A Level Mathematics. Sample Assessment Materials

Transcription

1 A Level Mathematics Sample Assessment Materials Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 017 First certification from 018 Issue 1

2 Edexcel, BTEC and LCCI qualifications Edexcel, BTEC and LCCI qualifications are awarded by Pearson, the UK s largest awarding body offering academic and vocational qualifications that are globally recognised and benchmarked. For further information, please visit our qualification website at qualifications.pearson.com. Alternatively, you can get in touch with us using the details on our contact us page at qualifications.pearson.com/contactus About Pearson Pearson is the world's leading learning company, with 35,000 employees in more than 70 countries working to help people of all ages to make measurable progress in their lives through learning. We put the learner at the centre of everything we do, because wherever learning flourishes, so do people. Find out more about how we can help you and your learners at qualifications.pearson.com References to third party material made in this sample assessment materials are made in good faith. Pearson does not endorse, approve or accept responsibility for the content of materials, which may be subject to change, or any opinions expressed therein. (Material may include textbooks, journals, magazines and other publications and websites.) All information in this document is correct at time of publication. Original Origami Artwork designed by Beth Johnson and folded by Mark Bolitho Origami photography: Pearson Education Ltd/Naki Kouyioumtzis ISBN All the material in this publication is copyright Pearson Education Limited 017

3 Contents Introduction 1 General marking guidance 3 Paper 1 sample question paper 5 Paper 1 sample mark scheme 37 Paper sample question paper 55 Paper sample mark scheme 81 Paper 3 sample question paper 97 Paper 3 sample mark scheme 13

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5 Introduction The Pearson Edexcel Level 3 Advanced GCE in Mathematics is designed for use in schools and colleges. It is part of a suite of AS/A Level qualifications offered by Pearson. These sample assessment materials have been developed to support this qualification and will be used as the benchmark to develop the assessment students will take. The booklet Mathematical Formulae and Statistical Tables will be provided for use with these assessments and can be downloaded from our website, qualifications.pearson.com. Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 1

6 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

7 General marking guidance All candidates must receive the same treatment. Examiners must mark the last candidate in exactly the same way as they mark the first. Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than be penalised for omissions. Examiners should mark according to the mark scheme not according to their perception of where the grade boundaries may lie. All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate s response is not worthy of credit according to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification/indicative content will not be exhaustive. However different examples of responses will be provided at standardisation. When examiners are in doubt regarding the application of the mark scheme to a candidate s response, a senior examiner must be consulted before a mark is given. Crossed-out work should be marked unless the candidate has replaced it with an alternative response. Specific guidance for mathematics 1. These mark schemes use the following types of marks: M marks: Method marks are awarded for knowing a method and attempting to apply it, unless otherwise indicated. A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned. B marks are unconditional accuracy marks (independent of M marks) Marks should not be subdivided.. Abbreviations These are some of the traditional marking abbreviations that may appear in the mark schemes. bod benefit of doubt ft follow through this symbol is used for correct ft cao correct answer only cso correct solution only. There must be no errors in this part of the question to obtain this mark isw ignore subsequent working awrt answers which round to SC: special case o.e. or equivalent (and appropriate) d dependent or dep indep independent dp decimal places sf significant figures The answer is printed on the paper or ag- answer given Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 3

8 or d The second mark is dependent on gaining the first mark 3. All M marks are follow through. All A marks are correct answer only (cao.), unless shown, for example, as A1 ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but answers that don t logically make sense e.g. if an answer given for a probability is >1 or <0, should never be awarded A marks. 4. For misreading which does not alter the character of a question or materially simplify it, deduct two from any A or B marks gained, in that part of the question affected. 5. Where a candidate has made multiple responses and indicates which response they wish to submit, examiners should mark this response. If there are several attempts at a question which have not been crossed out, examiners should mark the final answer which is the answer that is the most complete. 6. Ignore wrong working or incorrect statements following a correct answer. 7. Mark schemes will firstly show the solution judged to be the most common response expected from candidates. Where appropriate, alternative answers are provided in the notes. If examiners are not sure if an answer is acceptable, they will check the mark scheme to see if an alternative answer is given for the method used. If no such alternative answer is provided but deemed to be valid, examiners must escalate the response to a senior examiner to review. 4 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

