ACSI 2019 Promo Paper 1

FINAL EXAMINATION 2019 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 1 24 September 2019 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 in 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 HL and Further Mathematics HL formula booklet is required for this paper. • The maximum mark for this examination paper is [100 marks] This question paper consists of 10 printed pages including this cover page. Section A (50 Marks) Section B (50 Marks) Question Marks Question Marks 1 8 2 3 9 4 5 6 10 7 Subtotal Subtotal TOTAL / 100 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / 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 in the boxes provided. Working may be continued below the lines if necessary. 1. [Maximum mark: 5] Find the value of k if � 𝑘𝑘 2𝑟𝑟 = 5 ∞ 𝑟𝑟=0 . .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 2 2. [Maximum mark: 5] Consider the equation (ℎ − 1)𝑥𝑥2 − 𝑥𝑥ℎ + (ℎ + 1) = 0 , where ℎ is a real constant. Find the set of values of h for which this equation has real roots.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 3 3. [Maximum mark: 8] The sum of the first n terms of a sequence {un} is given by 𝑆𝑆𝑛𝑛 = 𝑛𝑛2 + 𝑟𝑟2𝑛𝑛 − 3 4 𝑛𝑛, where 𝑛𝑛 ∈ 𝑍𝑍+ and 𝑟𝑟 ∈ 𝑅𝑅 (a) Write down the value of 𝑢𝑢1in terms of r. [1] (b) Prove that {un} is an arithmetic sequence, stating clearly its common difference. [7]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 4 4. [Maximum mark: 8] Use Mathematical Induction to prove that (1 + 𝑥𝑥)𝑛𝑛 ≥ 1 + 𝑛𝑛𝑥𝑥 for {𝑛𝑛: 𝑛𝑛 ∈ 𝑍𝑍+} where 𝑥𝑥 > 1.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 5 5. [Maximum mark: 8] (a) Express the binomial coefficient �𝑛𝑛 + 1 𝑛𝑛 − 2� as a polynomial in n. [3] (b) Hence find the coefficient of 𝑥𝑥5−𝑛𝑛 in the expansion of �𝑥𝑥 + 1 𝑥𝑥� 𝑛𝑛+1 in terms of 𝑛𝑛. [5]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 6 6. [Maximum mark: 7] (a) Show that 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟2 𝑥𝑥 = 1 2 log𝑟𝑟 𝑥𝑥 . [2] (b) Hence express 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑥𝑥 + 𝑙𝑙𝑙𝑙𝑙𝑙9 𝑥𝑥 in terms of 𝑙𝑙𝑛𝑛(𝑥𝑥) and describe the transformation that will map 𝑦𝑦 = 𝑙𝑙𝑛𝑛(𝑥𝑥) into 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑥𝑥 + 𝑙𝑙𝑙𝑙𝑙𝑙9 𝑥𝑥. [5]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 7 7. [Maximum mark: 9] Consider the equations of the lines: 1 :l 11 3 2 1, 01 λλ −     = +− ∈       r  and 2 :l 2 113 x yz+ = −=− . (a) Determine whether or not 1l and 2l intersect. [6] (b) Find a vector that is perpendicular to the two lines. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 8 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. 