ACSI 2017 Promo Paper 2

FINAL EXAMINATION 2017 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 Monday 16th October 2017 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 12 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 / 2017/Final 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: 6] Solve the following equations ( ) 3 2 12 log 3 xy e xy e + = +=

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 2 2. [Maximum mark: 7] At the start of the first year, a student puts in 1000 dollars into a new savings account. At the end of that year, he earned a fixed interest of 1.2% after which he withdraws a sum of p dollars. The student continues with this practice of withdrawing p dollars annually for a total of 7 years. If the total balance in the account at the end of 7 years is $439.52, find p .

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 3 3. [Maximum mark: 6] Given that 1 34zi= + and ( )3 44 3zi = ++ , (a) Find 2z if 12 3zz z+= . [2] (b) Hence find the coordinates of a point B which is on a bearing of 330° and a distance of 4 km from point A, which has coordinates ( )3, 4 from a given origin. Leave your final answer in surd form. [4]

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 4 4. [Maximum mark: 5] How many four digit numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 which are even, without repeating the digits? [5]

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 5 5. [Maximum mark: 5] Given the function ( ) ( ) 22( ) ln 3 1f x x x ax= ++− + is an even function, find a .

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 6 6. [Maximum mark: 7] Given a function 2 2() 1 xefx x += + , find the range of values of x for which (a) ' () ()f x fx≤ . [4] (b) ()fx is concave downwards. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 7 7. [Maximum mark: 7] The plane 1π is given by 121 012 2 11 λµ        = ++      −    r (a) Find a unit vector ˆn normal to the plane and the equation of plane 1π in ˆ φ=rn form. [4] (b) Hence find the possible equations of plane 2π , given that it is parallel to 1π and a distance of 33 units from it. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 8 8. [Maximum mark: 7] A regular hexagon ABCDEF has a circumscribed circle of radius r and centre at O, and a second regular polygon is drawn such that its sides are tangential to the same circle. Adjacent vertices of both hexagons are concurrent with the centre of the circle. A drawing of the arrangement is shown below. (a) Find the perimeters of both hexagons, leaving your answer in terms of r and in surd form. [4] (b) Hence show that 3 23π≤≤ . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / 2017/Final Exam / Paper 2 9 Do not write solutions on this page. Section B Answer all questions in the answer sheets provided. Please start each question on a new page. 9. [Maximum mark: 18] (a) Solve ( ) ( ) 32 3 3 21 0z z iz i− +− − −= [4] (b) Given that cos sinzi θθ= + , (i) Show that sin cos *i izθθ+= , where *z is the complex conjugate of z . (ii) Expand ( ) 3 sin cosiθθ+ and hence show that 3sin 3 3sin 4sinθθ θ= − . [5] (c) A polynomial is given by the equation 8() 1fz z = + . (i) Find the zeros of ()fz , leaving your answers in ireθ form. (ii) Sketch all the zeros of ()fz in an Argand diagram. (iii) Hence show that 357cos cos cos cos 088 8 8 ππππ+++= . [9] 10. [Maximum mark: 17] (a) A triangle ABC is such that 10AC = m, 7BC = m and ˆ 40CAB= ° . (i) Find the possible length of the side AB and the area of the smaller triangle. (ii) Hence calculate the ratio of the area of the smaller to the larger triangle. [8]