SJI 2022 Year 6 HL MAA Prelim Examination Paper 1

ST JOSEPH’S INSTITUTION YEAR 6 PRELIMINARY EXAMINATION 2022 MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 1 Monday 15 August 2022 2 hours 1400 – 1600 hrs INSTRUCTIONS TO CANDIDATES  Write your session number in the boxes above.  Write your name and your teacher’s name in the spaces provided.  Do not open this examination paper until instructed to do so.  Section A: Answer all questions showing working and answers in the spaces provided in the exam paper.  Section B: Answer all questions using the writing paper provided.  The use of calculators is not permitted in this paper.  A clean copy of the Mathematics: Analysis and Approaches formula booklet is required for this paper.  Unless otherwise stated in the question, all numerical answers are to be giv en exactly or correct to three significant figures.  The maximum mark for this examination paper is [110 marks].  This question paper consists of 12 printed pages including the Cover Sheet.  Sections A and B are to be submitted separately. _______________________________________________________________________ FOR MARKER USE ONLY: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTAL /110 STUDENT NAME: TEACHER NAME:

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P1 2 Full marks are not necessarily awarded for a c orrect answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are advised to show all working. SECTION A (55 marks) Answer all questions in the spaces provided. 1 [Maximum mark: 5] Consider a sequence of equilateral triangles, where the length of a side of each triangle is half that of the previous triangle. The length of a side of Triangle 1 is x cm, and Triangles 1 and 2 are shown below. (a) Find the area of Triangle 1 in terms of x. [2] (b) The sequence of triangles continues to infinity. Given that the total area of all the triangles is 27 3 cm2, find the value of x. [3] ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… TURN OVER x x x Triangle 1 Triangle 2

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P1 3 2 [Maximum mark: 7] Anthony cycles on the weekends for leisure. Anthony has an equal chance of cycling or not on any given Saturday. If Anthony cycles on Saturday, then he has a probability of 0.4 of not cycling on Sunday. If he does not cycle on Saturday, then he has a probability of 0.2 of not cycling on Sunday as well. (a) Find the probability that Anthony cycles on exactly one day in a typical weekend. [2] (b) Find the probability that Anthony cycled on exac tly one day per weekend over all the four weekends in June. (You do not need to evaluate your answer.) [2] (c) On a typical weekend, given that Anthony cycled on Sunday, find the probability that he also cycled on Saturday. [3] ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… TURN OVER

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P1 4 3 [Maximum mark: 6] (a) Find   2d cosd xx . [2] (b) Find the area bounded by the curve 2 sin 2 2 cos xy x   , the x-axis, and the lines 4x  and 2x  . [4] ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… TURN OVER

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P1 5 4 [Maximum mark: 7] (a) Find the coefficient of 4x in the expansion of   52 1x  . [2] (b) Hence or otherwise, find the term in 4x in the expansion of   522 11xx . [4] (c) Write down the range of values of x for which the expansion in (b) is valid. [1] ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… TURN OVER

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P1 6 5 [Maximum mark: 11] Consider the function   2 1 ln xfx x  , defined on its maximal domain. (a) Find the maximal domain of f . [3] (b) Find the equation of the tangent to the curve  y f x at the point where ex . [5] (c) Find the range of values of x for which f is incre