9 Write your name here Surname Other names Pearson Edexcel Level 3 GCE Centre Number Mathematics Advanced Paper 1: Pure Mathematics 1 Candidate Number Sample Assessment Material for first teaching September 017 Time: hours You must have: Mathematical Formulae and Statistical Tables, calculator Paper Reference 9MA0/01 Total Marks Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Instructions Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided there may be more space than you need. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Answers should be given to three significant figures unless otherwise stated. Information A booklet Mathematical Formulae and Statistical Tables is provided. There are 15 questions in this question paper. The total mark for this paper is 100. The marks for each question are shown in brackets use this as a guide as to how much time to spend on each question. Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. If you change your mind about an answer cross it out and put your new answer and any working out underneath. Turn over S5459A 017 Pearson Education Ltd. 1/1/1/1/1/1/1/ *S5459A013* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 5

10 Answer ALL questions. Write your answers in the spaces provided. 1. The curve C has equation y = 3x 4 8x 3 3 dy (a) Find (i) dx (ii) d d y x (b) Verify that C has a stationary point when x = (c) Determine the nature of this stationary point, giving a reason for your answer. (3) () () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A03*

11 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued (Total for Question 1 is 7 marks) *S5459A033* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

12 . O 0.4 A O D 1cm Figure 1 B C 3cm The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre O joined to a sector AOB of a different circle, also centre O. Given that arc length CD = 3 cm, COD = 0.4 radians and AOD is a straight line of length 1 cm, (a) find the length of OD, (b) find the area of the shaded sector AOB. (3) () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question is 5 marks) 4 8 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A043*

13 3. A circle C has equation DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA x + y 4x + 10y = k where k is a constant. (a) Find the coordinates of the centre of C. () (b) State the range of possible values for k. () (Total for Question 3 is 4 marks) *S5459A053* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

14 4. Given that a is a positive constant and a t + 1 dt = ln 7 a t show that a = lnk, where k is a constant to be found. (4) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question 4 is 4 marks) 6 10 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A063*

15 5. A curve C has parametric equations DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA x = t 1, y = 4t t, t 0 Show that the Cartesian equation of the curve C can be written in the form y = x + ax + b, x 1 x + 1 where a and b are integers to be found. (3) (Total for Question 5 is 3 marks) *S5459A073* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

16 6. A company plans to extract oil from an oil field. The daily volume of oil V, measured in barrels that the company will extract from this oil field depends upon the time, t years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions: The initial daily volume of oil extracted from the oil field will be barrels. The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels. The daily volume of oil extracted will decrease over time. The diagram below shows the graphs of two possible models. V (0, ) O (4, 9000) Model A t V (0, ) O (4, 9000) Model B (a) (i) Use model A to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. (ii) Write down a limitation of using model A. (b) (i) Using an exponential model and the information given in the question, find a possible equation for model B. (ii) Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. t () (5) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8 1 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A083*

17 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 6 continued (Total for Question 6 is 7 marks) *S5459A093* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

18 7. B A C Figure Figure shows a sketch of a triangle ABC. Given AB = i + 3j + k and BC = i 9j + 3k, show that BAC = to one decimal place. (5) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A0103*

19 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued (Total for Question 7 is 5 marks) *S5459A0113* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

20 8. f ( x ) = ln( x 5) + x 30, x >.5 (a) Show that f ( x ) = 0 has a root α in the interval [3.5, 4] A student takes 4 as the first approximation to α. Given f(4) = and fʹ(4) = to 4 significant figures, (b) apply the Newton-Raphson procedure once to obtain a second approximation for α, giving your answer to 3 significant figures. (c) Show that α is the only root of f(x) = 0 () () () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1 16 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A013*

21 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 8 continued (Total for Question 8 is 6 marks) *S5459A0133* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

22 9. (a) Prove that nπ tanθ + cot θ cosec θ, θ, n Z (4) (b) Hence explain why the equation tanθ + cot θ = 1 does not have any real solutions. (1) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question 9 is 5 marks) Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A0143*

23 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 10. Given that θ is measured in radians, prove, from first principles, that the derivative of sin θ is cos θ You may assume the formula for sin(a ± B) and that as h 0, sin h 1 and cos h h h 1 0 (5) (Total for Question 10 is 5 marks) *S5459A0153* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

