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Mathematics: analysis andapproachesHigher level and Standard levelSpecimen papers 1, 2 and 3First examinations in 2021

CONTENTSMathematics: analysis and approaches higher levelpaper 1 specimen paperMathematics: analysis and approaches higher levelpaper 1 markschemeMathematics: analysis and approaches higher levelpaper 2 specimen paperMathematics: analysis and approaches higher levelpaper 2 markschemeMathematics: analysis and approaches higher levelpaper 3 specimen paperMathematics: analysis and approaches higher levelpaper 3 markschemeMathematics: analysis and approaches standard levelpaper 1 specimen paperMathematics: analysis and approaches standard levelpaper 1 markschemeMathematics: analysis and approaches standard levelpaper 2 specimen paperMathematics: analysis and approaches standard levelpaper 2 markscheme

SPEC/5/MATAA/HP1/ENG/TZ0/XXMathematics: analysis and approachesHigher levelPaper 1Specimen paperCandidate session number2 hoursInstructions to candidates Write your session number in the boxes above.Do not open this examination paper until instructed to do so.You are not permitted access to any calculator for this paper.Section A: answer all questions. Answers must be written within the answer boxes provided.Section B: a nswer all questions in the answer booklet provided. Fill in your session numberon the front of the answer booklet, and attach it to this examination paper and yourcover sheet using the tag provided. Unless otherwise stated in the question, all numerical answers should be given exactly orcorrect to three significant figures. A clean copy of the mathematics: analysis and approaches formula booklet is required forthis paper. The maximum mark for this examination paper is [110 marks].13 pages International Baccalaureate Organization 201916EP01

–2–SPEC/5/MATAA/HP1/ENG/TZ0/XXFull marks are not necessarily awarded for a correct answer with no working. Answers must besupported by working and/or explanations. Where an answer is incorrect, some marks may be givenfor a correct method, provided this is shown by written working. You are therefore advised to show allworking.Section AAnswer all questions. Answers must be written within the answer boxes provided. Working may becontinued below the lines, if necessary.1.[Maximum mark: 5]Let A and B be events such that P (A) 0.5 , P (B) 0.4 and P (A B) 0.6 .Find P (A B) . 16EP02

–3–2.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 5](a)Show that (2n - 1)2 (2n 1)2 8n2 2 , where n . [2](b)Hence, or otherwise, prove that the sum of the squares of any two consecutive oddintegers is even. [3] Turn over16EP03

–4–3.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 5]Let f ′( x) 8x2x2 1. Given that f (0) 5 , find f (x) . 16EP04

–5–4.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 5]The following diagram shows the graph of y f (x) . The graph has a horizontal asymptoteat y -1 . The graph crosses the x-axis at x -1 and x 1 , and the y-axis at y 2 .y43y f (x)21 4 3 21 1234x 1 2On the following set of axes, sketch the graph of y [ f (x)]2 1 , clearly showing anyasymptotes with their equations and the coordinates of any local maxima or minima.y654321 4 3 2 101234x 1 2Turn over16EP05

–6–5.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 5]The functions f and g are defined such that f ( x) x 3and g (x) 8x 5 .4(a)Show that ( g f )(x) 2x 11 . [2](b)Given that ( g f )-1(a) 4 , find the value of a . [3] 16EP06

–7–6.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 8](a)Show that log 9 (cos 2 x 2) log 3 cos 2 x 2 . [3](b)Hence or otherwise solve log3 (2 sin x) log9 (cos 2x 2) for 0 x . [5]2 Turn over16EP07

–8–7.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 7]A continuous random variable X has the probability density function f given by x x sin , 0 x 6f ( x) 36 6 . 0,otherwise Find P (0 X 3).

Mathematics: analysis and approaches standard level . paper 1 markscheme . Mathematics: analysis and approaches standard level . paper 2 specimen paper . Mathematics: analysis and approaches standard level . paper 2 markscheme . Candidate session number Mathematics: analysis and approaches Higher level Paper 1 13 pages Specimen paper 2 hours 16EP01 nstructions to candidates Write your session .

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