Mathematics HL Year 2 Paper 2 Exam Name: Dec 2015 Block: Marks: Test Grade: Mathematics Higher Level Year 2 Paper 2 Examination 1516 90 marks 90 minutes INSTRUCTIONS TO CANDIDATES ● ● ● ● ● ● ● Do not open this examination paper until instructed to do so. A calculator is required for this paper. Section A: answer all questions in the boxes provided Section B: answer all questions in the answer booklet provided. Write your name on the front of the answer booklet, and attach it to this examination paper. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. The maximum mark for this examination paper is [90 marks]. The time allowed will be 90 mins. Grade Boundaries NOTE: The following grade boundaries are only a general guideline. If your final marks are within +/- 2 points of a communicated grade boundary, your teacher will determine the final grade based on the grade descriptors and the level of achievement demonstrated on the test. Grade Lower Bound Upper Bound 1 0 13 2 14 26 3 27 38 4 39 49 5 50 59 6 60 70 7 71 90 Mathematics HL Year 2 Paper 2 Exam Dec 2015 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: 6] A student owns two Biology textbooks and six French textbooks. (a) In how many ways can he select one Biology and three French textbooks? (b) In how many ways can he arrange the books on the shelf so that the two biology textbooks are not next to each other? .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 1 of 9 Mathematics HL Year 2 2. Paper 2 Exam Dec 2015 [Maximum mark: 5] A particular breed of chicken produces eggs whose masses are known to follow a normal distribution with mean 50 g and standard deviation 4 g. It is found that the probability of any egg with mass in excess of 60 g being 'double-yolked' is 10%. Find the probability of seeing exactly 1 double-yolked egg in a random sample of 12 eggs. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 2 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 3. [Maximum mark: 9] Find the coordinates of the stationary points of x2 + y2 – 3 + 20 = 0. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 3 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 4. [Maximum mark: 8] (a) Find a, b ∈ ℝ such that 11 + 10x – x2 = a – (x – b)2 for all x. (b) Hence find c ∈ ℝ such that: c 5 1 11 10 x x 2 dx π 6 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 4 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 5. [Maximum mark: 8] cos cos Given that θ ∈ 0, and that vectors sin and sin are perpendicular, 2 sin cos (a) Show that tan 2θ = 2. (b) Hence find the exact value of tan θ. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 5 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 6. [Maximum mark: 6] The first three terms of the binomial expansion of (x + p)n are xn + 20xn – 1+ 180xn – 2. Find the values of n and p. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 6 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 Do NOT write solutions on this page. SECTION B Answer all questions in the answer booklet provided. Please start each question on a new page. 7. [Maximum mark: 7] Vesna and Stephen each played a game at a fairground. Vesna's game involved shooting an air-rifle at moving targets. She can be expected to hit, on average, one target every 40 seconds. If she hit at least five targets in 2 minutes, she won a prize. There was no restriction on the number of shots she could take in the 2 minute period. Stephen's game involved using a bow to shoot arrows at a target. He had a 0.7 probability of hitting on each shot (independent of previous results) and had 10 arrows. If he could hit the target eight or more times, he won a prize. Given that exactly one of them won a prize, determine the probability that Vesna won it. 8. [Maximum mark: 7] A water storage tank is in the shape of a triangular prism, as shown: Water drips into the tank at a constant rate of 600 cm3s−1. (a) Show that the volume of water in the tank (in m3) when the depth is h is given by V = 2 3h2 . (b) Find the rate of increase of the depth of water in the tank at the instant the tank is a quarter full. 7 of 9 Mathematics HL Year 2 Paper 2 Exam Dec 2015 Do NOT write solutions on this page. 9. [Maximum mark: 11] iq 3iq iq (2n-1)iq (a) Find an expression for the sum of the series e + e + e5 + ...+ e . cosq + cos3q + cos5q + ... + cos(2n-1)q = (b) Hence prove that sin(2nq ) 2sinq . (c) Find all solutions to the equation cosq + cos3q + cos5q = 0 when 0 < q < p . 10. [Maximum mark: 8] 𝑋 is a continuous random variable with a pdf: 2 x a xb f x x 1 0 otherwise (a) Sketch the graph of y = 2 x . x 1 (b) Using your sketch, determine a lower bound for 𝑎 and an upper bound for b. (c) Find the values of a and b if b – a = 1.8. 11. [Maximum mark: 15] 1 3 (a) Find the angle between the lines r = 2 t 1 and r = 2 1 k 1 2 s 1 . 3 2 (b) Find the value of k for which the lines l1 and l2 intersect. (c) 3 1 (i) Calculate 1 1 . 1 2 (ii) For the value of k found in part (b), find the Cartesian equation of the plane Π containing lines l1 and l2. (d) Find the exact distance of the point (5, 1, 2) from Π. 8 of 9
