ACSI 2017 Promo Paper 1

FINAL EXAMINATION 2017 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 1 11th 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.  No calculators are allowed 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 / Year 5 / 2017 Final Exam / Paper 1 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] (a) Find the term independent of x in the expansion of 6 2 2x x   . [3] (b) Prove that 11 nn nr rr r          . [ 3 ]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 2 2. [Maximum mark: 5] The diagram shows the graph of f . (a) On the same diagram, sketch the graph of 1f . [4] (b) Hence find the point of intersection of ()f x and 1()f x . [1]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 3 3. [Maximum mark: 5] The cubic polynomial ()f x and quadratic polynomial ()p x are related by the following equation: () ()( 1 ) 4fx p x x   . The roots of ()f x are at 2x and 3x . Find 32()f x x Ax Bx C   , where A , B and C are constants to be determined.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 4 4. [Maximum mark: 5] The graph of function  () c o s2 s i n2fx x x for 14 x  is shown below. (a) The function f can al so be expressed as () s i n2fx a x b where a  and 02 b . Find the value of a and b . [2] (b) Find '( )fx , hence show that  cos 2 sin 2 2 cos 2 8xx x      . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 5 5. [Maximum mark: 5] Determine the relationship between the following lines. Explain your answer clearly 1 11 2 :8 5 , 34 lk k              r  2 23 :1 1 , 95 lh h     r  .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 6 6. [Maximum mark: 5] Use mathematical induction to prove that 74 n is divisible by 3 for n  .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 7 7. [Maximum mark: 7] The functions f and g are defined by   2 1fx x   , x  and   1g xx , ,1x x . (a) State the ranges for f and g . [2] (b) Give a reason why fg exists. [2] (c) Find the composite function fg and state its domain. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 8 8. [Maximum mark: 12] (a) Given the equation of the curve 22 22 4xx y y  , find dy dx in terms of x and y . Hence find the value(s) of x when the tangent is parallel to the x-axis. [8] (b) Differentiate arctan x with respect to x . [ 4 ]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 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: 22] Consider the function () , ax bfx cx d   x  , dx c for non-zero constants a , b , c and d . (a) (i) Find an expression for  f x . (ii) Given that ad bc , show that  f x is an increasing function for all values of x  , dx c . [6] (b) Given 2a , 0b , 3c and 2d  , find   1f x for 2, 3xx  . (i) Hence or otherwise show that   2f xx and find   201f x . (ii) With the given values of ,,abc and d , solve the inequality ()f xx . [13] (c) The function h is defined by   sinhx p x q x r  , x  , where ,p q and r are real constants. Given that h is an odd function, find the value of r . [3] [Turn the page over]