Year 11 IB MATHEMATICS SL EXAMINATION PAPER 1

Transcription

1 Year 11 IB MATHEMATICS SL EXAMINATION PAPER 1 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. * 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 in the answer booklet 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 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. 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: 6 marks] The following diagram shows part of the graph of f, where f ( x) x x. a) Find both x-intercepts. (4) b) Find the x-coordinate of the vertex. () 1

3 . [Maximum mark: 5 marks] Let f ( x) x and g( x) x 3. a) Find g 1 ( x). () b) Find ( f g )(4). (3)

4 3. [Maximum mark: 7 marks] Let log3 p 6 and log3 q 7. (a) Find log3 p. () (b) Find log3 p q. () (c) Find log 3(9 p ). (3) 3

5 4. [Maximum mark: 6 marks] a) Given that m 8 and n 16, write down the value of m and of n. () x1 x3 b) Hence or otherwise solve (4). 4

6 5. [Maximum mark: 7 marks] Let f ( x) 3sin( x). a) Write down the amplitude of f( x ). (1) b) Find the period of f( x ). () c) On the following grid, sketch the graph of y f ( x), for 0 x 3. (4) 5

7 6. [Maximum mark: 7 marks] Let f ( x) ln( x 5) ln, for x 5. a) Find f 1 ( x). (4) x b) Let g() x e. Find ( g f )( x ), giving your answer in the form ax b, where ab,. (3). 6

8 7. [Maximum mark: 6 marks] a) Let f ( x) x and g( x) ( 1) x. The graph of g( x ) can be obtained from the graph of f( x ) using two transformations. Give a full geometric description of each of the two transformations. () b) The graph of g( x ) is translated by the vector 3 to give the graph of hx ( ). The point ( 1,1) on the graph of f( x ) is translated to the point P on the graph of hx ( ). Find the coordinates of P. (4) 7

9 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: 10] The line L1 shown on the set of axes below has equation 3x + 4y = 4. L1 cuts the x-axis at A and cuts the y- axis at B. y Diagram not drawn to scale L 1 B L M O C A x (a) Write down the coordinates of A and B. () M is the midpoint of the line segment [AB]. (b) Write down the coordinates of M. () The line L passes through the point M and the point C (0, ). (c) Write down the equation of L. () (d) Find the exact length of (i) MC; (ii) AC. (4) 8

10 9. [Maximum mark: 14] a) Let f ( x) 3x and 5 gx ( ) 3x, for x 0. Find f 1 ( x). () b) Show that ( g f )( x). () x c) Let hx ( ) x, for x 0. The graph of hx ( ) has a horizontal asymptote at y 0. Find the y -intercept of the graph of hx ( ). () d) Hence, sketch the graph of hx ( ). (3) e) For the graph of 1 h, write down the x -intercept; (1) f) For the graph of 1 h, write down the equation of the vertical asymptote. (1) g) Given that h 1 ( a) 3, find the value of a. (3) 9

11 10. [Maximum mark: 14] The following diagram represents a large Ferris wheel, with a diameter of 100 metres. Let P be a point on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant rate, in an anticlockwise (counter-clockwise) direction. One revolution takes 0 minutes. a) Write down the height of P above ground level after (i) 10 minutes; (ii) 15 minutes. () b) Let ht () metres be the height of P above ground level after t minutes. Some values of ht () are given in the table below. (i) Show that h(8) (ii) Find h (1). (4) c) Sketch the graph of ht (), for 0 t 40. (3) d) Given that h can be expressed in the form h( t) a cosbt c, find a, b and c. (5) END OF EXAM 10