ACSI 2021 Promo Paper 2

FINAL EXAMINATION 2021 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 FRIDAY 24 SEP 2021 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 12 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 / 2021 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: 6] An infinite geometric sequence has the first term 1. The sum of the first three terms is 2. The terms in the sequence are all positive. (a) Find the common ratio. [3] (b) Find the sum of the infinite sequence. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 3 2. [Maximum mark: 6] (a) Express ( )( ) 5 13 12xx+− in partial fractions. [2] (b) Hence expand ( )( ) 5 13 12xx+− as a series of ascending powers of x up to and including the term in 3.x [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 4 3. *[Maximum mark: 4] The moment magnitude scale, M, is a measure of the strength of an earthquake, defined by 𝑀𝑀 = 2 3 log10(𝐼𝐼) − 10.7, where 𝐼𝐼 is the intensity of the quake. The 2012 Aceh earthquake had a moment magnitude of 8.6. The 2004 Indian Ocean earthquake was 8 times as intense. (a) Find the intensity of the Aceh earthquake [2] (b) Find the magnitude of the Indian Ocean earthquake on the moment magnitude scale. Give your answer correct to 1 decimal place. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 5 4. [Maximum mark: 5] (a) How many ways can the letters of the word “LOGARITHMS” be arranged such that the vowels are separated? [2] (b) A bag contains 2 white pencils, 3 black pencils and 4 red pencils. In how many ways can 3 pencils be drawn from the bag, if at least one black pencil is to be included in the draw? [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 6 5. *[Maximum mark: 8] Two fair 8-sided dice are thrown and the number showing on each die is noted. The sum of these two numbers is S. Find the probability that (a) S is at least ‘14’, [2] (b) less than two dice shows a ‘7’, [3] (c) less than two dice shows a ‘7’, given that S is at least ‘14’. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 7 6. *[Maximum mark: 8] It is speculated that there is an association between the amount of time spent online (in hours) and the amount of sleep hours per day. The following paired data are collected. (a) Find the Pearson’s product moment correlation coefficient between the time spent online and sleep hours. Comment on your answer. [3] (b) Find the equation of the regression line of y on x , where x is the time spent online and y is sleep hours . [2] (c) Find the estimated sleep hours (to 1 decimal place) if the time spent online is 6.5 hours. [1] (d) To find the estimated time spent online per day f or a student who sleeps for 3. 9 hours, give 2 reasons why the estimated time spent online using (b) might not be reliable. [2] . Time spent online (in hours), x 7.5 8 8 9 6 5 5 4 9 8.5 Sleep hours (in hours), y 6 5.5 5 6 7 7.5 8 9 5.5 4

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 8 7. [Maximum mark: 5] A continuous random variable X has a probability density function given by the function ( )fx , where 2 , 0 1 ( ) |2 | , 1 3 0 , otherwise. kx x fx k x x  ≤<= − ≤≤  (a) Determine that 3 .4k = [2] (b) Hence find the median of X. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 9 8. [Maximum mark: 5] A complex number 2ikz ki += − where k is a positive real number. (a) Given that Re( ) 0.5,z = find the value of k. [3] (b) Determine arg( ),z giving your answer in radians. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2021 Final Examination / Paper 2 10 9. [Maximum mark: 8] The equation 32 0,x px qx c+ + += where , pq and ,c∈  has distinct roots , αβ and .γ (a) By expanding ( )( )( ),xxxαβγ−−− show that .q αβ βγ γα=++ [2] (b) Show that 2 2 22( ) 2( ).α β γ α β γ αβ βγ αγ++ = + + + + + [2] It is given that 2,p=− 3q= and that one of the roots is ,di where d∈  and 1.i= − (c) Using the result from part (b), calculate the value of 2 22 .αβγ++ [1] (d) Hence or otherwise, find the roots ,αβ and .γ [3]