24 11. An archer shoots an arrow. The height, H metres, of the arrow above the ground is modelled by the formula H = d 0.00d, d 0 where d is the horizontal distance of the arrow from the archer, measured in metres. Given that the arrow travels in a vertical plane until it hits the ground, (a) find the horizontal distance travelled by the arrow, as given by this model. (b) With reference to the model, interpret the significance of the constant 1.8 in the formula. (1) (c) Write d 0.00d in the form where A, B and C are constants to be found. A B(d C) It is decided that the model should be adapted for a different archer. The adapted formula for this archer is Hence or otherwise, find, for the adapted model H = d 0.00d, d 0 (d) (i) the maximum height of the arrow above the ground. (ii) the horizontal distance, from the archer, of the arrow when it is at its maximum height. () (3) (3) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 16 0 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A0163*

25 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 11 continued (Total for Question 11 is 9 marks) *S5459A0173* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

26 1. In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the experiment were counted. N and T are expected to satisfy a relationship of the form N = at b, where a and b are constants (a) Show that this relationship can be expressed in the form log 10 N = mlog 10 T + c giving m and c in terms of the constants a and/or b. log 10 N Figure log 10 T Figure 3 shows the line of best fit for values of log 10 N plotted against values of log 10 T (b) Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment. (c) Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds () (4) () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (d) With reference to the model, interpret the value of the constant a. (1) 18 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A0183*

27 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued *S5459A0193* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

28 Question 1 continued DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 0 4 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A003*

29 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued (Total for Question 1 is 9 marks) *S5459A013* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

30 13. The curve C has parametric equations x = cos t, y = 3 cos t, 0 t π (a) Find an expression for d y dx in terms of t. () The point P lies on C where t = π 3 The line l is the normal to C at P. (b) Show that an equation for l is x 3y 1 = 0 The line l intersects the curve C again at the point Q. (c) Find the exact coordinates of Q. You must show clearly how you obtained your answers. (5) (6) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A03*

31 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 13 continued *S5459A033* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

32 Question 13 continued DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4 8 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A043*

33 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 13 continued (Total for Question 13 is 13 marks) *S5459A053* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

34 14. y C S O 1 3 x Figure 4 Figure 4 shows a sketch of part of the curve C with equation y = x ln x 3 x + 5, x > 0 The finite region S, shown shaded in Figure 4, is bounded by the curve C, the line with equation x = 1, the x-axis and the line with equation x = 3 The table below shows corresponding values of x and y with the values of y given to 4 decimal places as appropriate. x y (a) Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of S, giving your answer to 3 decimal places. (b) Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S. (c) Show that the exact area of S can be written in the form a are integers to be found. b + ln c, where a, b and c (In part c, solutions based entirely on graphical or numerical methods are not acceptable.) (3) (1) (6) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6 30 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A063*

35 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 14 continued *S5459A073* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

36 Question 14 continued DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8 3 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A083*

37 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 14 continued (Total for Question 14 is 10 marks) *S5459A093* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

38 15. y O P Q Figure 5 Figure 5 shows a sketch of the curve with equation y = f ( x ), where f ( x ) = 4 sin x, 0 x π x 1 e The curve has a maximum turning point at P and a minimum turning point at Q as shown in Figure 5. (a) Show that the x coordinates of point P and point Q are solutions of the equation tan x = (b) Using your answer to part (a), find the x-coordinate of the minimum turning point on the curve with equation (i) y = f ( x ). (ii) y = 3 f ( x ). x (4) (4) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A0303*

39 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 15 continued *S5459A0313* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

40 Question 15 continued (Total for Question 15 is 8 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA TOTAL FOR PAPER IS 100 MARKS 3 36 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5459A033*

41 Paper 1: Pure Mathematics 1 Mark Scheme Question Scheme Marks AOs 1(a) (b) (c) Notes: (i) (ii) d d y 3 1x 4x x M1 A1 d y dx 1.1b 1.1b 36x 48x A1ft 1.1b Substitutes x = into their dy dx (3) M1 1.1b Shows d y 0 and states ''hence there is a stationary point'' A1.1 dx Substitutes x = into their d y dx () M1 1.1b d y 48 0 and states ''hence the stationary point is a minimum'' A1ft.a dx (a)(i) M1: Differentiates to a cubic form dy 3 A1: 1x 4x dx (a)(ii) d y A1ft: Achieves a correct d dy 36x 48x x x for their d (b) M1: Substitutes x = into their d y dx A1: Shows d y = 0 and states ''hence there is a stationary point'' All aspects of the proof dx must be correct (c) d y M1: Substitutes x = into their dx Alternatively calculates the gradient of C either side of x A1ft: For a correct calculation, a valid reason and a correct conclusion. Follow through on an incorrect d y dx () (7 marks) Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

