Mathematics (Project Maths – Phase 1)

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2012. M130Coimisiún na Scrúduithe StáitState Examinations CommissionLeaving Certificate Examination, 2012Mathematics(Project Maths – Phase 1)Paper 2Higher LevelMonday 11 JuneMorning 9:30 – 12:00300 marksExamination numberCentre stampFor examinerQuestionMark12345678GradeRunning totalTotal

InstructionsThere are two sections in this examination paper.Section AConcepts and Skills150 marks6 questionsSection BContexts and Applications150 marks2 questionsAnswer all eight questions, as follows:In Section A, answer:Questions 1 to 5 andeither Question 6A or Question 6B.In Section B, answer Question 7 and Question 8.Write your answers in the spaces provided in this booklet. You will lose marks if you do not do so.There is space for extra work at the back of the booklet. You may also ask the superintendent formore paper. Label any extra work clearly with the question number and part.The superintendent will give you a copy of the Formulae and Tables booklet. You must return it atthe end of the examination. You are not allowed to bring your own copy into the examination.Marks will be lost if all necessary work is not clearly shown.Answers should include the appropriate units of measurement, where relevant.Answers should be given in simplest form, where relevant.Write the make and model of your calculator(s) here:Leaving Certificate 2012Page 2 of 19Project Maths, Phase 1Paper 2 – Higher Level

Section AConcepts and Skills150 marksAnswer all six questions from this section.Question 1(a)(25 marks)Given the co-ordinates of the vertices of a quadrilateral ABCD, describe three different waysto determine, using co-ordinate geometry techniques, whether the quadrilateral is aparallelogram.method 1:method 2:method 3:(b)Using one of the methods you described, determine whether the quadrilateral with vertices(–4, –2), (21, –5), (8, 7) and (–17, 10) is a parallelogram.pageLeaving Certificate 2012Page 3 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Question 2(25 marks)The equations of two circles are:c1 : x 2 y 2 6 x 10 y 29 0c2 : x 2 y 2 2 x 2 y 43 0(a)Write down the centre and radius-length of each circle.centre of c1:radius-length of c1:(b)centre of c2:radius-length of c2:Prove that the circles are touching.Leaving Certificate 2012Page 4 of 19Project Maths, Phase 1Paper 2 – Higher Level

(c)Verify that (4, 7) is the point that they have in common.(d)Find the equation of the common tangent.pageLeaving Certificate 2012Page 5 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Question 3(25 marks)The circle shown in the diagram has, as tangents,the x-axis, the y-axis, the line x y 2 and theline x y 2k , where k 1 .Find the value of k.Leaving Certificate 2012Page 6 of 19Project Maths, Phase 1Paper 2 – Higher Level

Question 4(25 marks)A certain basketball player scores 60% of the free-throw shots she attempts. During a particulargame, she gets six free throws.(a)What assumption(s) must be made in order to regard this as a sequence of Bernoulli trials?(b)Based on such assumption(s), find, correct to three decimal places, the probability that:(i)she scores on exactly four of the six shots(ii)she scores for the second time on the fifth shot.pageLeaving Certificate 2012Page 7 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Question 5(25 marks)A company produces calculator batteries. The diameter of thebatteries is supposed to be 20 mm. The tolerance is 0·25 mm.Any batteries outside this tolerance are rejected. You mayassume that this is the only reason for rejecting the batteries.(a)The company has a machine that produces batteries withdiameters that are normally distributed with mean 20 mmand standard deviation 0·1 mm. Out of every 10 000batteries produced by this machine, how many, on average, are rejected?(b)A setting on the machine slips, so that the mean diameter of the batteries increases to20·05 mm, while the standard deviation remains unchanged. Find the percentage increase inthe rejection rate for batteries from this machine.Leaving Certificate 2012Page 8 of 19Project Maths, Phase 1Paper 2 – Higher Level

Question 6(25 marks)Answer either 6A or 6B.Question 6A(a)(i)Given the points B and C below, construct, without using a protractor or setsquare, apoint A such that ABC 60 .B(ii)(b)CHence construct, on the same diagram above, and using a compass and straight edgeonly, an angle of 15 .kIn the diagram, l1 , l2 , l3 , and l4 areparallel lines that make intercepts ofequal length on the transversal k. FG isparallel to k, and HG is parallel to ED.l1FHl2GProve that the triangles ΔCDE andΔFGH are congruent.l3CEl4DThere is space to continue your work on the next page.Leaving Certificate 2012Page 9 of 19pagerunningProject Maths, Phase 1Paper 2 – Higher Level

