AS Mathematics. q u al. io n. c r e dit. Sample Assessment Materials DRAFT

Transcription

1 AS Mathematics io n FT at q u al ac Of DR A c r e dit This draft qualification has not yet been accredited by Ofqual. It is published enable teachers have early sight of our proposed approach Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0). Further changes may be required and no assurance can be given at this time that the proposed qualification will be made available in its current form, or that it will be accredited in time for first teaching in September 07 and first award in 08. Sample Assessment Materials DRAFT Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) First teaching from September 07 First certification from 08

2 Edecel, BTEC and LCCI qualifications Edecel, 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 qualifications website at qualifications.pearson.com. Alternatively, you can get in uch 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 help people of all ages 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 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 change, or any opinions epressed therein. (Material may include tetbooks, journals, magazines and other publications and websites.) All information in this document is correct at time of publication. Original origami artwork: Mark Bolitho Origami phography: Pearson Education Ltd/Naki Kouyioumtzis ISBN All the material in this publication is copyright Pearson Education Limited 07

3 Contents Introduction General marking guidance 3 Paper sample question paper and mark scheme 5 Paper sample question paper and mark scheme 49 Mathematical formulae and statistical tables

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5 Introduction The Pearson Edecel Level 3 Advanced Subsidiary 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 support this qualification and will be used as the benchmark develop the assessment students will take. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

6 Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

7 General marking guidance All candidates must receive the same treatment. Eaminers must mark the last candidate in eactly 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. Eaminers should mark according the mark scheme not according their perception of where the grade boundaries may lie. All the marks on the mark scheme are designed be awarded. Eaminers should always award full marks if deserved, i.e. if the answer matches the mark scheme. Eaminers should also be prepared award zero marks if the candidate s response is not worthy of credit according the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and eemplification/indicative content will not be ehaustive. However different eamples of responses will be provided at standardisation. When eaminers are in doubt regarding the application of the mark scheme a candidate s response, a senior eaminer 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. These mark schemes use the following types of marks: M marks: Method marks are awarded for knowing a method and attempting 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 obtain this mark isw ignore subsequent working awrt answers which round 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 Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 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 eample, as A ft indicate that previous wrong working is 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 > 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 submit, eaminers should mark this response. If there are several attempts at a question which have not been crossed out, eaminers 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 be the most common response epected from candidates. Where appropriate, alternative answers are provided in the notes. If eaminers are not sure if an answer is acceptable, they will check the mark scheme see if an alternative answer is given for the method used. If no such alternative answer is provided but deemed be valid, eaminers must escalate the response a senior eaminer review. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 4

9 Pearson Edecel Level 3 GCE Mathematics Advanced Subsidiary Paper : Pure Mathematics Sample assessment material for first teaching September 07 Time: hours You must have: Mathematical Formulae and Statistical Tables Calcular Paper Reference(s) 8MA0/0 Candidates may use any calcular permitted by Pearson regulations. Calculars must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae sred 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 boes at the p of this page with your name, centre number and candidate number. Answer all the questions and ensure that your answers 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 make your methods clear. Answers without working may not gain full credit. Ineact answers should be given three significant figures unless otherwise stated. Information A booklet Mathematical Formulae and Statistical Tables is provided. There are 7 questions in this question paper. The tal mark for this paper is 00. The marks for each question are shown in brackets use this as a guide as how much time spend on each question. Advice Read each question carefully before you start answer it. Try 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. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 5

10 Answer ALL questions. Write your answers in the spaces provided.. The line l passes through the points A (3, ) and B (4, ). Find an equation for l. (3) (Total for Question is 3 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 6

11 . The curve C has equation y 6 Find the gradient of the curve at the point P (5, 6). (Solutions based entirely on graphical or numerical methods are not acceptable.) (4) (Total for Question is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 7

12 3. Given that the point A has position vecr 3i 7j and the point B has position vecr 8i + 3j, (a) find the vecr AB. () (b) Find AB. Give your answer as a simplified surd. () (Total for Question 3 is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 8

13 4. f() = (a) Use the facr theorem show that ( 3) is a facr of f(). () (b) Hence show that 3 is the only real root of the equation f() = 0 (4) (Total for Question 4 is 6 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 9

14 5. Given that find f( )d. f( ) 6 3, 0 (5) (Total for Question 5 is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 0

15 6. Prove, from first principles, that the derivative of 3 is 6. (4) (Total for Question 6 is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

16 7. (a) Find the first 3 terms, in ascending powers of, of the binomial epansion of 7, giving each term in its simplest form. (4) (b) Eplain how you would use your epansion give an estimate for the value of () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

17 (Total for Question 7 is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 3

