above Danni s hand? 80 m h NOT TO SCALE 55 (A) 46 m (B) 66 m (C) 98 m (D) 114 m 15 Ali is buying a speedboat at Betty s Boats. Betty s Boats Cash price $16 000 OR Terms 15% deposit plus $320 per month for 5 years What is the amount of interest Ali will have to pay if he chooses to buy the boat on terms? (A) $3200 (B) $5600 (C) $19 200 (D) $21 600 8 16 A bag contains some marbles. The probability of selecting a blue marble at random from 3 this bag is . 8 Which of the following could describe the marbles that are in the bag? (A) 3 blue, 8 red (B) 6 blue, 11 red (C) 3 blue, 4 red, 4 green (D) 6 blue, 5 red, 5 green 17 The diagram shows the position of Q, R and T relative to P. North R T P 165 Q NOT TO SCALE In the diagram, Q is SW of P R is NW of P QPT is 165 What is the bearing of T from P ? (A) 060 (B) 075 (C) 105 (D) 120 18 New car registration plates contain two letters followed by two numerals followed by two more letters eg AC 12 DC. Letters and numerals may be repeated. Which of the following expressions gives the correct number of car registration plates that begin with the letter M? (A) 263 102 (B) 253 102 (C) 264 102 (D) 254 102 9 19 The height of a particular termite mound is directly proportional to the square root of the number of termites. The height of this mound is 35 cm when the number of termites is 2000. What is the height of this mound, in centimetres, when there are 10 000 termites? (A) 16 (B) 78 (C) 175 (D) 875 20 A point P lies between a tree, 2 metres high, and a tower, 8 metres high. P is 3 metres away from the base of the tree. From P, the angles of elevation to the top of the tree and to the top of the tower are equal. Tower Tree x 2m 3 m P What is the distance, x, from P to the top of the tower? (A) 9 m (B) 9.61 m (C) 12.04 m (D) 14.42 m 10 8m NOT TO SCALE 21 A sphere and a closed cylinder have the same radius. The height of the cylinder is four times the radius. What is the ratio of the volume of the cylinder to the volume of the sphere? (A) 2 : 1 (B) 3 : 1 (C) 4 : 1 (D) 8 : 1 22 A die has faces numbered 1 to 6. The die is biased so that the number 6 will appear more often than each of the other numbers. The numbers 1 to 5 are equally likely to occur. The die was rolled 1200 times and it was noted that the 6 appeared 450 times. Which statement is correct? 1 . 7 (B) The number 6 is expected to appear 2 times as often as any other number. (A) The probability of rolling the number 5 is expected to be (C) The number 6 is expected to appear 3 times as often as any other number. (D) The probability of rolling an even number is expected to be equal to the probability of rolling an odd number. 11 Section II 78 marks Attempt Questions 23 28 Allow about 2 hours for this section Answer each question in the appropriate writing booklet. Extra writing booklets are available. All necessary working should be shown in every question. Marks Question 23 (13 marks) Use the Question 23 Writing Booklet. (a) You are organising an outside sporting event at Mathsville and have to decide which month has the best weather for your event. The average temperature must be between 20 C and 30 C, and average rainfall must be less than 80 mm. The radar chart for Mathsville shows the average temperature for each month, and the table gives the average rainfall for each month. Average temperature ( C) Jan 40 Dec Feb 30 Nov Mar 20 10 Oct Apr Sep May Aug Jun Jul Average rainfall (mm) Jan Feb Mar Apr May Jun (mm) 150 162 86 95 104 140 Jul Aug Sep Oct Nov Dec 59 81 60 72 70 90 (i) If you consider only the temperature data, there are a number of possible months for holding the event. Name ONE of these months. 1 (ii) If both rainfall and temperature data are considered, which month is the best month for the sporting event? 1 Question 23 continues on page 13 12 Marks Question 23 (continued) (b) The capacity of a bottle is measured as 1.25 litres correct to the nearest 10 millilitres. 1 What is the percentage error for this measurement? (c) An alcoholic drink has 5.5% alcohol by volume. The label on a 375 mL bottle says it contains 1.6 standard drinks. (i) How many millilitres of alcohol are in a 375 mL bottle? 1 (ii) It is recommended that a fully-licensed male driver should have a maximum of one standard drink every hour. 2 Express this as a rate in millilitres per minute, correct to one decimal place. 5x + 1 = 4 x 7. 2 (d) Solve (e) In a survey, 450 people were asked about their favourite takeaway food. The results are displayed in the bar graph. 3 2 Takeaway food Pizza Hamburgers Fish and chips How many people chose pizza as their favourite takeaway food? (f) Christina has completed three Mathematics tests. Her mean mark is 72%. What mark (out of 100) does she have to get in her next test to increase her mean mark to 73%? End of Question 23 13 2 Marks Question 24 (13 marks) Use the Question 24 Writing Booklet. (a) Bob is employed as a salesman. He is offered two methods of calculating his income. Method 1: Commission only of 13% on all sales Method 2: $350 per week plus a commission of 4.5% on all sales Bob s research determines that the average sales total per employee per month is $15 670. (i) 2 (ii) (b) Based on his research, how much could Bob expect to earn in a year if he were to choose Method 1? If Bob were to choose a method of payment based on the average sales figures, state which method he should choose in order to earn the greater income. Justify your answer with appropriate calculations. 