ACSI 2022 Promo Paper 1

FINAL EXAMINATION 2022 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS : ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 1 Tuesday 20 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. • You are not permitted access to any calculator for this paper. • Section A: answer all questions. Answers must be written within the 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 & 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 HL / Year 5 / 2022 Final Examination / Paper 1 1 Full marks are not necessarily awarded for a correct 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 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: 5] Solve the following equation. 9 12 3 27 0xx−  + =

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 2 2.* [Maximum mark: 6] Given that 33 2 9 9log 1 log lg lg 2lg 9 125 5xy= + = − + , find the values of x and y .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 3 3.* [Maximum mark: 6] (a) Show that ln x , 2ln x and 3ln x are 3 consecutive terms of an arithmetic progression. [2] (b) Solve 10 1 ln 55k k x = = . [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 4 4.* [Maximum mark: 7] (a) Find the term independent of x in the expansion of 6 2 1px x  + in terms of p . [5] (b) Given that the term independent of x is 60, find the values of p . [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 5 5. [Maximum mark: 7] (a) Factorise 32 99x x x− − + completely. [3] (b) Hence, solve the inequality 32 99 02 x x x x − − + − . [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 6 6. [Maximum mark: 4] Express 2 1 2 3 2 x xx + −− in partial fractions.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 7 7. [Maximum mark: 7] Consider the following system of equations where k . ( ) 2 3 2 4 2 3 10 x y z x y z x y k z k + − = + + = + + − = (a) Find the value of k for which the system of equations does not have any solution. [3] (b) Find the general solution if the system has an infinite number of solutions. [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 8 8. [Maximum mark: 6] Two unit vectors a and b are given such that 23+ = −a b b a . (a) Show that 1 2=ab . [4] (b) Hence find the angle between a and b . [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 1 9 9. [Maximum mark: 7] The above diagram shows the triangle ABC such that BD is perpendicular to AC, ABD x= and CBD y= . (a) Given that ( )arctanxa= and ( ) 2tan 2 abxy ab ++= − where ,ab + , find the value of tan y in terms of b . [4] (b) If 2tan 2 bx b −= + , show that 1 2 3arctan arctan 4ab    +=       . [3] y A B C D x