ACSI 2019 Promo Paper 2

FINAL EXAMINATION 2019 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 26th 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 11 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 / 2019/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: 5] Consider the ΔABC with 18BAC = ° , 6AC = units and 3BC = units. Find the possible values of the length of AB .

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 2 2. [Maximum mark: 6] (a) Given that * 22 13 iz i −= −+ , find the values of z and arg(z). [4] (b) Hence, find the value of arg(– z). [2]

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 3 3. [Maximum mark: 4] Consider the system of equations: 21 23 22 0 xy z xyz x y kz −+ = − +−= −+= (a) Given k = 2, find the solution of the system. [2] (b) Comment on the system, with clear explanation, when k = 4. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 4 4. [Maximum mark: 7] The height of the base of a cabin of the Ferris Wheel is given by ( ) 114 113cos( ),0 5 tht t kπ= − ≤≤ , where k is an integer, t is in minutes and h is in metres above the ground in one cycle. (a) State the value of k. [1] (b) Find the maximum and the minimum height of the base of this cabin. [2] (c) Find the duration when the height is at least 200m. [4] h (t)

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 5 5. [Maximum mark: 5] There are eight boys, six girls and five adults in a party. To play a game they need to form a team of five players. How many ways can they form the team, if (a) there are no restrictions, [2] (b) there is only one adult in the team. [3] .

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 6 6. [Maximum mark: 9] Given cosPR y= and sinQR y= where ,PQ and y∈  , show that (a) 222RPQ= + [2] (b) arctan Qy P= [1] (c) cos sin cos( )P xQ xR xy+≡ − [2] (d) Hence solve 3cos 4sin 5xx+= for 05 x≤≤ . [4]

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 7 7. [Maximum mark: 7] The function f is defined by () 4fx x x= − . (a) State the largest possible domain for f . [1] (b) Sketch the graph of ()y fx= and '( )y fx= on the same axes. Showing clearly all the intercepts and turning point(s). [3] (c) Hence solve ' () ()f x fx> . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/Final Exam / Paper 2 8 8. [Maximum mark: 7] The radius of the circle in the diagram below is 1 unit. D lies on the extension of OA. Points B and C are on the x-axis. Given that units, unitsOB BC ABαβ= = = and both AB and DC are perpendicular to the x-axis. y D A 1 β O α B x (a) Show that the area of the shaded region is 12 arcsin2αβ β− sq units. [4] (b) If αβ= , show that the area is 1 8 π− sq units. [3] A 1 β OB αα C

ACS (Independent) / Mathematics Department / Mathematics HL / 2019/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: 12] Let f be a function defined by ( ) cos arcsin 5 xfx x = −  for 55 x−< < and 0x ≠ . (a) Explain clearly whether f is an even or odd function. [2] (b) Sketch the graphs of '( )y fx= and ''( )yfx= on the same axes, showing clearly all the x–intercepts. [6] (c) Hence, state the x coordinates of the points of inflection. [2] (d) State the values of x for whichf is concave up. [2] 10. [Maximum mark: 21] The points A, B, C and D have position vectors given by 1 12 , 2, 1 14 OA a OB OC b −       = = =      −         and 11 1 1 OD  =   (a) Find the values of a and b if OA   is perpendicular to the plane OBC. [4] (b) Find the Cartesian equation of the plane, 1π , that passes through D and is parallel to the plane OBC. [4] (c) Find the coordinates of the point of intersection between 1π and the line passing through AC. [5] (d) Hence, or otherwise, find the distance between A and 1π . [5] (e) Given another plane with equation, 2 :2 5x yzπ − += , find the Cartesian equation of the line of intersection of 1π and 2π . [3]