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12YE ARHSCMathematicsRevision & ExamWorkbookFree-to-download Sample HSC Exams with answersGet the Results You Want!Lyn Baker

Chapter 10Sample HSC ExaminationsSample HSC Examination 1Time allowed: 3 hoursTotal marks 100Section IObjective-response questions1 mark each1 log3 3 A–1B122 To three decimal places, the value of sinA 0.206B 0.021C3pis813DC 0.924 13D 1.1783 A tank is being emptied. The volume of water, in litres, remaining in the tank t minutes after it started to beemptied is given by V t2 – 375t 35 100. At what rate (in litres/minute) is the water flowing from the tankafter 30 minutes?A 315B 3514 What is the period of the graph of y 3 cosA 3BC 357D 375pt?212C235 The diagram shows the graph of the derivative f ’(x) of a function.D 4yThe function f(x) has a minimum turning point at x A 0B 2C 4D 5y f ’(x)0245x Pascal Press ISBN 978 1 74125 010 76 sin q cos q cot q A tan qB sec qC cosec qD 1C stationaryD undefined7 When x 1, the curve y 5x3 – 6x2 7x – 8 isA increasingB decreasing8 The displacement of a moving particle is given by x 2et – 3t – 4.The initial displacement isA 0204B –2C –4D –5EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 19 The probability that a person selected at random has a gene linked with a particular disease is 0.12.Of people with that gene, the probability that they will get that disease in their lifetime is 0.4.What is the probability that a person selected at random has the gene but will not get the disease?A 0.6B 0.48C 0.72D 0.07210 The volume of the solid formed by rotating the section of theycurve y x between y 0 and y 3 about the y-axis is given by2AC3 πxdxBπ y 2 dyD0 30433 π y dyy x203 πy40dyxTotal for Section I: 10 marksSection IIQuestion 11aFind a primitive of ex – x2 marksbFind the exact length of the arc PQ.2 marksP8 cmO5π128 cmQFind the second derivative of y (3x – 4)7dDifferentiate the following functions:iloge (2x – 1)CHAPTER 10 – Sample HSC Examinations2 marks2 marksiix sin x2 marksEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK205 Pascal Press ISBN 978 1 74125 010 7c

Sample HSC Examination 1Question 11eA particle is moving in a straight line from a fixed point O.At time t seconds its displacement from O, x m, is given by x 18t2 – t3iAt what times is the particle stationary?2 marksiiFind the displacement at the time when the acceleration is zero.3 marksTotal for question 11: 15 marksQuestion 12aA box contains 10 unlabelled CDs, 2 of which are blank and the remainder used.Jack chooses one at random, puts it aside and then chooses another. Pascal Press ISBN 978 1 74125 010 7iDraw a tree diagram to show the possibleoutcomes.1 markiiFind the probability that both CDsare blank.iii Find the probability that at least one CD isblank.2 marksivJack plays one of the CDs and finds that it isblank. What is the probability that the otherCD is also blank?1 mark2061 markEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 1Question 12bFind the exact value of:ic π3π4sec2 x dx3 marks iie1Find all solutions over the domain 0 x 2p for which 2 cos x dxx3 marks4 marks3 0Total for question 12: 15 marksQuestion 13aUse Simpson’s rule with the table of values to estimate 5333.03.52.642.34.52.051.8 Pascal Press ISBN 978 1 74125 010 7xf(x)3 marksf ( x ) dxCHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK207

