ACSI 2018 Promo Paper 1

FINAL EXAMINATION 2018 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 1 8 October 2018 2 hours _________________________________________________________________________ INSTRUCTIONS TO CANDIDATES • Write your session number in the boxes above. • Do not open this examination paper until instructed to do so. • No calculators are allowed 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 information booklet is required for this paper. • The maximum mark for this examination paper is [100 marks] This question paper consists of 11 printed pages including this cover page. Section A (50 Marks) Section B (50 Marks) Question Marks Question Marks 1 9 2 3 10 4 5 6 11 7 8 Subtotal Subtotal TOTAL / 100 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2018 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. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. 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 in the boxes provided. Working may be continued below the lines if necessary. 1. [Maximum mark: 5] Find the term independent of x in the expansion of ( 1 √𝑥 − 1) 4 (1 − 𝑥)2. [5] ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………….

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2018 Final Examination Paper 1 2 2. [Maximum mark: 5] (i) Show that 𝟏 √𝟒𝐫−𝟑+√𝟒𝐫+𝟏 = √𝟒𝐫+𝟏−√𝟒𝐫−𝟑 𝟒 [3] (ii) Hence find the exact value of ∑ 𝟏 √𝟒𝐫−𝟑+√𝟒𝐫+𝟏 𝟏𝟎𝟎 𝐫=𝟏 . [2] ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………….

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2018 Final Examination Paper 1 3 3. [Maximum mark: 5] Consider the geometric series 1 + 𝑡𝑎𝑛𝜃 + 𝑡𝑎𝑛2𝜃 + 𝑡𝑎𝑛3𝜃+. . .. such that − π 4 < θ < π 4 . Find the values of 𝜃 for which the sum to infinity S∞ < 𝟑+√𝟑 𝟐 . ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………….

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2018 Final Examination Paper 1 4 4. [Maximum mark: 6] The equation of a curve is given by 4𝑥3 + 𝑥𝑦2 = 5𝑥𝑦. The curve meets the line 𝑦 = 𝑥 at the origin and the point P. Find the equation of the tangent to the curve at P. [6] ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………… …………………………………………………………………………………………………………….

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2018 Final Examination Paper 1 5 5. [Maximum mark: 9] (a) There are 4 married couples, 1 single man and 1 single woman. Find the number of ways these 10 people could be arranged in a row if the couples are with their spouse. [3] (b) By mathematical induction, prove that the formula for sum to n terms of a Geometric Progression whose first term is a and common ratio, r is given by 𝑺𝒏 = 𝒂(𝒓𝒏−𝟏) 𝒓−𝟏 [6] ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………….