18 8. Figure A triangular lawn is modelled by the triangle ABC, shown in Figure. The length AB is be 30 m long. Given that angle BAC = 70 and angle ABC = 60, (a) calculate the area of the lawn 3 significant figures. (b) Why is your answer unlikely be accurate the nearest square metre? (4) () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 4

19 Question 8 continued (Total for Question 8 is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 5

20 9. Solve, for , sin 7 cos 3 0 Give your answers one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 6

21 Question 9 continued (Total for Question 9 is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 7

22 0. The equation k 4k 3 0, where k is a constant, has no real roots. Prove that 3 0 k 4 (4) (Total for Question 0 is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 8

23 . (a) Prove that for all positive values of and y y y () (b) Prove by counter eample that this is not true when and y are both negative. () (Total for Question is 3 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 9

24 . A student was asked give the eact solution the equation 4 9( ) 0 The student s attempt is shown below: 4 9( ) 0 4 9( ) 0 Let y y 9y80 ( y8)( y) 0 y 8ory So 3 or 0 (a) Identify the two errors made by the student. () (b) Find the eact solution the equation.... () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 0

25 Question continued (Total for Question is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

26 3. (a) Facrise completely () (b) Sketch the curve with equation 3 y 0 5 showing the coordinates of the points at which the curve cuts or uches the -ais. () The point with coordinates ( 3, 0) lies on the curve with equation where a is a constant. y a a a 3 ( ) 0( ) 5( ) (c) Find the two possible values of a. (3) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07

27 Question 3 continued (Total for Question 3 is 7 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 3

28 4. Figure t A wn s population, P, is modelled by the equation P ab, where a and b are constants and t is the number of years since the population was first recorded. The line l shown in Figure illustrates the linear relationship between t and log0 P for the population over a period of 00 years. The line l meets the vertical ais at (0, 5) as shown. The gradient of l is 00. (a) Write down an equation for l. () (b) Find the value of a and the value of b. (4) (c) With reference the model interpret (i) the value of the constant a, (ii) the value of the constant b () (d) Find the population predicted by the model when t = 00, giving your answer the nearest hundred thousand. (e) State two reasons why this may not be a realistic population model. () () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 4

29 Question 4 continued (Total for Question 4 is marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 5

30 5. Figure 3 The line with equation y = 6 + cuts the curve with equation y = at the points A and B, as shown in Figure 3. (a) Find the coordinates of A and the coordinates of B. The shaded region S is bounded by the line and the curve, as shown in Figure 3. (5) (b) Find the eact area of S. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 6

31 Question 5 Continued (Total for Question 5 is 0 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 7

32 6. Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool ABCDEA consists of a rectangular section ABDE joined a semicircular section BCD as shown in Figure 4. Given that AE = metres, ED = y metres and the area of the pool is 50 m, (a) show that the perimeter, P metres, of the pool is given by 50 P (4) (b) Eplain why () Given that the pool is designed have minimum perimeter, (c) find the minimum perimeter of the pool, giving your answer 3 significant figures. (4) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 8

33 Question 6 continued (Total for Question 6 is 0 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 9

34 7. A circle C with centre at (, 6) passes through the point (0, ). (a) Show that the circle C also passes through the point (0, ). (3) The tangent the circle C at the point (0, ) meets the y ais at the point P and the tangent the circle C at the point (0, ) meets the y ais at the point Q. (b) Show that the distance PQ is 58 eplaining your method clearly. (7) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 30

35 Question 7 Continued Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 3

36 Question 7 Continued (Total for Question 7 is 0 marks) TOTAL FOR PAPER IS 00 MARKS Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 3

37 Paper : Pure Mathematics Mark Scheme Question Scheme Marks AOs (Way ) Uses y = m + c with both (3,) and (4, ) and attempt find m or c M.b m = 3 A.b c = 0 so y = o.e. A.b Or (Way ) Uses y y y y (3) with both (3,) and (4, ) M.b Gradient simplified 3 (may be implied) A.b Or (Way 3) y = o.e. A.b Uses a + by + k = 0 and substitutes both = 3 when y = and = 4 when y = with attempt solve find a, b or k in terms of one of them (3) M.b Obtains a = 3b, k = 0b or 3k = 0a A.b Obtains a = 3, b =, k = 0 Or writes 3 + y 0 = 0 o.e. A.b (3) (3 marks) Notes M: Need correct use of the given coordinates A: Need fractions simplified 3 (in ways and ) A: Need constants combined accurately N.B. Answer left in the form (y ) = 3( 3) or (y ( ))= 3( 4) is awarded MAA0 as answers should be simplified by constants being collected Note that a correct answer implies all three marks in this question. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 33