3 Three-digit numbers are formed from five cards labelled 1, 2, 3, 4 and 5. (i) 1 (ii) If one of these numbers is selected at random, what is the probability that it is odd? 1 (iii) How many of these three-digit numbers are even? 1 (iv) (c) How many different three-digit numbers can be formed? What is the probability of randomly selecting a three-digit number less than 500 with its digits arranged in descending order? 2 Heidi s funds in a superannuation scheme have a future value of $740 000 in 20 years time. The interest rate is 4% per annum and earnings are calculated six-monthly. 3 What single amount could be invested now to produce the same result over the same period of time at the same interest rate? 14 Marks Question 25 (13 marks) Use the Question 25 Writing Booklet. (a) The number of penguins, P, after t years in a new colony can be found using the following formula. P = a 2t (i) If there are 24 penguins after two years, find the value of a . 2 (ii) How many years will it take for the number of penguins to first exceed 1500? 2 (b) In a drawer there are 30 ribbons. Twelve are blue and eighteen are red. Two ribbons are selected at random. (i) Copy and complete the probability tree diagram. 1 B 12 30 18 30 B R B R R (ii) What is the probability of selecting a pair of ribbons which are the same colour? Question 25 continues on page 16 15 2 Marks Question 25 (continued) (c) Pieces of cheese are cut from cylindrical blocks with dimensions as shown. 40 15 cm NOT TO SCALE 7 cm Twelve pieces are packed in a rectangular box. There are three rows with four pieces of cheese in each row. The curved surface is face down with the pieces touching as shown. NOT TO SCALE (i) 4 What are the dimensions of the rectangular box? To save packing space, the curved section is removed. 40 NOT TO SCALE 15 cm 7 cm Curved section (ii) What is the volume of the remaining triangular prism of cheese? Answer to the nearest cubic centimetre. End of Question 25 16 2 Marks Question 26 (13 marks) Use the Question 26 Writing Booklet. (a) Cecil invited 175 movie critics to preview his new movie. After seeing the movie, he conducted a survey. Cecil has almost completed the two-way table. Aged < 40 Movie critics who liked the movie Aged 40 65 Totals 102 Movie critics who did not like the movie 31 A Totals 175 (i) Determine the value of A . 1 (ii) A movie critic is selected at random. 2 What is the probability that the critic was less than 40 years old and did not like the movie? (iii) Cecil believes that his movie will be a box office success if 65% of the critics who were surveyed liked the movie. 1 Will this movie be considered a box office success? Justify your answer. (b) The retirement ages of two million people are displayed in a table. Retirement age 36 40 41 45 46 50 51 55 56 60 61 65 66 70 71 75 76 80 Number of people (thousands) 5 10 20 35 180 700 500 400 150 (i) What is the relative frequency of the 51 55 year retirement age? 1 (ii) Describe the distribution. 1 Question 26 continues on page 18 17 Marks Question 26 (continued) (c) Joel is designing a game with four possible outcomes. He has decided on three of these outcomes. Chance of occurring Outcome 1 Result 10% Win $12 30% Outcome 2 Win $3 40% Outcome 3 Outcome 4 2 Win $6 What must be the value of the loss in Outcome 4 in order for the financial expectation of this game to be $0? (d) The graph shows the predicted population age distribution in Australia in 2008. Population pyramids for Australia in 2008 Male 1.2 1.0 0.8 0.6 0.4 0.2 Age group 80+ 75 79 70 74 65 69 60 64 55 59 50 54 45 49 40 44 35 39 30 34 25 29 20 24 15 19 10 14 5 9 0 4 0.0 0.0 Female 0.2 0.4 0.6 0.8 1.0 1.2 Population (in millions) Source: U.S. Census Bureau, Population Division (i) How many females are in the 0 4 age group? 1 (ii) What is the modal age group? 1 (iii) How many people are in the 15 19 age group? 2 (iv) Give ONE reason why there are more people in the 80 + age group than in the 75 79 age group. 1 End of Question 26 18 Marks Question 27 (13 marks) Use the Question 27 Writing Booklet. (a) An aircraft travels at an average speed of 913 km/h. It departs from a town in Kenya (0 , 38 E) on Tuesday at 10 pm and flies east to a town in Borneo (0 , 113 E). (i) 2 (ii) How long will the flight take? (Answer to the nearest hour.) 1 (iii) (b) What is the distance, to the nearest kilometre, between the two towns? (Assume the radius of Earth is 6400 km.) What will be the local time in Borneo when the aircraft arrives? (Ignore time zones.) 