ACSI 2023 Promo Paper 2

FINAL EXAMINATION 2023 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS : ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 2 Monday 25 September 2023 2 hours _________________________________________________________________________ INSTRUCTIONS TO CANDIDATES • Write your session number in the boxes above. • Do not open this examination paper until instructed to do so. • A graphic display calculator is required for this paper. • Section A: answer all questions. Answers must be written within the answer 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 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 [110 marks] This question paper consists of 11 printed pages including this cover page. Section A (55 Marks) Section B (55 Marks) Question Marks Question Marks 1 9 2 3 4 10 5 6 7 11 8 Subtotal Subtotal TOTAL / 110 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 2 Full marks are not necessarily 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. 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: 7] Consider the geometric sequence 2,8,32,… (a) Write down the common ratio of the sequence. [1] (b) Find the largest term of the sequence that has only four digits. [3] (c) The sum of n terms of this geometric series is less than 100000. Find the largest possible value of n. [3]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 3 2. [Maximum mark: 7] (a) Prove that 2log log . aa mm= [3] (b) Hence, solve the equation 2log 3log 2 5, xx+= giving your answer in exact form. [4]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 4 3. [Maximum mark: 8] A quadratic function 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 − 𝜆𝜆)(𝑥𝑥 − 2) has axis of symmetry 𝑥𝑥 = 4.5 and 𝑦𝑦-intercept at (0, −4). (a) Find the value of 𝜆𝜆. [2] (b) Find the value of 𝑎𝑎. [2] (c) Find the possible values of k, for which the line 𝑦𝑦 = 𝑘𝑘𝑥𝑥 + 9 is a tangent to the graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). [4]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 5 4. [Maximum mark: 4] The equation 22 3 60xx− −= has roots and .αβ The equation 2 0x px q+ += has roots 2 and 2 .αβ Find p and q.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 6 5. [Maximum mark: 4] Prove that ( )1arcsin arccosec xx  ≡ . Hence or otherwise, solve ( )( ) 1sin arccosec 2 , ye −= leaving your answer in terms of e.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 7 6. [Maximum mark: 9] Consider the following functions ( ) 2sin(4 ) 0 2 () , ft t t t gt t k k π= + ≤≤ = +∈  (a) For 2,k =− find the value(s) of t, where () () , 0 2 .f t gt t π= ≤≤ [4] (b) Determine the range of k such that the graph of ()y gt= intersects the graph of ( ).y ft= [2] (c) Find the value of k where the graph of ()y gt= has the most number of intersections with the graph of ( ).y ft= State the total number of intersections.[3]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 8 7. [Maximum mark: 7] Consider a complex number ,z x iy= + such that ( )arg 1 . 5zi π−+ = (a) Find the equation of the straight line formed, in terms of its gradient and its intercept on the y-axis. [5] (b) The same line is rotated anti-clockwise by 060 about (1,-1) and results in an equation ( )arg 1 .zi β−+ = Find .β [2]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 9 8. [Maximum mark: 9] Consider two points A and B with position vectors a and b such that a 2 1 2 − =− and |b| 10.= (a) Determine the vector equation of a line passing through A that is perpendicular to both a and to the line r 31 4 3 , . 71 λλ −     = +− ∈      [4] (b) Deduce that the minimum value of |a + b| is 7 and find the maximum value of |a + b|. [3] (c) Given that |a + b| is a minimum, find b. [2]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 2 10 Do not write solutions on this page. Section B Answer all questions in the answer sheets provided. Please start each question on a new page. 9. [Maximum Mark: 25] Consider the functions: 1() 2 , . 2 x xfx x = +∈  2 1( ) , , 1 , 1 .1gx x x xx= ∈ ≠ ≠−−  (a) Show that f is an even function. [2] (b) Determine whether f has an inverse, stating your reasons clearly. [2] (c) By sketching the graphs, solve the inequality () () .f x gx> [5] (d) Find .gg [3] (e) Describe the sequence of transformations that maps the graph of ()y gx= to the graph of 2 1 5.22y x= +−+ [4] (f) Given that the domain of g is restricted to 1x<− , find the value of ( ) 1 2.g− [3] A function h is defined by 2 2() , 1 xxhx x += + where , 1.xx∈ ≠− (g) Sketch the graph of ( ),y hx= showing clearly any asymptote(s), intercept(s) and stationary point(s). [4] (h) Given a straight line ( )1,y mx= + determine the range of m such that there would be at least one intersection with the graph of ( ).y hx= [2] 10. [Maximum Mark: 11] Consider the complex numbers 2 cos sin77zi ππ= + and 228 cos sin ,77 kkwi ππ= + where .k +∈  (a) Find the modulus of zw. [1] (b) Find the argument of zw in terms of k. [2] Suppose that zw is purely real, (c) Find the least value of k for .k +∈  [3] (d) Given 4 8,pz = use de Moivre’s theorem to find all the values of p in the form , where .ireθ πθπ−<≤ [5]