38 Question Scheme Marks AOs Attempt differentiate M.a dy 4 d A.b Substitutes 5 dy... d M.b dy 8 d Aft.b (4 marks) Notes M : Differentiation implied by one correct term A : Correct differentiation M : Attempts substitute = 5 in their derived function Aft: Substitutes = 5 in their derived function correctly i.e. Correct calculation of their f (5) so follow through slips in differentiation Question Scheme Marks AOs 3(a) Attempts ABOBOA or similar M.b AB 5i 0j A.b (b) Finds length using 'Pythagoras' () AB (5) (0) M.b AB 5 5 Aft.b () (4 marks) Notes (a) M: Attempts subtraction but may omit brackets A: cao (allow column vecr notation) (b) M: Correct use of Pythagoras theorem or modulus formula using their answer (a) Aft: AB 5 5 ft from their answer (a) Note that the correct answer implies MA in each part of this question Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 34

39 Question Scheme Marks AOs 4(a) States or uses f( 3) 0 M.b (b) 3 4(3) (3) (3) and so ( 3) is a facr Begins division or facrisation so ( 3)(4...) ( 3)(4 ) A.b () M. A.b Considers the roots of their quadratic function using completion of square or discriminant (4 ) = 0 has no real roots with a reason (e.g. negative number does not have a real square root, or So = 3 is the only real root of f() = 0 * 4 0for all ) M. A*.4 Notes (a) M: States or uses f (+3) = 0 A: See correct work evaluating and achieving zero, gether with correct conclusion (b) M: Needs have ( 3) and first term of quadratic correct A: Must be correct may further facrise ( 3)( ) M: Considers their quadratic for no real roots by use of completion of the square or consideration of discriminant then A*: a correct eplanation. (4) (6 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 35

40 Question Scheme Marks AOs f( ) 6 3 B.b Attempts integrate M.a 6 3 d =... 3 (...) A.b 3 = A.b 3 = 4 3 c A.b Notes B: Correct function with numerical powers n n M: Allow for raising power by one. A: Correct two terms A: Correct fractional terms (may be unsimplified at this stage) A: Completely correct, simplified and including constant of integration. Simplification is epected for full marks. (5 marks) Question Scheme Marks AOs 6 3( h) 3 Considers h B. Epands 3( h) 3 6h 3h M.b 6h 3h 6 3( ) so gradient = 6 3h or 6 3 h A.b States as h 0, gradient 6 so in the limit derivative 6 * A*.5 B: gives correct fraction as in the scheme above or (4 marks) Notes 3( ) 3 M: Epands the bracket as above or 3( ) 3 6 3( ) A: Substitutes correctly in earlier fraction and simplifies A*: Completes the proof, as above ( may use 0), considers the limit and states a conclusion with no errors Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 36

41 Question Scheme Marks AOs 7(a) (b) (...) Solve.995 so = 0.0 and state that 0.0 would be substituted for in the epansion M.b B.b A.b A.b (4) B.4 () (5 marks) Notes (a) M: Need correct binomial coefficient with correct power of and correct power of. Coefficients may be given in any correct form; e.g., 7, or 7 C 0, 7 C, 7 C or equivalent B: Correct answer, simplified as given in the scheme. A: Correct answer, simplified as given in the scheme. A: Correct answer, simplified as given in the scheme. (b) B: Needs a full eplanation i.e. state = 0.0 and that this would be substituted and that it is a solution of.995 Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 37

42 Question Scheme Marks AOs 8(a) (b) Finds third angle of triangle Finds third angle of triangle and uses or states and uses or states M. 30 y 30 sin 60 sin"50 " sin 70 sin"50 " 30sin 60 30sin 70 So ( 33.9) So y sin50 sin 50 ( 36.8) A.b Area = 30 sin 70 or 30 y sin 60 M 3. = 478 m Aft.b (4) Plausible reason e.g. Because the angles and the side length are not given four significant figures B 3.b Or e.g. The lawn may not be flat () (5 marks) Notes (a) M: Uses sine rule with their third angle find one of the unknown side lengths A: finds epression for, or value of either side length M: Completes method find area of triangle Aft: Obtains a correct answer for their value of or their value of y. (b) B: As information given in the question may not be accurate 4sf or the lawn may not be flat so modelling by a plane figure may not be accurate. Question Scheme Marks AOs 9 Uses sin cos ( cos ) 7cos 3 0 M 3.a cos 7 cos 0 A.b Uses solution of quadratic give cos = M.b Uses inverse cosine on their values, giving two correct follow through values (see note) M.b 430.5, A.b (5 marks) Notes M: Uses correct identity A: Correct three term quadratic M: Solves their three term quadratic give values for cos ( The correct answers are cos 3 or 4but this is not necessary for this method mark) M: Uses inverse cosine on their values, giving two correct follow through values- may be outside the given domain A: Two correct answers in the given domain Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 38

