Mathematics: Analysis And Approaches Higher Level Paper 1

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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). 16EP08

–9–8.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 7]The plane П has the Cartesian equation 2x y 2z 3 . 3 1 The line L has the vector equation r 5 µ 2 , µ , p . The acute angle between 1 p the line L and the plane П is 30 .Find the possible values of p . Turn over16EP09

– 10 –9.SPEC/5/MATAA/HP1/ENG/TZ0/XX[Maximum mark: 8]The function f is defined by f (x) e2x - 6ex 5 , x , x a . The graph of y f (x) isshown in the following diagram.y542 202x 2 4(a)Find the largest value of a such that f has an inverse function. [3](b)For this value of a , find an expression for f -1 (x) , stating its domain. [5](This question continues on the following page)16EP10

– 11 –SPEC/5/MATAA/HP1/ENG/TZ0/XX(Question 9 continued) Turn over16EP11

– 12 –SPEC/5/MATAA/HP1/ENG/TZ0/XXDo not write solutions on this page.Section BAnswer all questions in the answer booklet provided. Please start each question on a new page.10.[Maximum mark: 16]Let f ( x) (a)ln 5 xwhere x 0 , k .kxShow that f ′( x) 1 ln 5 x. [3]kx 2The graph of f has exactly one maximum point P .(b)Find the x-coordinate of P . [3]The second derivative of f is given by f ′′( x) point of inflexion Q .(c)Show that the x-coordinate of Q is2 ln 5 x 3. The graph of f has exactly onekx 31 32e . [3]5The region R is enclosed by the graph of f , the x-axis, and the vertical lines through themaximum point P and the point of inflexion Q .yPQRx(d)Given that the area of R is 3 , find the value of k . [7]16EP12

– 13 –SPEC/5/MATAA/HP1/ENG/TZ0/XXDo not write solutions on this page.11.[Maximum mark: 18](a)Express 3 3i in the form reiθ , where r 0 and -π θ π . Let the roots of the equation z 3 3 (b)[5]3i be u , v and w .Find u , v and w expressing your answers in the form reiθ , where r 0 and -π θ π . [5]On an Argand diagram, u , v and w are represented by the points U , V and W respectively.(c)Find the area of triangle UVW . (d)By considering the sum of the roots u , v and w , show thatcos12.[4]5 7 17 cos cos 0. [4]181818[Maximum mark: 21]The function f is defined by f (x) esin x .(a)Find the first two derivatives of f (x) and hence find the Maclaurin series for f (x) up toand including the x2 term. [8](b)Show that the coefficient of x3 in the Maclaurin series for f (x) is zero. [4](c)Using the Maclaurin series for arctan x and e3x - 1 , find the Maclaurin seriesfor arctan (e3x - 1) up to and including the x3 term. [6](d)Hence, or otherwise, find limx 0f ( x) 1. [3]arctan ( e3 x 1) 16EP13

Please do not write on this page.Answers written on this pagewill not be marked.16EP14

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

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