Year 11 IB MATHEMATICS SL EXAMINATION PAPER 2

Transcription

1 Year 11 IB MATHEMATICS SL EXAMINATION PAPER Semester Question and Answer Booklet STUDENT NAME: TEACHER(S): Mr Rodgers, Ms McCaughey TIME ALLOWED: Reading time 5 minutes Writing time 90 minutes INSTRUCTIONS * 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 in the answer booklet provided. * Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. * A clean copy of the Mathematics SL formula booklet is required for this paper. STRUCTURE OF BOOKLET / MARKINGSCHEME Number of questions Number of questions to be answered Total marks Students are not permitted to bring mobile phones and / or any other unauthorized electronic devices into the examination room.

2 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 the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 7 marks] a) Let f ( x) x 8x 9. (i) Write down the coordinates of the vertex. (ii) Hence or otherwise, express the function in the form f ( x) ( x h) k. (4) b) Solve the equation f( x) 0. (3) 1

3 . [Maximum mark: 7 marks] a) Describe two transformations whose composite transforms the graph of y f ( x) to the x graph of y 3 f. () x b) Sketch the graph of y 3ln () x c) Sketch the graph of y 3ln 1, marking clearly the positions of any asymptotes and x - intercepts. (3)...

4 3. [Maximum mark: 8 marks] Jose takes medication. After t minutes, the concentration of medication left in his bloodstream 0.014t is given by At ( ) 10(0.5), where A is in milligrams per litre. a) Write down A (0). (1) b) Find the concentration of medication left in his bloodstream after 50 minutes. () c) At 13:00, when there is no medication in Jose s bloodstream, he takes his first dose of medication. He can take his medication again when the concentration of medication reaches milligrams per litre. What time will Jose be able to take his medication again? (5) 3

5 4. [Maximum mark: 6 marks] Consider the line L with equation y + x = 3. The line L 1 is parallel to L and passes through the point (6, 4). (a) Find the gradient of L 1. () (b) Find the equation of L 1 in the form y = mx + b. () (c) Find the x-coordinate of the point where line L 1 crosses the x-axis. () 4

6 5. [Maximum mark: 7 marks] Consider f ( x) is given below. x, for x and g( x) sin e x, for x. The graph of f( x ) a) On the diagram above, sketch the graph of g. (3) b) Solve f ( x) g( x). () c) Write down the set of values of x such that f ( x) g( x). () 5

7 6. [Maximum mark: 5 marks] Solve the equation x e 4sin x, for 0 x. (5) 7. [Maximum mark: 5 marks] The quadratic equation kx k x ( 3) 1 0 has two equal real roots. Find the possible values of k. (5)... 6

8 Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 16 marks] x f( x) log log 16 log 4, for x 0. Let a) Show that f ( x) log3 x. () b) Find the value of f (0.5) and of f (4.5). (3) c) The function f can also be written in the form (i) Write down the value of a and of b. ln ax f( x). ln b (ii) Hence on graph paper, sketch the graph of f( x ), for 5 x 5, 5 y 5, using a scale of 1 cm to 1 unit on each axis. (iii) Write down the equation of the asymptote. (6) d) Write down the value of f 1 (0). (1) e) The point A lies on the graph of f( x ). At A, x 4.5. On your diagram, sketch the graph of f 1 ( x), noting clearly the image of point A. (4) 7

9 9. [Maximum mark: 9 marks] Consider the function 3x 5 f( x) x. a) Write down the equation of the horizontal asymptote of the graph y f ( x). (1) 1 b) Show that f( x) 3 x () 1 c) Hence describe a translation vector which transforms the graph of y to the graph of x y f ( x). () d) Find an expression for f 1 ( x) and state the domain. (3) e) Describe the transformation which transforms the graph of y f ( x) to the graph of f 1 ( x). (1) 10. [Maximum mark:8] A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by h t t t 0 5, 0 (a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). () (b) Show that the height of the ball after one second is 17 metres. () (c) At a later time the ball is again at a height of 17 metres. (i) Write down an equation that t must satisfy when the ball is at a height of 17 metres. (ii) Solve the equation algebraically. (4) END OF EXAM 8