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=arcsin5m2−42m+1331413+m2()⎛⎝⎜⎜⎞⎠⎟⎟

Yr6/5/MATAA/HP3/Aug2022 - 4 - 2. [Maximum mark: 27] This question investigates different proof techniques to lead to the proof for infinite primes. The kth Fermat number, , is defined by (a) Find and . [3] Consider the relation given by . ----------------------------- () [ means ] (b) When , the relation () becomes . Verify that this result is true. [1] (c) Show that the relation () is true for and . [4] (d) Prove by induction that where . [7] You will now deduce that no two Fermat numbers have a common factor greater than 1. Suppose is a common factor of two Fermat numbers and , where . Then and , where . (e) (i) Explain why p is also a factor of . (ii) Hence, deduce that p is a factor of . [2] (f) Use part (e) to write down the relationship between p , and . [1] (g) (i) Justify that the relationship in part (f) is impossible. (ii) Hence, using proof by contradiction, write down the conclusion. [3] (This question continues on the following page) FkFk=22k()+1,k=0,1,2,3,...F0,F1,F2F3F0F1F2...Fn−1=Fn−2∗F0F1F2...Fn−1=Fn−2F0×F1×F2×...×Fn−1=Fn−2n=1∗F0=F1−2∗n=2n=3F0F1F2...Fn−1=Fn−2Fn=22n()+1,n∈!+p>1FjFmj<mFj=apFm=bpa,b∈ZF0F1...Fj...Fm−1Fm−2FmFm−2

Yr6/5/MATAA/HP3/Aug2022 Turn over - 5 - (Question 2 continued) You will now use the result above that “no two Fermat numbers have a common factor greater than 1” to prove that there are infinite primes. (h) By considering and , list down the number of distinct positive factors of these Fermat numbers. [2] (i) Using the result “no two Fermat numbers have a common factor greater than 1”, write down the number of common factors between two Fermat numbers. [1] It is given that any positive integer can be written as a product of two distinct prime numbers. (j) Using part (i) and the above given result, deduce the conclusion about the occurrence of a prime number in a Fermat number. [2] (k) Hence, deduce that there are infinitely many primes, justifying your answer. [1] F1F2

Yr6/5/MATAA/HP3/Aug2022 - 6 - Please do not write on this page. Answers written on this page will not be marked.