SJI 2024 Year 6 HL MAA Prelim Examination Paper 2

ST. JOSEPH’S INSTITUTION YEAR 6 PRELIMINARY EXAMINATION 2024 MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 2 Tuesday 20 August 2024 2 hours 1300 – 1500 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 a scientific or examination graphical calculator is permitted in this paper. • TI-Nspire calculators must be in Press-to-Test mode and cleared of all previous data. • TI-84+ graphical calculators must only have permitted apps and be ram cleared. • 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 given exactly or correct to three significant figures. • The maximum mark for this examination paper is [110 marks]. • This question paper consists of 14 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 Q11 TOTAL /110 STUDENT NAME: TEACHER NAME:

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 2 Full marks are not necessarily awarded for a c orrect answer with no working. Answers must be supported by working and/or explanations. In particular, 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 th is 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: 6] The weights, W, of newborn babies in Singapore are normally distributed with a mean of 3.24 kg and standard deviation 375 g. A newborn baby is considered a big baby if it weighs more than w kg. 10% of newborn babies are considered big babies. (a) Find the value of w. [3] (b) A particular newborn baby is considered a big baby. Find the probability that the baby weighs at least 4 kg. [3] ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 3 2 [Maximum mark: 4] Given that a and b are two non-zero vectors such that 2  =  abvb b , find the value of ( )−a v b and hence show that ( )−  = −a v b a v b . ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. TURN OVER

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 4 3 [Maximum mark: 5] The Student Council comprises 30 councillors, out of which a 6-member executive committee (EXCO) is formed. The EXCO includes a President and a Vice-President of different genders. (a) Given that 4 female councillors and 4 male councillors were shortlisted for the President and Vice-President, in how many ways can the EXCO be formed from the 30 councillors? [3] (b) During a n EXCO photo-taking session, the EXCO is seated in a line such that the President and Vice -President are in the centre. In how many ways can the EXCO be arranged for the photo? [2] ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 5 4 [Maximum mark: 5] Consider the polynomial function ( ) 3 4 3( 3)( 6 10)P x x x x x= + − + − . Its expanded form is given by ( ) 7 6 5 4 3 2P x x Ax Bx Cx Dx Ex Fx G= + + + + + + + where each coefficient is a constant real number. (a) Find the value of A . [2] (b) Find the product of the roots of ( ) 0Px = . [2] (c) Find the product of the roots of ( ) 1Px = . [1] ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. TURN OVER

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 6 5 [Maximum mark: 8] Let ( ) 2 πsinπ2f x x =  , where 0x . The nth minimum point on the graph of f has x-coordinate nx where n + . (a) The graph of ( )2 sinπyx= undergoes a horizontal scaling with scale factor k, resulting in the graph of ( )y f x= . Write down the exact value of k. [1] (b) Given that 1nnx x a+ =+ , find the value of a. [3] (c) Hence find the least value of N such that 1 500 N n n x =  . [4] ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P2 7 6 [Maximum mark: 6] Let ( ) 2 7f x x x= + − for xa where a is a positive real number. (a) Find exactly, the smallest possible value of a such that the function is defined. [2] Using the value of a found in part (a), a solid, S , is formed when the region bounded by ( )y f x