Sample HSC Examination 1Question 13bcdThe first term of an arithmetic series is 17 and the common difference is –3. Find:ithe ninth term1 markiiiFind the equation of the tangent to the curve y 2ex 1 at the point where x 1.iiAt what point does the tangent in part i cut the y-axis?2 marksthe sum of the first nine terms3 marks1 markThe diagram shows the graph of y sin x and y –cos x.iShow that x 3πis a solution of the equation4sin x cos x 01 mark10y sin xπ2π3π22πy –cos x Pascal Press ISBN 978 1 74125 010 7–1iiFind the area shaded in the diagram.4 marksTotal for question 13: 15 marks208EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 1Question 14aSolve 2 ln x ln (6 x)3 marksbThe mass, M, in grams of a radioactive substance is expressed as M 175e–kt where k is a positive constantand t the time in days. The mass of the substance halved in 6 days.iFind the value of k correct to 5 decimal places.iiFind the mass of the substance remaining after 10 days.1 mark2 marksiii At what rate is the mass disintegrating after 10 days?In the diagram, AB is parallel to DC and AB DC.BADCiBy proving triangle ABC congruent to triangle CDA, prove that AD BC.4 marksiiIf CA bisects3 marksBAD show that ABCD is a rhombus.Total for question 14: 15 marksCHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK209 Pascal Press ISBN 978 1 74125 010 7c2 marks

Sample HSC Examination 1Question 15aFind the limiting sum of the series 10 5 2.5 1.25 .2 marksbFor what values of x is the curve y x3 – 5x2 10x 7 concave up?3 markscA and B are the points (3, –1) and (6, 5) respectively.iFind the midpoint of AB.1 markiiFind the gradient of AB.1 mark3 marks Pascal Press ISBN 978 1 74125 010 7iii Find the equation of the perpendicular bisector of AB.210EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 1Question 15dAt the beginning of each year, Jennifer invests 3000 into an account earning 5% p.a. interest compoundedannually.iFind, to the nearest dollar, the amount to which the first 3000 accumulated at the end of twenty years.(Assume the last amount of interest has just been paid.)1 markiiFind the total value of the investment at the end of 20 years.4 marksTotal for question 15: 15 marksQuestion 16The graph of y f(x) passes through the point (2, –3). f ’(x) 6x – 1. Find f(x).3 marks Pascal Press ISBN 978 1 74125 010 7aCHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK211

Sample HSC Examination 1Question 16bBz ABC is right-angled at B.DEFG is a rectangle.AB 120 m.BC 160 m.FG x m, GD y m, BG z m.iFind the length of AC.iii Find an expression for z in terms of x.vFGyA1 mark2 marksFind an expression for y in terms of z.2 marksECDShow that FBG is similar to ABC.1 markiv Show that GDC is similar to ABC.1 markiivi Show that y 96 vii Find the area of the largest possible rectangle DEFG. Pascal Press ISBN 978 1 74125 010 7x12 x251 mark4 marksTotal for question 16: 15 marksTotal for Section II: 90 marksTotal marks: 100212EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 2Time allowed: 3 hoursTotal marks 100Section IObjective-response questions1 mark each1 Correct to two decimal places, the value ofA 0.73B 1.619.6 2.8is3.1 1.7C 4.94D 11.842 Which number line shows the values of x for which x – 2 45678C2x 3x2 93x1 x 9 x 32A3x 3B3x 3D2x 3x2 94 A bag contains 5 green, 7 yellow and 8 pink pegs. Two pegs are chosen at random. What is the probabilitythat both pegs are green?A576B119C116D2215 A car was bought for 24 950 five years ago. If it depreciates at 15% p.a., which is the best approximation toA 11 1006B 13 900C 18 700D 6200B 100 C 150 D 200 B loga10C loga32D loga62 Pascal Press ISBN 978 1 74125 010 7its value now?5pradians 9A 75 7 3 loga4 – loga2 A loga6CHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK213