2 Julie takes out a $290 000 home loan. The terms of the loan are 8.25% per annum over 30 years with monthly repayments. (i) Show that the minimum monthly repayment is $2178.67, to the nearest cent. 2 (ii) Determine the total amount paid for the loan over 30 years. 1 (iii) Each month, Julie decides to pay $250 more than the minimum monthly repayment. 3 Would she be able to pay off the loan in 20 years? Justify your answer by showing all calculations. (c) A plasma TV depreciated in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $2023, using the declining balance method. What was the purchase price of the plasma TV? 19 2 Marks Question 28 (13 marks) Use the Question 28 Writing Booklet. (a) The following graph indicates z-scores of height-for-age for girls aged 5 19 years. Height-for-age for girls aged 5 19 years (z-scores) 190 3 180 2 170 z-scores 1 0 160 1 2 Height (cm) 150 3 140 130 120 110 100 90 5 6 7 8 9 10 11 12 13 Age (years) 14 15 16 17 18 19 Source: U.S. Census Bureau, Population Division (i) What is the z-score for a six year old girl of height 120 cm? (ii) 1 Rachel is 10 1 years of age. 2 (1) If 2.5% of girls of the same age are taller than Rachel, how tall is she? (2) Rachel does not grow any taller. At age 15 1 , what percentage of girls 2 of the same age will be taller than Rachel? (iii) 1 2 What is the average height of an 18 year old girl? 1 Question 28 continues on page 21 20 Marks Question 28 (continued) (iv) For adults (18 years and older), the Body Mass Index is given by m where m = mass in kilograms and h = height in metres. B= h2 The medically accepted healthy range for B is 21 B 25. 2 What is the minimum weight for an 18 year old girl of average height to be considered healthy? (v) The average height, C, in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with equation C = 6A + 79 where A is the age in years. 1 (1) For this line, the gradient is 6. What does this indicate about the heights of girls aged 6 to 11? (2) Give ONE reason why this equation is not suitable for predicting heights of girls older than 12. 1 (b) A tunnel is excavated with a cross-section as shown. a h b h a h h (i) Find an expression for the area of the cross-section using TWO applications of Simpson s rule. 2 (ii) The area of the cross-section must be 600 m2. The tunnel is 80 m wide. 2 If the value of a increases by 2 m, by how much will b change? End of paper 21 BLANK PAGE 22 BLANK PAGE 23 BLANK PAGE 24 Board of Studies NSW 2008 2 008 HIGHER SCHOOL CER TIFIC ATE EXAMINATION General Mathematics FORMULAE SHEET Area of an annulus Surface area A = R2 r 2 Sphere A = 4 r 2 Closed cylinder A = 2 rh + 2 r 2 ( ) R = radius of outer circle e r = radius of inner circle r = radius h = p erpendicular height Area of an ellipse n A = ab Volume Cone 1 Ah 3 V= 43 r 3 r = radius h = p erpendicular height A = area of base Arc length of a circle l= 2 r 3 60 = n umber of degrees in central angle Simpson s rule for area approximation h A d + 4 dm + dl 3 f ( V= Sph here = n umber of degrees in central angle g V = r 2h Pyramid Area of a sector 2 r A= 360 1 V = r 2h 3 Cylinder a = length of semi-major axis b = length of semi-minor axis ) Sine rule a b c = = sin A sin B sin C Area of a triangle 1 a b sin C 2 h = d istance between successive measurements d f = first measurement A= dm = middle measurement Cosine rule dl = last measurement c 2 = a 2 + b 2 2ab cos C or a2 + b2 c2 cos C = 2ab 373 25 FORMULAE SHEET Declining balance formula for depreciation Simple interest I S = V0 (1 r ) = Prn P = initial quantity r = p ercentage interest rate per period, c expressed as a decimal n = n umber of periods n S = salvage value of asset after n p eriods r = p ercentage interest rate per perio d, o expressed as a decimal Mean of a sample Compound interest A = P (1 + r ) A P n r = = = = final balance initial q u antity n umber of compounding periods p ercentage interest rate per compounding perio d, expressed as a decimal o Future value ( A ) of an annuity (1 + r )n 1 A = M r M = contribution per period, paid at the end of the period Present value ( N ) of an annuity n (1 + r ) 1 N = M n r (1 + r ) or N= x= n A n (1 + r ) Straight -line formula for depreciation t S = V0 Dn S = salvage value of asset after n p eriods V0 = p urchase price of the asset c D = amount of depreciation apportioned per period n = n umber of periods x= x x n f = = = = x n fx f mean individual score n umber of scores r frequency Formula for a z-score z= x x s s = standard deviation Gradient of a straight line m= vertical change in position h o rizontal change in position Gradient inte cept form of a straight line y = mx + b m = g radie ent b = y-intercept Probability of an event Th e probability of an event where outcomes are equally likely is given by: P (event) = 26 numb er of favourable outcomes total number of ou tcomes u

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Additional Info : New South Wales Higher School Certificate General Mathematics 2008
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