ACSI 2020 Promo Paper 2

FINAL EXAMINATION 2020 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 FRIDAY 25 SEP 2020 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 must 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] • Questions with an asterisk (*) are common to both HL and SL papers This question paper consists of 14 printed pages including this cover page. Section A (55 Marks) Section B (55 Marks) Question Marks Question Marks 1 10 2 3 4 11 5 6 7 12 8 9 Subtotal Subtotal TOTAL / 110 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 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: 4] Solve the simultaneous equations 3 24 8 log log 3 xy xy = +=

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 3 2. *[Maximum mark: 3] Prove that the product of two odd numbers is odd.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 4 3. *[Maximum mark: 9] It is speculated that there is an association between the height of a student, x centimetres and the time taken, y seconds, to do a 100 m sprint. The following paired data are collected.. (a) Find the Pearson’s product moment correlation coefficient between the height and the time taken. Comment on your answer. [3] (b) Find the equation of the regression line of y on x , where x is the height of the students and y is the time taken to do a 100 m sprint. [2] (c) (i) Find the expected time taken if the height of the student is 168 cm. [1] (ii) Find the expected height of the student if he takes 11.8s to run 100m. [3] Height ( cm) 160 171 157 172 183 161 182 156 166 191 Time ( s) 16.0 15.2 14.0 15.1 12.0 12.4 11.5 13.7 13.1 11.2 x y

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 5 4. [Maximum mark: 5] Eight women and six men form a queue. Find the number of ways this can be done if (a) a particular man and woman have to stand next to each other. [2] (b) no two men are to stand next to each other [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 6 5. [Maximum mark: 5] During recess, Ryan will eat either chicken rice or chicken noodles but not both. If he eats chicken rice, the probability of him feeling full is 0.6, while if he eats chicken noodles, the probability of him feeling full is 0.7. The probability that Ryan feels full on any given day is 0.64. (a) Find the probability that Ryan eats chicken rice. [3] (b) Find the probability that Ryan eats chicken rice given that does not feel full. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 7 6. [Maximum mark: 7] The probability that an archer hits the bullseye is 0.68. For a training set, the archer will shoot six arrows at the target. On a training day, the archer will go through eight training sets. (a) Find the probability that the archer will hit the target more than thrice in a training set.[2] (b) Find the expected number of training sets in a day where the archer will hit the target more than thrice in the set. [3] (c) The archer now decides to shoot n arrows in a training set. Find the minimum value of n such that the probability of hitting the target in a training set more than thrice is more than 0.90. [2] .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 8 7. *[Maximum mark: 7] X is the random variable that denotes the weights of 17 year old male students in a school and it is normally distributed with mean µ and standard deviation .σ It is known that ( )61 0.25PX >= and ( )41 0.21.PX <= (a) Find the value of µ and .σ [5] (b) A male student is considered severely overweight if his weight lies within the top 1% of the population. Find the minimum value of X for a student to be considered severely overweight. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 9 8. [Maximum mark: 7] Let the complex number zi αβ= + where ,αβ are non-zero real numbers. (a) The complex number 2 *z z is purely imaginary. Find z in terms of α only. [4] (b) If ( )arg 0,z > find the possible values of ( )arg .z [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2020 Final Examination / Paper 2 10 9. [Maximum mark: 8] A circle with radius 8 cm and another with radius 6 cm are located such that they touch at one point. A piece of string is to be wrapped around the perimeter of the diagram below. (a) Show that 192CD= cm. [2] (b) Find the obtuse angle CBE∠ in radians. [3] (c) Hence, or otherwise, determine the length of string needed to go around the perimeter of the diagram, such that it is just sufficiently tight. [3] C E F A D B 8 cm 6 cm