2019 Prelim Exam Y6 HL Paper 1 (Questions)

PRELIMINARY EXAMINATION 2019 YEAR 6 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 1 Tuesday 17 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. • You are not permitted access to any calculator 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 / Year 6 / 2019 Prelim 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. 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] The discrete random variable X has probability density function ( ) 2 5 x P X x k ==  for 2,3,x= (a) Show that 15 .4k = [2] (b) Find the cumulative distribution function ( )P X x . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 2 2. [Maximum mark: 6] Given that ( ) sinf x x = , for  0,2x  and ( ) 2 2g x x =+ for  0,1x , (a) Determine if ( )fg x exists and find ( )fg x , stating its domain clearly. [3] (b) Sketch the graph of ( )y fg x= , indicating clearly the positions of the axial intercepts and turning points. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 3 3. [Maximum mark: 6] The points P , Q and R are ( ) ( )2,5,1 , 0,2, 1− and ( )3,7, 2− respectively. (a) Find the cartesian equation of the plane .PQR [4] (b) Find the volume of the pyramid formed by the origin and the points ,,PQ and .R [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 4 4. [Maximum mark: 6] The curve C has equation 2 2xye + = . Find the equation of the normal to C at the point where 0x= .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 5 5. [Maximum mark: 5] Consider the three planes below, where k is a constant. 0 2 3 3 2 3 10 1 x y z x y z x y kz + + = + + = − + =− (a) Find the value of k for which the three planes do not meet at a unique point. [3] (b) If 9,k=− find the coordinates of the point of intersection of the three planes. [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 6 6. [Maximum mark: 6] By using a suitable substitution, find cos x x dx .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 7 7. [Maximum mark: 9] (a) Use mathematical induction to prove that ( ) ( ) 2 1 111 1 1 !! n rn r r r n rn= + + +− = − − for any positive integer .n [6] (b) Hence, find the value of ( ) 20 2 3 11 ! r r rr r= ++− in the form ! A CB + . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim Exam / Paper 1 8 8. [Maximum mark: 7] A continuous random variable X has probability density function defined by ( ) , 0 1 1 , 1 32 0, otherwise xex f x x e −  =     . (a) Find ( )02PX  , leaving your answer in terms of .e [3] (b) Hence, explain why the median lies in 0 1.x [2] (c) Find the median of .X [2]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2019 Prelim 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: 17] Let 3 1 x xy e − −= for 0x . (a) Prove that 3 .xdy eydx −=− [3] (b) Find the maximum and minimum values of y in the interval  1,4 . [4] (c) Find the coordinates of the point of inflexion on the graph of 3 1 x xy e − −= . [4] [You do not need to justify that it is a point of inflexion] (d) (i) State the range of values of x where the graph is concave down. (ii) Give a reason why the graph of 3 1 x xy e − −= is decreasing for 2.x (iii) Find the equation of the horizontal asymptote of 3 1 x xy e − −= . [6]