ORQuestion 6BThe incircle of the triangle ABC has centre O and touches the sides at P, Q and R, as shown.1Prove that PQR ( CAB CBA ) .2CRPOALeaving Certificate 2012QPage 10 of 19BProject Maths, Phase 1Paper 2 – Higher Level

pageLeaving Certificate 2012Page 11 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Section BContexts and Applications150 marksAnswer Question 7 and Question 8.Question 7(75 marks)To buy a home, people usually take out loans called mortgages. If one of the repayments is notmade on time, the mortgage is said to be in arrears. One way of considering how much difficultythe borrowers in a country are having with their mortgages is to look at the percentage of allmortgages that are in arrears for 90 days or more. For the rest of this question, the term in arrearsmeans in arrears for 90 days or more.The two charts below are from a report about mortgages in Ireland. The charts are intended toillustrate the connection, if any, between the percentage of mortgages that are in arrears and theinterest rates being charged for mortgages. Each dot on the charts represents a group of peoplepaying a particular interest rate to a particular lender. The arrears rate is the percentage in arrears.7September 20096Standard variableinterest rate (Sept 2011)Standard variableinterest rate (Sept 2009)754321September 201165430246890 day arrears rateSeptember 2009, owner occupier, % balance0510152090 day arrears rateSeptember 2011, owner occupier, % balance(Source: Goggin et al.Variable Mortgage Rate Pricing in Ireland, Central Bank of Ireland, 2012)(a)Paying close attention to the scales on the charts, what can you say about the change fromSeptember 2009 to September 2011 with regard to:(i)the arrears rates?(ii)the rates of interest being paid?Leaving Certificate 2012Page 12 of 19Project Maths, Phase 1Paper 2 – Higher Level

(iii) the relationship between the arrears rate and the interest rate?(b)What additional information would you need before you could estimate the median interestrate being paid by mortgage holders in September 2011?(c)Regarding the relationship between the arrears rate and the interest rate for September 2011,the authors of the report state: “The direction of causality is important” and they go on todiscuss this.Explain what is meant by the “direction of causality” in this context.pageLeaving Certificate 2012Page 13 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

(d)A property is said to be in “negative equity” if the person owes more on the mortgage than theproperty is worth. A report about mortgaged properties in Ireland in December 2010 has thefollowing information: Of the 475 136 properties examined, 145 414 of them were in negative equity. Of the ones in negative equity, 11 644 were in arrears. There were 317 355 properties that were neither in arrears nor in negative equity.(i)What is the probability that a property selected at random (from all those examined) willbe in negative equity?Give your answer correct to two decimal places.(ii)What is the probability that a property selected at random from all those in negativeequity will also be in arrears?Give your answer correct to two decimal places.(iii) Find the probability that a property selected at random from all those in arrears will alsobe in negative equity.Give your answer correct to two decimal places.Leaving Certificate 2012Page 14 of 19Project Maths, Phase 1Paper 2 – Higher Level

(e)The study described in part (d) was so large that it can be assumed to represent thepopulation. Suppose that, in early 2012, researchers want to know whether the proportion ofproperties in negative equity has changed. They analyse 2000 randomly selected propertieswith mortgages. They discover that 552 of them are in negative equity. Use a hypothesis testat the 5% level of significance to decide whether there is sufficient evidence to conclude thatthe situation has changed since December 2010.Be sure to state the null hypothesis clearly, and to state the conclusion clearly.pageLeaving Certificate 2012Page 15 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Question 8(75 marks)The diagram is a representation of a robotic arm that canmove in a vertical plane. The point P is fixed, and so arethe lengths of the two segments of the arm. The controllercan vary the angles α and β from 0 to 180 .QβRαP(a)Given that PQ 20 cm and QR 12 cm, determine the values of the angles α and β so as tolocate R, the tip of the arm, at a point that is 24 cm to the right of P, and 7 cm higher than P.Give your answers correct to the nearest degree.Leaving Certificate 2012Page 16 of 19Project Maths, Phase 1Paper 2 – Higher Level

(b)In setting the arm to the position described in part (a), which will cause the greater error in thelocation of R: an error of 1 in the value of α or an error of 1 in the value of β?Justify your answer. You may assume that if a point moves along a circle through a smallangle, then its distance from its starting point is equal to the length of the arc travelled.(c)The answer to part (b) above depends on the particular position of the arm. That is, in certainpositions, the location of R is more sensitive to small errors in α than to small errors in β,while in other positions, the reverse is true. Describe, with justification, the conditions underwhich each of these two situations arises.pageLeaving Certificate 2012Page 17 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

(d)Illustrate the set of all possible locations of the point R on the coordinate diagram below.Take P as the origin and take each unit in the diagram to represent a centimetre in reality.Note that α and β can vary only from 0 to 180 52025303540-5-10-15Leaving Certificate 2012Page 18 of 19Project Maths, Phase 1Paper 2 – Higher Level

You may use this page for extra work.pageLeaving Certificate 2012Page 19 of 19runningProject Maths, Phase 1Paper 2 – Higher Level

Leaving Certificate 2012 – Higher LevelMathematics (Project Maths – Phase 1) – Paper 2Monday 11 JuneMorning 9:30 – 12:00