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ACEEXAMPAPERStudent name:YEAR 12YEARLYEXAMINATIONPAPER 1Mathematics AdvancedGeneralInstructions Working time - 180 minutesWrite using black penNESA approved calculators may be usedA reference sheet is provided at the back of this paperIn questions 11-16, show relevant mathematical reasoning and/orcalculationsTotal marks:100Section I – 10 marks Attempt Questions 1-10 Allow about 15 minutes for this sectionSection II – 90 marks Attempt questions 11-16 Allow about 2 hours and 45 minutes for this section1

Year 12 Mathematics AdvancedSection I10 marksAttempt questions 1 - 10Allow about 15 minutes for this sectionUse the multiple-choice answer sheet for questions 1-101. What is the solution to the equation 2cos % & 1 0 in the domain 0 & 2π ?(A)& π 11π,6 6(B)& π 7π,4 4(C)& π 5π 7π 11π, , ,4 4 4 4(D)& π 3π 5π 7π, , ,4 4 4 42.Which of the following properties matches the above graph?(A)3′(&) 0 and 3′′(&) 0(B)3′(&) 0 and 3′′(&) 0(C)3′(&) 0 and 3′′(&) 0(D)3′(&) 0 and 3′′(&) 02

Year 12 Mathematics Advanced3.A factory produces bags of cashews. The weights of the bags are normally distributed,with a mean of 900 g and a standard deviation of 50 g. What is the best approximation forthe percentage of bags that weigh more than 1000 g?(A)0%(B)2.5%(C)5%(D)16%C4. What is the value of ( ?@ 1)B& ?D(A) ?(B)1 ? 3(C)1 ?( 1)3(D)1 ?( 2)35. What is the gradient to the curve F (& G)(& % 1) at the point when x –2?(A) 3G 6(B) 5G 1(C)4G 11(D)5G 46.What is the correlation between the variables in this scatterplot?(A)Weak negative(B)Weak Positive(C)Moderate negative(D)Moderate positive3

Year 12 Mathematics Advanced7. A section of the graph F 3(&) is shown below.Which of the following is the correct function for the above graph?(A)1π3(&) tan I J& KL24(B)π3(&) tan I2 J& KL4(C)1π3(&) tan I J& KL22(D)π3(&) tan I2 J& KL28. The graph of the derivative function is shown below.Where is the function F 3(&) increasing?(A){& & 0}(B){& & 2}(C){& 3 & 2}(D){& & 3} or {& & 2}4

Year 12 Mathematics Advanced9.The table below shows the present value of a 1 annuity.Present value of 1End of 6.46326.2098What is the present value of an annuity where 12,000 is contributed each year for sixyears into an account earning 3% per annum compound interest?(A) 15 183.83(B) 54 956.40(C) 65 006.40(D) 72 000.0010. Which of the following graphs could not represent a probability density function f(x)?(A)(B)(C)(D)5

Year 12 Mathematics AdvancedSection II90 marksAttempt questions 11 - 16Allow about 2 hours and 45 minutes for this sectionAnswer each question in the spaces provided.Your responses should include relevant mathematical reasoning and/or calculations.Question 11 (2 marks)MarksDifferentiate the following functions with respect to x.(a)3(&) sin& & %1(b)3(&) ln(& % 1)?1Question 12 (3 marks)For the arithmetic sequence 4, 9, 14, 19, .(a)Write the rule to describe the nth term.1(b)What is the 25th term?1(c)Find the sum of the first 100 terms.16

Year 12 Mathematics AdvancedQuestion 13 (4 marks)MarksA continuous random variable X has a function f given by 3 & 2 & 43(&) R0otherwise(a)Find X(2 Y 3.5)2(b)Find X(2 Y 2.5)2Question 14 (4 marks)Differentiate(a)2 @ cos&2(b)tan&&2Question 15 (1 mark)Find (2& 3)CD B&17

Year 12 Mathematics AdvancedQuestion 16 (2 marks)MarksTran’s industrial unit produces aluminium rods. In the past week the industrial unithas produced aluminium rods with a mean weight of 12.5 kilograms and a standarddeviation of 0.5 kilograms.(a)Quality control requires any aluminium rod with a z-score less than –1 to berejected. What is the minimum weight that will be accepted?1(b)Aluminium rods with a z-score greater than 2 are also rejected. What is themaximum weight that will be accepted?1Question 17 (2 marks)What is the area enclosed between the curves F & % 1 and F 3& 1 ?82