42 Question Scheme Marks AOs (a) Uses sr 3 r 0.4 M1 1. (b) Uses angle AOB 0.4 Notes: Uses area of sector OD 7.5 cm A1 1.1b or uses radius is (1 7.5 ) cm M1 3.1a 1 1 (1 7.5) ( 0.4) r M1 1.1b = 7.8cm A1ft 1.1b () (3) (5 marks) (a) M1: Attempts to use the correct formula s r with s 3and 0.4 A1: OD = 7.5 cm (An answer of 7.5cm implies the use of a correct formula and scores both marks) (b) M1: AOB 0.4 may be implied by the use of AOB = awrt.74 or uses radius is (1 their 7.5 ) M1: Follow through on their radius (1 their OD) and their angle A1ft: Allow awrt 7.8 cm. (Answer ). Follow through on their (1 their 7.5 ) Note: Do not follow through on a radius that is negative. 38 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

43 Question Scheme Marks AOs 3(a) Attempts x y 5... M1 1.1b Centre (, 5) A1 1.1b (b) Sets k 5 0 M1.a () Notes: (a) k 9 A1ft 1.1b M1: Attempts to complete the square so allow x y 5... A1: States the centre is at (, 5). Also allow written separately x, y 5 (, 5) implies both marks (b) M1: Deduces that the right hand side of their x y A1ft: k 9 Also allow is > 0 or 0 k Follow through on their rhs of x y () (4 marks) Question Scheme Marks AOs 4 Notes: Writes t 1 1 dt 1 dt and attempts to integrate M1.1 t t tln t c M1 1.1b a a a a ln ln ln 7 M1 1.1b 7 a ln with 7 k A1 1.1b (4 marks) M1: Attempts to divide each term by t or alternatively multiply each term by t -1 1 M1: Integrates each term and knows dt ln t. The + c is not required for this mark t M1: Substitutes in both limits, subtracts and sets equal to ln7 A1: 7 7 Proceeds to a ln and states k or exact equivalent such as 3.5 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

44 Question Scheme Marks AOs 5 Attempts to substitute x 1 into Attempts to write as a single fraction y (x5)( x1) 6 ( x 1) x 1 6 y y 4 7 ( x 1) M1.1 M1.1 Notes: x 3x1 y a3, b1 x 1 M1: x 1 3 Score for an attempt at substituting t or equivalent into y 4t 7 + t M1: Award this for an attempt at a single fraction with a correct common denominator. x 1 Their 4 7 term may be simplified first A1: Correct answer only y x 3x1 x 1 a3, b1 A1 1.1b (3 marks) 40 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

45 Question Scheme Marks AOs 6 (a)(i) barrels B1 3.4 (ii) Gives a valid limitation, for example The model shows that the daily volume of oil extracted would become negative as t increases, which is impossible States when t 10, V 1500 which is impossible 64 States that the model will only work for 0 t 7 B1 3.5b (b)(i) Suggests a suitable exponential model, for example V Ae kt M Uses 0,16000 and 4,9000 in e k dm1 3.1b () 1 9 k ln awrt M1 1.1b 1 9 ln t 4 16 V 16000e or V 16000e 0.144t A1 1.1b (ii) Uses their exponential model with t 3 V awrt barrels B1ft 3.4 Notes: (a)(i) B1: 10750barrels (a)(ii) B1: See scheme (b)(i) M1: Suggests a suitable exponential model, for example V Ae kt t, V Ar or any other suitable function such as V Ae kt bwhere the candidate chooses a value for b. dm1: Uses both 0,16000 and 4,9000 in their model. With V Ae kt candidates need to proceed to t With V Ar candidates need to proceed to With V Ae kt b e k r candidates need to proceed to 4 (5) (7 marks) b e k b where b is given as a positive constant and Ab M1: Uses a correct method to find all constants in the model. A1: Gives a suitable equation for the model passing through (or approximately through in the case of decimal equivalents) both values 0,16000 and 4,9000. Possible equations for the model could be for example V 0.144t 16000e V t (b)(ii) B1ft: Follow through on their exponential model V 0.146t 15800e 00 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

46 Question Scheme Marks AOs 7 Attempts AC AB BC i3jki9j3k3i6j4k M1 3.1a Attempts to find any one length using 3-d Pythagoras M1.1 Finds all of AB 14, AC 61, BC 91 A1ft 1.1b Notes: cos BAC M angle BAC * A1* 1.1b (5) (5 marks) M1: Attempts to find AC by using AC AB BC M1: Attempts to find any one length by use of Pythagoras' Theorem A1ft: Finds all three lengths in the triangle. Follow through on their AC M1: Attempts to find BAC using cos BAC AB AC BC AB AC Allow this to be scored for other methods such as cos BAC AB. AC AB AC A1*: This is a show that and all aspects must be correct. Angle BAC = Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