43 Question Scheme Marks AOs 0 Realises that k = 0 will give no real roots as equation becomes 3 = 0 (proof by contradiction) (For k 0) quadratic has no real roots provided b 4ac so6k k So 0 k 3 4 B 3.a M.4 4 k(4k3) 0with attempt at solution M.b, which gether with k = 0 gives 3 0 k 4 * A*. Notes B : Eplains why k = 0 gives no real roots M : Considers discriminant give quadratic inequality does not need the k 0 for this mark M : Attempts solution of quadratic inequality A* : Draws conclusion, which is a printed answer, with no errors (dependent on all three previous marks) (4 marks) Question Scheme Marks AOs (a) Way Way Longer method (b) Since and y are positive, their square roots are real and so ( y) 0 giving y y0 y y provided and y are positive and so y y * Since ( y) 0 for real values of and y, y y 0 and so 4y y y i.e. 4 y( y) y y provided and y are positive and so y y * Let = 3 and y = 5 then LHS = 5 and RHS= 4 so as 5 4 result does not apply M. A*.a () M. A*.a () B.4 (3 marks) Notes (a) M : Need two stages of the three stage argument involving the three stages, squaring, square rooting terms and rearranging. A*: Need all three stages making the correct deduction achieve the printed result. (b) B : Chooses two negative values and substitutes, then states conclusion () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 39

44 Question Scheme Marks AOs (a) 4 is wrong in line - it should be In line 4, (b) Way 4 9( ) 0 4 9( ) 0 Let y 6y 9 y 0 9 y 6 or y 0 9 log 6 9 So log 6 or log o.e. with no second answer. 4 B.3 4 has been replaced by 8 instead of by 6 B.3 Way ( 4)log log9 log 0 Notes (a) B: Lists error in line (as above) B : Lists error in line 4 (as above) (b) M: Correct work with powers reaching this equation A : Correct answer here there are many eact equivalents () M. = log9 4 log o.e. A.b () (4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 40

45 Question Scheme Marks AOs 3(a) ( 0 5) M.b (b) ( 5) A.b A cubic with correct orientation () M.b (c) Aft.b () Curve has been translated a the left M 3.a Curve passes through the origin (0, 0) and uches at ( 5, 0) (see note below for ft) a = Aft 3.a a = 3 Aft.b (7 marks) Notes (a) M: Takes out facr A: Correct facrisation - allow ( +5)( +5) (b) M: Correct shape Aft: Curve passes through the origin (0, 0) and uches at ( 5, 0) allow follow through from incorrect facrisation (c) M: May be implied by one of the correct answers for a or by a statement Aft: ft from their cubic as long as it meets the -ais only twice. Aft : ft from their cubic as long as it meets the -ais only twice. (3) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 4

46 Question Scheme Marks AOs 4(a) log0 P = mt + c M.b log0 P t 5 00 A.b () (b) Way : Way : t As P ab then t As log 0 P 5 then log Ptlog b log a 00 M. (c) (d) log t t P b or log0 a a 0 or b 0 M.b so a = or b =.06 A.b both a = and b =.06 (awrt.0) A.b (4) (i)the initial population B 3.4 (ii) The proportional increase of population each year B 3.4 () nearest hundred thousand B 3.4 () (e) Any two valid reasons- e.g. 00 years is a long time and population may be affected by wars and disease Inaccuracies in measuring gradient may result in widely 3.5b B different estimates 3.5b Population growth may not be proportional population size The model predicts unlimited growth () ( marks) Notes (a) M: Uses a linear equation relate logp and t A: Correct use of gradient and intercept give a correct line equation (b) M: Way : Uses logs correctly give log equation; Way Uses powers correctly undo log equation and epresses as product of two powers M: Way : Identifies log b or log a or both; Way : identifies a or b as powers of 0 A: Correct value for a or b A: Correct values for both (c) (i) B: Accept equivalent answers e.g. The population at t = 0 (ii) B: So accept rate at which the population is increasing each year or scale facr.0 or increase of % per year (d) B: cao (e) As given in the scheme any two valid reasons Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 4