Year 12 Mathematics AdvancedQuestion 18 (3 marks)Find the value of k if F \@ sin& andMarksBF 3F \@ cos&.B&3Question 19 (3 marks)The diagram below shows a native garden. All measurements are in metres.(a)Use the Trapezoidal Rule with 4 intervals to find an approximate value for thearea of the native garden.2(b)If 25 millimetres of rain fell overnight, how many litres of rain fell on thenative garden? Assume 1 m? 1000 L.19

Year 12 Mathematics AdvancedQuestion 20 (3 marks)MarksConsider the functions F & % and F & % 3& 2.(a)Sketch the two functions on the same axes.2(b)Hence or otherwise find the values of x such that & % (& 1)(& 2).1Question 21 (2 marks)State the amplitude and period of the function 3(&) 4 3cos J10π&K22

Year 12 Mathematics AdvancedQuestion 22 (2 marks)MarksThe normal distribution shows the results of a mathematics assessment task. It hasa mean of 60 and a standard deviation of 10.(a)What is the mathematics assessment result with a z-score of –2?1(b)What is the z-score of a mathematics assessment result of 65?1Question 23 (2 marks) 2Find (sec % 2&)B&DQuestion 24 (2 marks)How many solutions does the equation cos(2&) 1 have for 0 & 2π?112

Year 12 Mathematics AdvancedQuestion 25 (5 marks)MarksA function 3(&) is defined by 3(&) & % (3 &) .(a)Find the stationary points for the curve F 3(&) and determine their nature.2(b)Sketch the graph of F 3(&) showing the stationary points and x-intercepts.2(c)Find the equation of the tangent to the curve at the point X(1,2) .1Question 26 (2 marks)Construct a recurrence relation in the form abcC ab (1 e) f to model thebalance of a loan of 58 000 borrowed at 6% per annum, compounding monthly,with payments of 810 per month.122

Year 12 Mathematics AdvancedQuestion 27 (4 marks)MarksTen kilograms of chlorine is placed in water and begins to dissolve. After t hours theamount A kg of undissolved chlorine is given by g 10 h\i(a)Calculate the value of k given that A 3.6 when t 5. Answer correct to threedecimal places.2(b)After how many hours does one kilogram of chlorine remain undissolved?Answer correct to one decimal place.2Question 28 (2 marks)The third and seventh terms of a geometric series are 1.25 and 20 respectively.What is the first term?132

Year 12 Mathematics AdvancedQuestion 29 (5 marks)MarksThe table below shows forearm length and hand length.Forearm (in cm) 25.0 25.6 26.0 26.6 27.0 27.4 28.0 28.6 29.0 29.2Hand (in cm)17.2 17.6 18.2 18.4 19.0 19.0 19.8 19.8 20.4 20.6(a)Draw a scatterplot using the above table.1(b)Draw a line of best fit on the scatterplot.1(c)Charlotte has a forearm whose length is 27.8 cm. What is her expected handlength?1(d)Calculate the value of the Pearson’s correlation coefficient. Answer correct tofour decimal places.214

Year 12 Mathematics AdvancedQuestion 30 (3 marks)MarksFlorence left 1000 in her will for World Vision. Her instructions were that thismoney be invested at 5% interest, compounded annually.(a)How much money would be given to World Vision after 100 years? Give youranswer to the nearest dollar.1(b)Florence has requested her family invest a further 1000 at the beginning ofeach subsequent year at the same interest rate. How much money would begiven to World Vision after 100 years if her family followed Florence’sinstructions? Give your answer to the nearest dollar.2Question 31 (3 marks)Evaluate the following definite integrals.(a)% & % 1B&1hC(b)k2 3& 4B&hC15