47 Question Scheme Marks AOs 8 (a) f (3.5) = 4.8, f (4) = (+)3.1 M1 1.1b (b) (c) Change of sign and function continuous in interval [3.5, 4] Root * Attempts x f( x ) 0 1 x0 f( x0 ) A1*.4 () x1 4 M1 1.1b x 1 = 3.81 A1 1.1b y = ln(x 5) () y 30 x M1 3.1a Notes: Attempts to sketch both y = ln(x 5) and y = 30 x States that y = ln(x 5) meets y = 30 x in just one place, therefore y = ln(x 5) = 30 x has just one root f (x) = 0 has just one root A1.4 () (6 marks) (a) M1: Attempts f(x) at both x = 3.5 and x = 4 with at least one correct to 1 significant figure A1*: f (3.5) and f(4) correct to 1 sig figure (rounded or truncated) with a correct reason and conclusion. A reason could be change of sign, or f (3.5) f (4) 0 or similar with f(x) being continuous in this interval. A conclusion could be 'Hence root' or 'Therefore root in interval' (b) f( x0 ) M1: Attempts x1 x0 evidenced by x1 4 f( x0 ) A1: Correct answer only x (c) M1: For a valid attempt at showing that there is only one root. This can be achieved by Sketching graphs of y = ln(x 5) and y = 30 x on the same axes Showing that f(x) = ln(x 5) + x 30 has no turning points Sketching a graph of f(x) = ln(x 5) + x 30 A1: Scored for correct conclusion Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

48 Question Scheme Marks AOs 9(a) sin cos tan cot cos sin M1.1 Notes: (a) sin cos A1 1.1b sincos 1 1 sin (b) States tancot 1sin AND no real solutions as 1 sin 1 M1: Writes sin tan cos and cos cot sin A1: sin cos Achieves a correct intermediate answer of sincos M1: Uses the double angle formula sin sincos A1*: Completes proof with no errors. This is a given answer. M1.1 cosec * A1* 1.1b (4) B1.4 (1) (5 marks) Note: There are many alternative methods. For example 1 tan 1 sec 1 1 tan cot tan then as the tan tan tan sin cos cossin cos main scheme. (b) B1: Scored for sight of sin and a reason as to why this equation has no real solutions. Possible reasons could be 1 sin 1...and therefore sin or sin arcsin which has no answers as 1 sin 1 44 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

49 Question Scheme Marks AOs 10 Notes: Use of sin( h) sin ( h) Uses the compound angle identity for sin( A B) with A, B h sin( h) sincosh+ cossin h Achieves B1.1 M1 1.1b sin( h ) sin sincos h+ cossin h sin A1 1.1b h h sin h cos h1 cos sin h h Uses h 0, sin h 1 h and cos h 1 0 h sin( h) sin Hence the limith0 cos and the gradient of ( h) dy the chord gradient of the curve cos * d B1: States or implies that the gradient of the chord is M1.1 A1*.5 sin( h ) sin or similar such as h sin( ) sin for a small h or M1: Uses the compound angle identity for sin(a + B) with A, B h or A1: Obtains sincos h+ cossin h sin or equivalent h M1: Writes their expression in terms of sin h h A1*: Uses correct language to explain that d y d cos 1 and h h cos For this method they should use all of the given statements h 0, sin h 1 h, cos h 1 sin( h) sin 0 meaning that the limith cos h 0 ( h) and therefore the gradient of the chord gradient of the curve dy cos d (5 marks) Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

50 Question Scheme Marks AOs 10alt sin( h) sin Use of ( h) h h h h sin sin sin( h) sin Sets ( h) h and uses the compound angle identity for sin(a + B) and h h sin(a B) with A, B sin( ) sin Achieves h h h h h h h h h h sin cos + cos sin sin cos cos sin h h sin h cos h h sin h h Uses h 0, 0 hence 1and cos cos h sin( h) sin Therefore the limith0 cos and the gradient of ( h) dy the chord gradient of the curve cos * d B1.1 M1 1.1b A1 1.1b M1.1 A1*.5 (5 marks) Additional notes: A1*: Uses correct language to explain that d y cos d. For this method they should use the h sin h h (adapted) given statement h 0, 0 hence 1with cos cos h sin( h) sin meaning that the limith0 cos and therefore the gradient of the ( h) dy chord gradient of the curve cos d 46 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

