SJI HL Math EOY papers 2018-2022 COMBINE

STUDENT NAME: TEACHER NAME: ____________________________ ST. JOSEPH’S INSTITUTION YEAR 5 END OF YEAR EXAMINATION 2018 MATHEMATICS HIGHER LEVEL PAPER 1 Wednesday 26th September 2018 1 hr 30 mins 0800 – 0930 hrs INSTRUCTIONS TO CANDIDATES  Write your session number in the boxes above.  Write your name and teacher’s name in the spaces provided.  Do not open this examination paper until instructed to do so.  Section A: Answer all questions showing working and answers in the spaces provided in the exam paper.  Section B: Answer all questions using the foolscap paper provided.  The use of calculators is not permitted in this paper.  A clean copy of the Mathematics HL Formulae Booklet is required for this paper.  Unless otherwise stated in the question all numerical answers are to be given exactly.  The maximum mark for this examination paper is [80 marks].  This question paper consists of 10 printed pages including the Cover Sheet.  Sections A and B are to be submitted separately. _______________________________________________________________________ FOR MARKER USE ONLY: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTAL /80

Year 5 HL Maths End of Year Exam 2018/P1 2 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 advised to show all working. SECTION A (40 marks) Answer all questions in the spaces provided. 1 [Maximum mark: 4] The complex numbers 23ui and 32vi satisfy the equation 1 1 10 u v w , w . Express w in the form a ib where ,ab  and 2 1i  . ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 3 2 [Maximum: 5 marks] Let 𝑔: ℕ ⟶ ℤ+ be a piecewise function defined as 𝑔(𝑛) = { 1, if 𝑛 = 0 𝑔 (1 2𝑛) , if 𝑛 is even 1 + 𝑔(𝑛 − 1), if 𝑛 is odd (a) Find g(3). [3] (b) Does g have an inverse? Justify your answer. [2] ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 4 3 [Maximum mark: 8] (a) The term independent of x in the expansion 6 2 4 3 2xx   can be written in the form 2 3 5p q r , where ,,pqr  . Find the values of p, q and r. [4] (b) Determine    5 Im 1 2 i , where 2 1i  , leaving your answer in the form 2a , where a . [4] ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 5 4 [Maximum mark: 5] Find the coordinates of the point of inflection of the function  f where xx xe x , justifying your answer. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 6 5 [Maximum mark: 5] Determine the series of transformations that transform the circle with equation 22 4 6 8 0x x y y     into the ellipse with equation   22 4 3 20xy   . ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 7 6 [Maximum: 4 marks] Solve for x : sin (arcsin ( 1 5) + arccos(𝑥)) = 1. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. ……………………………………………………………………………………………………. TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 8 7 [Maximum mark: 9] The function g is given by   11g , , 2 1 2 xx x x x    . (a) Write down the equations of the horizontal and vertical asymptotes of g. [2] (b) In the space below, sketch the graph of   1y g x , indicating the asymptotes, critical point and point(s) of intersection with the axes. [6] (c) State the range of   1y g x . [1] TURN OVER

Year 5 HL Maths End of Year Exam 2018/P1 9 Do NOT write solutions on this page. SECTION B (40 marks) Answer all questions on the foolscap paper provided. Please start each question on a new page. 8 [Maximum mark: 12] (a) (i) Express 4cos 2𝑥 − 3sin2𝑥 in the form 𝑅cos(2𝑥 + 𝜃), where R > 0 and 𝜃 is acute, giving the exact value of R and 𝜃. [2] (ii) Hence, write down the greatest and least value of 2 4cos2𝑥−3sin2𝑥+7 . [2] (b) A curve has equation 𝑥 − 𝑦 = (𝑥 + 𝑦)2. Find 𝑑𝑦 𝑑𝑥 in terms of x and y. [4] (c) Consider another curve with equation 𝑦 = log3 𝑥 . If 𝑑2𝑦 𝑑𝑥2 + ( 𝑑𝑦 𝑑𝑥) 2 = 𝑘 𝑥2 , find the value of k in the form 1−𝑝 𝑝2 , where 𝑝 ∈ ℝ. [4] 9 [Maximum mark: 15] (a) (i) Evaluate ∑ [sin2𝑟 (5𝜋 6 )] ∞ 𝑟=0 [3] (ii) For what values of x, where 𝑥 ∈ (0, 𝜋) does the geometric series sin𝑥 + sin2𝑥 + 4sin𝑥cos2𝑥 + ⋯ exist? [5] (b) A computer password is to