Year 12 Mathematics AdvancedQuestion 32 (2 marks)MarksThe table below shows the future value of a 1 annuity.Future value of 1End of 123.183.253.3144.254.374.514.64(a)What would be the future value of a 32 000 per year annuity at 8% perannum for 4 years, with interest compounding annually?1(b)An annuity of 6300 is invested every six months at 8% per annum,compounded biannually for 2 years. What is the future value of the annuity?1Question 33 (4 marks)Consider the function 3(&) 11 &%(a)Find the value of 3′(&).2(b)Find the coordinates of the point on the curve F 3(&) at which the tangentis parallel to the x-axis.216

Year 12 Mathematics AdvancedQuestion 34 (7 marks)MarksAn object is moving in a straight line and its velocity is given by;o 1 2sin2p for p 0where v is measured in metres per second and t in seconds.Initially the object is at the origin.(a)Find the displacement x, as a function of t.2(b)What is the position of the object when p ?1(c)Find the acceleration a, as a function of t.1(d)Sketch the graph of a, as a function of t, for 0 p π.2(e)What is the maximum acceleration of the object?1?17

Year 12 Mathematics AdvancedQuestion 35 (3 marks)MarksSketch the follow graphs on the same number plane.F &,F & 1,3F & 1Question 36 (2 marks)Class A has 24 students and achieved a mean on an assessment task of 75.5%.Class B has 28 students and achieved a mean on the same assessment task of80.5%. What was the mean mark for both classes. Answer correct to one decimalplaces.182

Year 12 Mathematics AdvancedQuestion 37 (5 marks)MarksA V8 supercars racetrack consists of two semi-circular curves and two straights.The dimensions of the racetrack are shown below. The total length of the racetrackis 4.8 km.(a)Let x km represent the length of the straight and y km represent the diameterof the smaller semicircle. Show that:9.6 4&F 3π2(b)The average speed of a V8 supercar on this racetrack is dependent on thelength of the straight. It is given by:3s 200 I&? π FL27 6What is the length of the straight that maximizes the speed?19

Year 12 Mathematics AdvancedQuestion 38 (4 marks)Marks(a)Sketch the graph F 2& 4 .2(b)Using the graph from part (a), or otherwise, find all values of m for which theequation 2& 4 t& 1 has exactly one solution.2Question 39 (2 marks)The Pearson’s correlation coefficient between students assessment result and theirheight was 0.12. What is the meaning of this correlation?2Question 40 (2 marks)The heights of a group of friends are normally distributed with a mean of 167 cmand a standard deviation of 12 cm. What percentage of the group are more than179 cm tall?End of paper202

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Year 12 Mathematics AdvancedACE Examination Paper 1Year 12 Mathematics Advanced Yearly ExaminationWorked solutions and Marking guidelinesSection I1.2.3.4.5.6.7.8.9.10.Solution2cos % 𝑥 1 011cos % 𝑥 or cos𝑥 2 2π 3π 5π 7π𝑥 , , ,4 4 4 4𝑓′(𝑥) 0 (increasing)Criteria1 Mark: D1 Mark: A𝑓′′(𝑥) 0 (concave down)𝑥 𝑥̅𝑧 𝑠1000 900 50 295% of scores have a z-score between –2 and 2\ 2.5% have a z-score greater than 2.MM1 FGFGD (𝑒 1)𝑑𝑥 J 𝑒 𝑥K3LL1 F1 N 𝑒 1O 331 F (𝑒 2)3𝑦 (𝑥 𝑎)(𝑥 % 1) 𝑥 F 𝑎𝑥 % 𝑥 𝑎𝑑𝑦 3𝑥 % 2𝑎𝑥 1𝑑𝑥Gradient at the point when x –2𝑚 3 ( 2)% 2𝑎 ( 2) 1 4𝑎 11Correlation between 0.5 and 0.74.\Moderate positive.π πPeriod 𝑛 2 𝑛 2XAlso a translation of Y in the positive x-direction.π𝑓(𝑥) tan [2 \𝑥 ] 4Increasing function 𝑓′(𝑥) 0{𝑥 3 𝑥 2}Intersection value is 5.4172 (3% and 6 years)𝑃𝑉 5.4172 12 000 65 006.40Fundamental property of a probability density is that for anyvalue of x, the value of f(x) is non-negative.\Graph (A) has 𝑓(𝑥) 011 Mark: B1 Mark: D1 Mark: C1 Mark: D1 Mark: B1 Mark: C1 Mark: C1 Mark: A