Mathematics Standard level - crunchy math

1 Mathematics Standard level Specimen papers 1 and 2 For first examinations in 2014 CONTENTS Mathematics Standard level paper 1 specimen paper Mathematics Standard level paper 1 specimen markscheme Mathematics Standard level paper 2 specimen paper Mathematics Standard level paper 2 specimen markscheme SPEC/5/MATME/SP1/ENG/TZ0/XXmathematicsst aNDaRD levelPaPeR 1 SPECIMENINSTruCTIoNS 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 answer sheets provided. Write your session number on each answer sheet, and attach them to this examination paper and your cover sheet using the tag provided.

2 At the end of the examination, indicate the number of sheets used in the appropriate box on your cover sheet. 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. The maximum mark for this examination paper is [90 marks].11 pages1 hour 30 minutes International Baccalaureate organization 2012 Examination codeXXXX XXXXC andidate session number000112 SPEC/5/MATME/SP1/ENG/TZ0/XX 2 0212 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 A (44 Marks)Answer all questions in the boxes provided.

3 Working may be continued below the lines if [Maximum mark: 7]In the following diagram, AB =u and BD = midpoint of AD is E and BD1DC3=.Express each of the following vectors in terms of u and v . (a) AE [3 marks] (b) EC [4 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 3 Turn over 03122. [Maximum mark: 5] There are nine books on a shelf. For each book, x is the number of pages, and y is the selling price in pounds ( ). Let r be the correlation coefficient. (a) Write down the possible minimum and maximum values of r .[2 marks] (b) Given that , which of the following diagrams best represents the data.[1 mark] (c) For the data in diagram D , which two of the following expressions describe the correlation between x and y ?perfect, zero, linear, strong positive, strong negative,weak positive, weak negative[2 marks].

4 XxABCD yyxyxySPEC/5/MATME/SP1/ENG/TZ0/XX 4 04123. [Maximum mark: 6] A toy car travels with velocity 1m sv for six seconds. This is shown in the graph (metres per second)yxv01234123456t (seconds) (a) Write down the car s velocity at 3t=.[1 mark] (b) Find the car s acceleration at [2 marks] (c) Find the total distance travelled.[3 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 5 Turn over 05124. [Maximum mark: 5] A data set has a mean of 20 and a Standard deviation of 6. (a) Each value in the data set has 10 added to it. Write down the value of (i) the new mean; (ii) the new Standard deviation.[2 marks] (b) Each value in the original data set is multiplied by 10. (i) Write down the value of the new mean. (ii) Find the value of the new variance.[3 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 6 06125. [Maximum mark: 7] (a) Find ed1 exxx+.

5 [3 marks] (b) Find sin 3 cos 3 dxx x .[4 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 7 Turn over 07126. [Maximum mark: 7]The expression 6 sin cosxx can be expressed in the form sinabx. (a) Find the value of a and of b .[3 marks] (b) Hence or otherwise, solve the equation 36 sin cos2xx=, for 42x .[4 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 8 08127. [Maximum mark: 7]Given that 1( )f xx=, answer the following. (a) Find the first four derivatives of ( )f x.[4 marks] (b) Write an expression for ( )( )nfx in terms of x and n .[3 marks] ..SPEC/5/MATME/SP1/ENG/TZ0/XX 9 Turn over 0912Do NOT write solutions on this B (46 Marks)Answer all the questions on the answer sheets provided. Please start each question on a new [Maximum mark: 15] Let 2( ) 3 (1)12f xx= + . (a) Show that 2( ) 369f xxx= +.

6 [2 marks] (b) For the graph of f (i) write down the coordinates of the vertex; (ii) write down the y-intercept; (iii) find both x-intercepts.[7 marks] (c) Hence sketch the graph of f .[3 marks] (d) Let 2( )g xx=. The graph of f may be obtained from the graph of g by the following two transformationsa stretch of scale factor t in the y-direction,followed by a translation of pq . Write down pq and the value of t .[3 marks]SPEC/5/MATME/SP1/ENG/TZ0/XX 10 1012Do NOT write solutions on this [Maximum mark: 14] Two Standard six-sided dice are tossed. A diagram representing the sample space is shown on second diescore on first die654321654321 Let X be the sum of the scores on the two dice. (a) (i) Find P (6)X=. (ii) Find P (6)X>. (iii) Find P (7 |6)XX=>.[6 marks] (b) Elena plays a game where she tosses two the sum is 6, she wins 3 points.

7 If the sum is greater than 6, she wins 1 point. If the sum is less than 6, she loses k points. Find the value of k for which the game is fair.[8 marks] SPEC/5/MATME/SP1/ENG/TZ0/XX 11 1112Do NOT write solutions on this [Maximum mark: 17] Let ( ) cos3 sinf xxx=+, 02x . The following diagram shows the graph of f .ABxy(0 , 1) The y-intercept is at (0 , 1), there is a minimum point at A (),p q and a maximum point at B. (a) Find ( )f x .[2 marks] (b) Hence (i) show that 2q= ; (ii) verify that A is a minimum point.[10 marks] (c) Find the maximum value of ( )f x.[3 marks] The function ( )f x can be written in the form cos ()rx a . (d) Write down the value of r and of a .[2 marks]Please do not write on this written on this page will not be do not write on this written on this page will not be 11 pages MARKSCHEME SPECIMEN Mathematics Standard level Paper 1 2 SPEC/5/MATME/SP1/ENG/TZ0/XX/M Instructions to Examiners Abbreviations M Marks awarded for attempting to use a correct Method.

8 A Marks awarded for an Answer or for Accuracy; often dependent on preceding M marks. R Marks awarded for clear Reasoning. N Marks awarded for correct answers if no working shown. AG Answer given in the question and so no marks are awarded. Using the markscheme 1 General Award marks using the annotations as noted in the markscheme eg M1, A2. 2 Method and Answer/Accuracy marks Do not automatically award full marks for a correct answer; all working must be checked, and marks awarded according to the markscheme. It is generally not possible to award M0 followed by A1, as A mark(s) depend on the preceding M mark(s), if any. An exception to this rule is when work for M1 is missing, as opposed to incorrect (see point 4). Where M and A marks are noted on the same line, M1A1, this usually means M1 for an attempt to use an appropriate method ( substitution into a formula ) and A1 for using the correct values.

9 Where there are two or more A marks on the same line, they may be awarded independently; so if the first value is incorrect, but the next two are correct, award A0A1A1. Where the markscheme specifies M2, N3, etc., do not split the marks, unless there is a note. Once a correct answer to a question or part-question is seen, ignore further working. 3 N marks If no working shown, award N marks for correct answers. In this case, ignore mark breakdown (M, A, R). Do not award a mixture of N and other marks. There may be fewer N marks available than the total of M, A and R marks; this is deliberate as it penalizes candidates for not following the instruction to show their working. If a candidate has incorrect working, which somehow results in a correct answer, do not award the N marks for this correct answer.

10 However, if the candidate has indicated (usually by crossing out) that the working is to be ignored, award the N marks for the correct answer. 3 SPEC/5/MATME/SP1/ENG/TZ0/XX/M 4 Follow through marks (only applied after an error is made) Follow through (FT) marks are awarded where an incorrect answer from one part of a question is used correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must be working present and not just a final answer based on an incorrect answer to a previous part. However, if the only marks awarded in a subpart are for the answer ( there is no working expected), then FT marks should be awarded if appropriate. Within a question part, once an error is made, no further A marks can be awarded for work which uses the error, but M marks may be awarded if appropriate.