51 Question Scheme Marks AOs 11(a) Notes: (a) Sets H d d M1 3.4 Solves using an appropriate method, for example d 0.00 dm1 1.1b Distance awrt 04 m only A1.a (b) States the initial height of the arrow above the ground. B1 3.4 (c) d d d d M1 1.1b 0.00 ( d 100) M1 1.1b ( d 100) A1 1.1b (d) (i).1 metres B1ft 3.4 (ii) 100 metres B1ft 3.4 M1: Sets H d0.00d 0 M1: Solves using formula, which if stated must be correct, by completing square (look for d (3) (1) (3) () (9 marks) d..) or even allow answers coming from a graphical calculator A1: Awrt 04 m only (b) B1: States it is the initial height of the arrow above the ground. Do not allow '' it is the height of the archer'' (c) M1: Score for taking out a common factor of 0.00 from at least the d and d terms M1: For completing the square for their d 00d A1: ( d 100) or exact equivalent (d) B1ft: For their ' ' =.1m B1ft: For their 100m term Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

52 Question Scheme Marks AOs 1 (a) b b N at log N log a log T M log10 N log10 a blog10 T so 10 m band c log a A1 1.1b (b) intercept Uses the graph to find either a or b a 10 or b = gradient M1 3.1b () Uses the graph to find both a and b intercept a 10 and b = gradient M1 1.1b Uses T 3in N b at with their a and b M1 3.1b Number of microbes 800 A1 1.1b (c) N log N 6 M (4) (d) We cannot extrapolate the graph and assume that the model still holds States that 'a' is the number of microbes 1 day after the start of the experiment A1 3.5b () B1 3.a (1) (9 marks) 48 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

53 Question 1 continued Notes: (a) M1: Takes logs of both sides and shows the addition law M1: Uses the power law, writes log10 N log10 ablog10 T and states m band c log10 a (b) M1: Uses the graph to find either a or b the sight of 1.8 b.3 or a10 63 M1: Uses the graph to find both a and b by the sight of 1.8 b.3 and a10 63 intercept a 10 or b = gradient. This would be implied by intercept a 10 and b = gradient. This would be implied b M1: Uses T 3N at with their a and b. This is implied by an attempt at A1: Accept a number of microbes that are approximately 800. Allow following correct work. There is an alternative to this using a graphical approach. M1: Finds the value of log T from T =3. Accept as T 3 log T 0.48 M1: Then using the line of best fit finds the value of log10 Accept log10 N.9 N from their ''0.48'' '.9' M1: Finds the value of N from their value of log N log N.9 N 10 A1: Accept a number of microbes that are approximately 800. Allow following correct work (c) M1 For using N = and stating that log10 N 6 A1: Statement to the effect that ''we only have information for values of log N between 1.8 and 4.5 so we cannot be certain that the relationship still holds''. ''We cannot extrapolate with any certainty, we could only interpolate'' There is an alternative approach that uses the formula. M1: Use N = in their N 63T log10 63 log10 T A1: The reason would be similar to the main scheme as we only have log10 T values from 0 to 1.. We cannot extrapolate the graph and assume that the model still holds (d) B1: Allow a numerical explanation T 1 N a1 b N a giving a is the value of N at T =1 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

54 Question Scheme Marks AOs 13(a) dy dy Attempts dt dx dx dt M1 1.1b dy 3 sin t dx sin t 3 cost A1 1.1b () (b) Substitutes t 3 dy 3 sin t M1.1 dx sin t in Uses gradient of normal = d 3 d M1.1 3 Coordinates of P = 1, B1 1.1b Correct form of normal y x M1.1 3 Completes proof x 3y1 0 * A1* 1.1b (5) (c) Substitutes x = cos t and y = 3 cos t into x 3y1 0 M1 3.1a Uses the identity Substitutes their cos t cos t 1 to produce a quadratic in cost M1 3.1a Finds 1cos t4cost5 0 A1 1.1b 5 1 cos t, M cost into x = cos t, y = 3 cos t, M1 1.1b 6 Q 5 7, A1 1.1b (6) (13 marks) 50 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

