SOTA 2022 Year 6 MAAHL Prelim Paper 3

Yr6/5/MATAA/HP3/Aug2022 5 pages © School of the Arts, Singapore Year 6 Mathematics: analysis and approaches Higher Level Paper 3 Preliminary Examinations Tuesday 30 August 2022 1 hour Instructions to candidates • Do not open this examination paper until instructed to do so. • A graphic display calculator is required for this paper. • Answer all questions on the answer sheets provided. Write your name and class on each answer sheet, and attach them to this examination paper. • Unless otherwise stated in the question, all numerical answers should 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 [55 marks]. Question Marks 1 2 Name: ____________________________________ Class: ____________________________________ Index: ____________________________________ 55

Yr6/5/MATAA/HP3/Aug2022 - 2 - Answer all questions on the answer sheets provided. Please start each question on a new page. Full marks are not necessary 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. 1. [Maximum mark: 28] This question asks you to explore the angle between two vectors using the scalar product and the vector product. The angle between the vector and the vector is given by p in degrees. You will use the scalar product to find the angle between the two vectors. (a) Show that . [4] (b) (i) Write down the sign of when . (ii) Show that when , . (iii) Hence, deduce that there are no valid values of m for which . [6] (c) By considering the graph of p, find the smallest angle between the two vectors and state the corresponding value of m for which this occurs. [2] (d) (i) Let . Find , expressing your answer as where . (ii) Hence, find , giving your answer to one decimal place. (iii) Deduce . [5] (e) Using part (c) and part (d), deduce the range of values of p. [2] (This question continues on the following page) a=i−2j+3kb=3i−2j+mk,m∈!p=arccos7+3m1413+m2⎛⎝⎜⎞⎠⎟cospm<−3p=1503m2−84m+175=0p=150gm()=7+3m1413+m2limm→∞gm()sts,t∈Z+limm→∞plimm→−∞p

Yr6/5/MATAA/HP3/Aug2022 Turn over - 3 - (Question 1 continued) You will use the vector product to find the angle between the two vectors. (f) By considering , show that . [5] (g) By considering the graph of p in part (f), find the range of values of p. [2] (h) Explain the difference between the answers you obtained in part (e) and part (g). [2] a×bp=arcsin5m2−4