ACSI 2019 Prelim Exam Y6 HL Paper 2 (Questions)

PRELIMINARY EXAMINATION 2019 YEAR 6 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 24 September 2019 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 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 formula booklet is required for this paper. • The maximum mark for this examination paper is [100 marks] This question paper consists of 13 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 6 / 2019 Prelim Exam / Paper 2 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] ACD is a triangle and B is a point on AC such that BC x= cm and ( )2AB x=+ cm . 48 and 50ooBCD BDCÐ= Ð= . If the area of ∆ 23.06ABD = cm2, find the value of x . B A C D

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 2 2. [Maximum mark: 7] Find the value of n if the coefficient of 𝑥𝑥3 in the expansion of (2 + 3𝑥𝑥 + 𝑥𝑥2)𝑛𝑛 is 1560 .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 3 3. [Maximum mark: 5] Solve the following system of equations: ln 2xy= − ln 2yx= −

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 4 4. [Maximum mark: 8] (a) Prove that cos 3 sin 2cos 3ttt π+=−  . [3] (b) Particle P travels with velocity = cos 3 sinvt t + m/s for 50 6t π≤≤ . (i) Find the maximum velocity. (ii) Find the exact time when it is first travelling with velocity 2 m/s. [3] (c) Another particle, Q travels with velocity v such that 1 cos 3 sin v tt = + for 0 2t π≤≤ . Find the total distance travelled by particle Q. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 5 5. [Maximum mark: 6] Let w be one of the non-real solutions to 3 1z = . (a) Determine the value of 21 ww++ . [2] (b) Hence evaluate( )( ) 2222w ww w-- . [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 6 6. [Maximum mark: 7] (a) Show that ( )( ) 2 22 5 23 2 1 1 211 21 xx x xxxx -+ - =+ -+-+ . [2] (b) Hence find ( )( ) 2 2 5 23 1 21 xx dx xx -+ -+ò . [5]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 7 7. [Maximum mark: 6] (a) Find the value(s) of k for which 44 23 x kx − =+ has no solutions. [3] (b) Find the values of x such that 4413 23 x x −<< + . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 8 8. [Maximum mark: 6] 6 men and 4 women participated in a sports competition which consisted of earlier elimination rounds. (a) In the first round the 10 contestants are grouped into two groups of 5. In how many ways can the two groups be formed if each group must have at least one woman? [3] (b) A particular male contestant was eliminated in the first round and the remaining 9 people were asked to do a photo shoot. 9 chairs were arranged with 4 chairs in the first row and 5 chairs in the second row. In how many ways can the 9 remaining contestants be arranged such that the men and women occupy alternate seats? [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 2 9 Do not write solutions on this paper. Section B Answer all questions in the answer sheets provided. Please start each question on a new page. 9. [Maximum mark: 18] (a) The number of cars sold in a car showroom in one week has a Poisson distribution with mean 3.2. (i) Show that the probability that at least 7 cars are sold in a week is 0.0446, to 3 significant figures. (ii) A random sample of 150 weeks is taken. Find the probability that at least 7 cars are sold per week in no more than 4 of the weeks. (iii) For two randomly chosen weeks, find the probability that 1 car is sold in one of the weeks given that a total of 3 cars are sold in the two weeks. [12] (b) The price of cars follows a normal distribution with mean µ and variance 30 0002. The top 10% of cars that cost above $200 000 are classified as ‘expensive’, the next 75% are classified as ‘moderate’ and the rest are classified as ‘value for money’. (i) Find the value of µ to the nearest dollar. (ii) Find, to the nearest dollar, the minimum price of a car for it to be classified as ‘moderate’. [6]