55 Question 13 continued Notes: (a) dy dy M1: Attempts dt dx dx and achieves a form sin t k Alternatively candidates may apply the sin t dt sin tcost double angle identity for cos t and achieve a form k sin t 3 sin t A1: Scored for a correct answer, either or 3 cost sin t (b) M1: For substituting t 3 in their d y d which must be in terms of t x M1: Uses the gradient of the normal is the negative reciprocal of the value of d y. This may be dx seen in the equation of l. B1: States or uses (in their tangent or normal) that P = 1, M1: Uses their numerical value of normal at P dy 1 with their d x 1, 3 3 to form an equation of the A1*: This is a proof and all aspects need to be correct. Correct answer only x 3y1 0 (c) M1: For substituting x = cos t and y = 3 cos t into x 3y1 0 to produce an equation in t. Alternatively candidates could use y Ax B. M1: Uses the identity cos t cos t 1 cos t cos t 1 In the alternative method it is for combining their an equation in just one variable to set up an equation of the form to produce a quadratic equation in cost A1: For the correct quadratic equation 1cos t4cost5 0 Alternatively the equations in x and y are 3x x 5 0 y Ax B with x 3y 1 0 to get 1 3y 4y M1: Solves the quadratic equation in cost (or x or y) and rejects the value corresponding to P. M1: 5 5 Substitutes their cost or their t arccos in x = cos t and y = cos t If a value of x or y has been found it is for finding the other coordinate. A1: Q 5 7, Allow 5 7 x, y 3 but do not allow decimal equivalents Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

56 Question Scheme Marks AOs 14(a) Uses or implies h = 0.5 B1 1.1b For correct form of the trapezium rule = M1 1.1b (b) (c) 0.5 A1 1.1b (3) Any valid statement reason, for example Increase the number of strips B1.4 Decrease the width of the strips Use more trapezia (1) For integration by parts on ln xdx M1.1 3 x x ln x dx A1 1.1b 3 3 All integration attempted and limits used Area of S = x 5d x x 5 x c B1 1.1b x3 x ln x x x 1 x5 dx ln x x 5x x1 M1.1 Uses correct ln laws, simplifies and writes in required form M1.1 Area of S = 8 ln 7 ( 8, 7, 7) 7 a b c A1 1.1b (6) (10 marks) 5 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

57 Question 14 continued Notes: (a) 0.5 B1: States or uses the strip width h = 0.5. This can be implied by the sight of... in the trapezium rule M1: For the correct form of the bracket in the trapezium rule. Must be y values rather than x values first y value last y value sum of other y values A1: (b) B1: See scheme (c) M1: Uses integration by parts the right way around. Look for 3 x ln xdx Ax ln x Bx dx A1: 3 x x ln x dx 3 3 B1: Integrates the x 5 term correctly x 5x M1: All integration completed and limits used a M1: Simplifies using ln law(s) to a form ln c b A1: Correct answer only 8 ln 7 7 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

58 Question Scheme Marks AOs 15(a) Attempts to differentiate using the quotient rule or otherwise M1.1 Notes: (a) e 8cosx4sinx e x1 x1 x x1 A1 1.1b f() e 1 Sets f() x 0and divides/ factorises out the e x terms M1.1 Proceeds via sin x 8 to tan cos x 4 x * A1* 1.1b (b) (i) Solves tan 4x and attempts to find the nd solution M1 3.1a (4) x 1.0 A1 1.1b (ii) Solves tan x and attempts to find the 1 st solution M1 3.1a x A1 1.1b M1: Attempts to differentiate by using the quotient rule with u 4sin x and v 1 x alternatively uses the product rule with u 4sin x and v e A1: For achieving a correct f() x. For the product rule f ( x) e 8cos x4sin x e 1 x 1 x (4) 1 e x or (8 marks) M1: This is scored for cancelling/ factorising out the exponential term. Look for an equation in just cos x and sin x A1*: Proceeds to tan x. This is a given answer. (b) (i) M1: Solves tan 4x attempts to find the nd solution. Look for x arctan 4 Alternatively finds the nd solution of tan x and attempts to divide by A1: Allow awrt x 1.0. The correct answer, with no incorrect working scores both marks (b)(ii) M1: Solves tan x attempts to find the 1 st solution. Look for x arctan A1: Allow awrt x The correct answer, with no incorrect working scores both marks 54 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017