Sample HSC Examination 28 ABDE is a square. Triangle BCD is equilateral.ABWhat is the size of EAC?CEA 60 B 65 C 70 D 75 D9 x 4 cos 2t x A cos 2tB –cos 2tC 16 cos 2tD –16 cos 2t10 For a particular curve y f(x) it is known that f ’(a) 0 and f ’’(a) 0.At x a the curve isA increasing and concave upB increasing and concave downC decreasing and concave upD decreasing and concave downTotal for Section I: 10 marksSection IIQuestion 11a2 marksiisin xx2 marksiex ln xiFind the coordinates of Q, the point of intersection of k: y 3x 9 and l: x 3y – 17 0 Pascal Press ISBN 978 1 74125 010 7bDifferentiate with respect to xQ2 marksx 3y – 17 0P(5, 4)y 3x 9RS(3, –2)214EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 2Question 11b1 markiii Find the coordinates of M, the midpoint of Q and S(3, –2).2 marksivFind the equation of the line joining P(5,4) and R.2 marksvAre P, M and R collinear? Justify your answer.1 markLeah opens a bank account and deposits 10 in the first month, 30 in the second month and 50 in thethird month. If she can continue to increase the amount of her deposit by 20 each month, how much willshe have saved by the end of eighteen months?3 marks Pascal Press ISBN 978 1 74125 010 7cii If R is the point where k cuts the x-axis, find the coordinates of R.Total for question 11: 15 marksCHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK215

Sample HSC Examination 2Question 12aAB BD, AE ED, AB DE 4 cm, AE 8 cm.BAECDbiShow that triangles ABC and DEC are congruent.2 marksiiFind the length of AC.3 marksFind the exact value of: Pascal Press ISBN 978 1 74125 010 7i216 21(3 x 2)5 dx2 marksii 10e 4 x dx2 marksEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 2Question 12cPoints P, Q and R are the vertices of a triangle. Q is 28 km from P on a bearing of 065 , and R is 15 km fromP on a bearing of 120 .iNQFind the size of QPR.1 mark28 kmP15 kmRiid2 marksFind the area of the triangle PQR. Give the answer to the nearest km2.Solve the equation 2x 1 9, giving the answer correct to four significant figures.3 marksTotal for question 12: 15 marksQuestion 13iSolve –x2 13x – 36 0iiFind the equation of the tangent to the parabola y –x2 13x – 36 at the point where x 6.CHAPTER 10 – Sample HSC Examinations1 markEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK3 marks217 Pascal Press ISBN 978 1 74125 010 7a

Sample HSC Examination 2Question 13aiii Draw a diagram showing the parabola and the tangent. Shade the region bounded bythe parabola, the tangent and the x-axis.2 marksyxFind the area shaded in the diagram.iFactorise 2m2 – 3m – 2iiFind all values of x, 0 x 2p, for which 2 cos 2x – 3 cos x 23 marks1 mark3 marks Pascal Press ISBN 978 1 74125 010 7biv218EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 2Question 13cFor what values of m does the series 1 m2 m4 m6 have a limiting sum?416 642 marksTotal for question 13: 15 marksQuestion 14aiWhat was the initial temperature of the object?1 markiiAfter approximately how many minutes will the temperature of the object be 50 ?A curve, y f(x) has a turning point at (0, 3). If f ’’(x) ex e–x find the equation of the curve.2 marks3 marks Pascal Press ISBN 978 1 74125 010 7bThe temperature T( C) of a cooling object at time t minutes (t 0) is given by T 120e–0.009t – 20CHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK219

Sample HSC Examination 2Question 14cA circular sector OABC has radius r m and angle AOC measures q radians.If the length of the chord AC is x m:iShow that x 2r (1 – cos q)22Arm2 marksxmOBrmCii2pand x 6 3 find the exact length of the arc ABC.33 marksA bag holds 20 red and 30 green apples. Two apples are drawn at random from the bag. What is theprobability that:iboth apples are red?2 marksiione apple is red and one is green?2 marks Pascal Press ISBN 978 1 74125 010 7dIf q Total for question 14: 15 marks220EXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK

Sample HSC Examination 2Question 15aConsider the function f(x) –x4 4x3 – 16iFind the stationary points of the curve and determine their nature.4 marksiiFind any points of inflection.2 marksiii What value does y approach as x ?1 markivWhat value does y approach as x– ?1 markvSketch the curve y –x4 4x3 – 162 marksviWhat is the range of the function? 1 mark Pascal Press ISBN 978 1 74125 010 7yxCHAPTER 10 – Sample HSC ExaminationsEXCEL HSC MATHEMATICS REVISION AND EXAM WORKBOOK221