ACSI 2022 Y6 HL Mathematics Prelim Exam Paper 2 (Questions)

PRELIMINARY EXAMINATION 2022 YEAR 6 IB DIPLOMA PROGRAMME MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 2 Wednesday, 14 September 2022 2 hours _________________________________________________________________________ INSTRUCTIONS TO CANDIDATES  Write your session number in the boxes above.  Do not open this examination paper until instructed to do so.  A graphic display calculator is required for this paper.  Section A: answer all questions. Answers must be written within the answer boxes provided.  Section B: answer all questions on the writing paper provided. Fill in your session number on each answer sheet, and attach them to this examination paper using the string provided.  Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures.  A clean copy of the Mathematics: Analysis and Approaches formula booklet is required for this paper.  The maximum mark for this examination paper is [110 marks]  Questions with an asterisk (*) are common to both HL and SL papers This question paper consists of 14 printed pages including this cover page. Section A (55 Marks) Section B (55 Marks) Question Marks Question Marks 1 10 2 3 4 11 5 6 7 12 8 9 Subtotal Subtotal TOTAL / 110 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 1 Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. Section A Answer all questions. Answers must be written within the answer boxes provided. Working may be continued below the lines, if necessary. 1. *[Maximum mark: 4] (a) In a circle centre O, radius r, ABis any chord and AOB  where 0    . Find an expression for the area of the minor segment cut off by ABin terms of r and . [1] (b) If this area is one quarter of the area of the circle, show that 2 2sin    . [1] (c) Find the value of AOB . [2]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 2 2. *[Maximum mark: 4] The derivative of a function f is given by ' sin cosf x x x , where x. The graph of f passes through the point ,03     . Find f x in exact form.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 3 3. *[Maximum mark: 6] Two events Xand Yare not independent and their respective probabilities of occurring are given by 0.6P X and 0.7P Y . It is given that 0.95P X Y  . Calculate a) P X Y [2] b)  P Y X [2] c) 'P X Y [2]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 4 4. *[Maximum mark: 7] The table below shows the number of hours spent on the computer in a week for 10 adults. Adult A B C D E F G H I J No. of hours 71 21 82 35 11 75 43 73 59 33 (a) Find the mean hours and variance for these ten adults. [2] (b) A box and whisker plot for the data is drawn. Write down the values of a, b and c. [3] (c) An error was spotted in the data recorded and q hours are subtracted from each timing recorded. State the new mean and standard deviation of the weekly hours spent on the computer in terms of q. [2] 11 a b c 82

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 5 5. *[Maximum mark: 7] A particle is moving along a straight line, and t seconds after it passes the point O of the line, its velocity is v m/s where ln( )v A t B   and A and Bare constants. It is also known that when 10t , the acceleration of the particle is1 20 m/s2 and when 100t , the particle comes to rest. (a) Find the exact values of Band of A. [4] (b) Find the distance travelled in the 4th second. [3]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 6 6. [Maximum mark: 5] There are 13 people in a tour group and they are assigned the above seats on the plane they will be taking. Find the number of ways the tour group can be seated on the plane if (a) there are no restrictions; [2] (b) the Lee family (comprising Mr & Mrs Lee and their 3 children) is to be seated in a row next to one another (having aisles in between them is fine) with the 3 children between Mr & Mrs Lee. [3] A I S L E A I S L E

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 7 7. [Maximum mark: 9] Consider the function  ln(sec )xf x x for 0x . (a) Use l’Hopital’s rule to evaluate 0limx f x . [2] (b) Show that the Maclaurin series for ln(sec )x , as far as the term in 4x is 2 41 1 2 12x x . [4] (c) The region R is bounded by the xaxis, the curve  ln(sec )xf x x and the line 2y x  . Find the volume of the solid of revolution formed by rotating R completely about the xaxis. [3]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 8 Q7 continues here.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 6 / 2022 Preliminary Examination / Paper 2 9 8. [Maximum mark: 6] Consider the differential equation 2d d yy x ex with boundary condition 1, 0.5x y  . (a) By Euler’s method with step size 0.1h , find an approximate value of y when 1.3x .[3] (b) Find the particular solution of the differential equation. [3]