59 Write your name here Surname Other names Pearson Edexcel Level 3 GCE Centre Number Mathematics Advanced Paper : Pure Mathematics Candidate Number Sample Assessment Material for first teaching September 017 Time: hours You must have: Mathematical Formulae and Statistical Tables, calculator Paper Reference 9MA0/0 Total Marks Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Instructions Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided there may be more space than you need. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Answers should be given to three significant figures unless otherwise stated. Information A booklet Mathematical Formulae and Statistical Tables is provided. There are 16 questions in this question paper. The total mark for this paper is 100. The marks for each question are shown in brackets use this as a guide as to how much time to spend on each question. Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. If you change your mind about an answer cross it out and put your new answer and any working out underneath. Turn over S5460A 017 Pearson Education Ltd. 1/1/1/1/1/1/ *S5460A016* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited

60 Answer ALL questions. Write your answers in the spaces provided. 1. f ( x ) = x 3 5 x + ax + a Given that (x + ) is a factor of f ( x ), find the value of the constant a. (3) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question 1 is 3 marks) 56 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A06*

61 . Some A level students were given the following question. DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Solve, for 90 < θ < 90, the equation cos θ = sin θ The attempts of two of the students are shown below. Student A cos θ = sin θ tan θ = θ = 63.4 (a) Identify an error made by student A. Student B cos θ = sin θ cos θ = 4 sin θ 1 sin θ = 4sin θ sin θ = 1 sinθ =± Student B gives θ = 6.6 as one of the answers to cos θ = sin θ. (b) (i) Explain why this answer is incorrect. (ii) Explain how this incorrect answer arose θ =± 6.6 (1) () (Total for Question is 3 marks) *S5460A036* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

62 3. Given y = x(x + 1) 4, show that dy dx = ( x + 1) n ( Ax + B) where n, A and B are constants to be found. (4) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4 58 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A046* (Total for Question 3 is 4 marks)

63 4. Given f (x) = e x, x DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA g (x) = 3lnx, x > 0, x (a) find an expression for gf(x), simplifying your answer. () (b) Show that there is only one real value of x for which gf(x) = fg(x) (3) (Total for Question 4 is 5 marks) *S5460A056* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

64 5. The mass, m grams, of a radioactive substance, t years after first being observed, is modelled by the equation According to the model, m = 5e 0.05t (a) find the mass of the radioactive substance six months after it was first observed, (b) show that d m = km, where k is a constant to be found. dt () () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question 5 is 4 marks) 6 60 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A066*

65 6. Complete the table below. The first one has been done for you. DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case. Statement The quadratic equation ax + bx + c = 0, (a ¹ 0) has real roots. (i) When a real value of x is substituted into x 6x + 10 the result is positive. (ii) If ax > b then x > (iii) b a The difference between consecutive square numbers is odd. () () () Always True Sometimes True Never True Reason It only has real roots when b 4ac > 0. When b 4ac = 0 it has 1 real root and when b 4ac < 0 it has 0 real roots. (Total for Question 6 is 6 marks) *S5460A076* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

66 7. (a) Use the binomial expansion, in ascending powers of x, to show that where k is a rational constant to be found. 1 ( 4 x) = x + kx A student attempts to substitute x = 1 into both sides of this equation to find an approximate value for 3. (b) State, giving a reason, if the expansion is valid for this value of x. (4) (1) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8 6 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A086*

67 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued (Total for Question 7 is 5 marks) *S5460A096* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

68 8. y A O B D x C Figure 1 Figure 1 shows a rectangle ABCD. The point A lies on the y-axis and the points B and D lie on the x-axis as shown in Figure 1. Given that the straight line through the points A and B has equation 5y + x = 10 (a) show that the straight line through the points A and D has equation y 5x = 4 (4) (b) find the area of the rectangle ABCD. (3) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A0106*

69 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 8 continued (Total for Question 8 is 7 marks) *S5460A0116* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

70 9. Given that A is constant and 1 4 ( ) = 3 x + A dx A show that there are exactly two possible values for A. (5) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (Total for Question 9 is 5 marks) 1 66 Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A016*

71 10. In a geometric series the common ratio is r and sum to n terms is S n DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Given 8 S = S 7 show that r = ± 1, where k is an integer to be found. k (4) 6 (Total for Question 10 is 4 marks) *S5460A0136* Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited Turn over

72 11. y y = f ( x ) O x Figure Figure shows a sketch of part of the graph y = f ( x ), where f ( x ) = 3 x + 5, x 0 (a) State the range of f (1) (b) Solve the equation f ( x ) = 1 x + 30 (3) Given that the equation f ( x ) = k, where k is a constant, has two distinct roots, (c) state the set of possible values for k. () DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 April 017 Pearson Education Limited 017 *S5460A0146*