8. [Maximum mark: 16] i) (a) Expand and simplify (𝑥𝑥 − 1)(𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1). [2] (b) Given that w is a root of the equation 𝑧𝑧6 − 1 = 0 which does not lie on the real axis in the Argand diagram, show that 𝑤𝑤5 + 𝑤𝑤4 + 𝑤𝑤3 + 𝑤𝑤2 + 𝑤𝑤 + 1 = 0. [2] (c) Find the roots of the equation 𝑧𝑧6 = 1, giving your answers exactly in the form 𝑟𝑟𝑒𝑒𝑖𝑖𝑖𝑖, where −𝜋𝜋 < 𝜃𝜃 ≤ 𝜋𝜋 . [3] ii) Consider the complex numbers 𝑧𝑧1 = 1 − √3𝑖𝑖 and 𝑧𝑧2 = 2 − 2𝑖𝑖 . (a) Find the exact value of the argument of 𝑧𝑧1 𝑧𝑧2 . [3] (b) Express 𝑧𝑧1 𝑧𝑧2 in the form 𝑎𝑎 + 𝑖𝑖𝑖𝑖, where 𝑎𝑎, 𝑖𝑖 ∈ 𝑅𝑅. [3] (c) Using your results, find the exact value of 𝑡𝑡𝑎𝑎𝑛𝑛 −𝜋𝜋 12, giving your answer in the form √𝑎𝑎 − 𝑖𝑖, 𝑎𝑎, 𝑖𝑖 ∈ 𝑍𝑍+. [3] 9. [Maximum mark: 16] (a) The curve C is defined by equation 2𝑥𝑥2 − 𝑦𝑦2 + 𝑥𝑥𝑦𝑦 = 4. (i) Find 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 in terms of x and y. [3] (ii) Explain clearly whether C has any turning point/s. [3] (iii) Find the equation of the tangent to the curve C at the point 𝑃𝑃�√2, 0�. [2] (iv) The tangent to the curve C at the point 𝑃𝑃�√2, 0� meets the y-axis at the point A. Find the area of triangle PAO. [2] (b) Consider the function 𝑓𝑓 defined by 𝑓𝑓(𝑥𝑥) = 𝑒𝑒𝑑𝑑𝑐𝑐𝑙𝑙𝑐𝑐𝑥𝑥, 0 ≤ 𝑥𝑥 < 𝜋𝜋. (i) Find an expression for 𝑓𝑓′(𝑥𝑥) . [2] (ii) Express 𝑓𝑓′′(𝑥𝑥) in the form 𝐴𝐴𝑒𝑒𝑑𝑑𝑐𝑐𝑖𝑖𝑛𝑛𝑥𝑥, where A is a constant. [2] (iii) Find the coordinates of the point of inflexion/s of the graph of 𝑓𝑓. (The proof of inflexion is not required.) [2] [Turn the page over]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 9 10. [Maximum mark: 18] (a) The functions 𝑓𝑓 and 𝑙𝑙 are defined by ( ) 2 f : 2 4,xx −− + , 2∈≤xx , ( ) 2 1g : , 74 x x−−  x∈  , x ≤ k, x ≠ 5. (i) Sketch the graphs of 𝑓𝑓, 𝑓𝑓−1 and 𝑓𝑓𝑓𝑓−1 on the same axes, showing the relationship between the three graphs and their domains. [4] (ii) If 𝑓𝑓(𝑎𝑎) = 𝑓𝑓−1(𝑎𝑎), show that 𝑎𝑎2 − 3𝑎𝑎 = 0. [3] (iii) State the maximum value of k such that the inverse of 𝑙𝑙 exists. [1] (iv) Hence find the inverse of 𝑙𝑙, stating its domain. [4] (b) A rational function is defined by 𝑓𝑓(𝑥𝑥) = 𝑎𝑎 (𝑑𝑑−𝑏𝑏)(𝑑𝑑−𝑐𝑐) where the parameters 𝑖𝑖, 𝑐𝑐 > 0 and 𝑎𝑎, 𝑖𝑖 and 𝑐𝑐 𝜖𝜖 𝑍𝑍. The following diagram represents the graph of 𝑦𝑦 = 𝑓𝑓(|𝑥𝑥|). 𝑓𝑓(|𝑥𝑥|) has a maximum point (0, -0.45) and minimum points (-7, 2) and (7, 2). The horizontal asymptote of 𝑓𝑓(|𝑥𝑥|) is 𝑦𝑦 = 0 and the vertical asymptotes of 𝑓𝑓(|𝑥𝑥|) are 𝑥𝑥 = −10, 𝑥𝑥 = −4, 𝑥𝑥 = 4 and 𝑥𝑥 = 10. (i) Using the information on the graph, find the values of a, b and c. [3] (ii) Sketch the graph of 1 𝑓𝑓(|𝑥𝑥|). [3] END OF PAPER 10 -10 - 4 4 (-7, 2) (7, 2) -0.45 𝑦𝑦 = 𝑓𝑓(|𝑥𝑥|)