47 Question Scheme Marks AOs 5(a) Way : Equates y6 and y = reach e.g Solves their quadratic reach two values for e.g. ( )(4 +0) = 0 so = Solves give and (b) Way (b) Way (may be used by some who have been taught along with A level candidates) 0 4 Way : Eliminates give y y y Solves their quadratic reach two values for y e.g. (y 6)(y 7) = 0 so y = M.b M.b Solves give y 6and y 7 A.b Attempts find y value from Attempts find value from their value their y value 5 Coordinates of B and A are, 7 and,6 M.b Aft.b Attempts integrate 4 n n + + by raising power M 3.a d c Aft.b Uses both their limits from (a) give area beneath curve M.b Uses the area of a trapezium find area under l with their limits (It should be l ) then subtracts from previous area give shaded area 4 (5) M 3.a Area of S = A.b (5) Subtracts 6 + from before integrating M 3.a Attempts integrate their by raising power n n M.b d c Aft.b Use both limits from (a) and subtracts M 3.a Area of S = A.b (5) Notes (a) M: Eliminates one variable reach quadratic in the other variable M: Attempts solve their quadratic give two values A: = and or equivalent (Way ) or y 6 and y 7 (Way ) 0 4 M: Substitutes back in one of the equations find other variable A: Two correct ft answers using 6 + for their values of. (b) Way M: Attempts integrate Aft: Correct integration (no need for constant of integration) (0 marks) (continued on net page) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 43

48 5 Question 5(b) continued M : Uses their values of as limits and subtracts (It should be () 6() () M : Uses area of trapezium, or triangle plus rectangle, or equivalent (e.g. integration of line equation with appropriate limits) and completes a correct method find required area (i.e. by subtraction of areas) A : cao (b) Way M : Subtracts linear epression minus quadratic epression before integrating (either way round at this stage) M : Attempts integrate their 4 n n 6 +0 by raising power Aft: Correct integration of their quadratic (terms may not have been combined and there may be sign slips) no need for +c M : Uses both limits from (a) and subtracts A : cao ) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 44

49 Question Scheme Marks AOs 6(a) (b) (c) Sets y 50 B. 50 Obtain y and substitute in P M.b Use Py with their y substituted M P * A*.b (4) > 0 and y > 0 (distance) 50 0or50 0o.e. M As and y are distances they are positive so 0 * A* 3.b Differentiates P with negative inde correct in d P d M 3.4 dp 50 d A.b Sets d P 0 d M Substitutes their in P give A.b perimeter = 59.8m. (4) () (0 marks) Notes (a) B : Correct area equation M : Rearranges their area equation make y the subject of the formula and attempt use with an epression for P M : Use correct equation for perimeter with their y substituted A*: Completely correct solution obtain and state printed answer (b) M : States > 0 and y > 0 and uses their epression from (a) form inequality A*: Eplains that and y are positive because they are distances, and uses correct epression for y give the printed answer correctly. (c) M: Attempt differentiate P (deals with negative power of correctly) A : Correct differentiation M : Sets derived function equal zero and obtains = 500 A: The value of may not be seen (it is sf or 4 ). Need see awrt 59.8m with units included for the perimeter. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 45

50 Question Scheme Marks AOs 7 (a) Way : Finds circle equation ( ) ( y6) (b) (0 ( )) ( 6) Checks whether (0,) satisfies their circle equation Obtains ( ) ( y6) 3 and checks that (0 ) (6) 3 so states that (0,) lies on C * Way : Finds distance between (,6) and (0, ) Finds distance between (,6) and (0, ) Concludes that as distance is the same (0, ) lies on the circle C * M.b M 3. A*. 6 6 Finds radius gradient or 0 ( ) 0 ( ) (m) M 3. Finds gradient perpendicular their radius using m M.b Finds (equation and ) y intercept of tangent (see note below) M.b Obains a correct value for y intercept of their tangent i.e. 35 or 3 A.b Way : Deduces gradient of Way : Deduces midpoint of second tangent PQ from symmetry ( (0,6)) M.b Finds (equation and ) y intercept of second tangent Uses this find other intercept M.b So obtains distance PQ=35+3=58* A*.b (0 marks) Notes (a) Way and Way : M : Starts use information in question find equation of circle or radius of circle M : Completes method for checking that (0, ) lies on circle A*: Completely correct eplanation with no errors concluding with statement that circle passes through (0, ) (b) 6 6 M: Calculates or (m) 0 ( ) 0 ( ) M: Finds (correct answer is or ) This is referred as m in the net note. m 5 5 M: Attempts y their ( 0) or y their ( 0) and puts = 0, or uses 5 5 y vecrs find intercept e.g. m 0 A: One correct intercept 35 or 3 (continued on net page) (3) (7) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 46

51 Qu 7(b) continued Way : M: Uses the negative of their previous tangent gradient or uses a correct or 5 5 M: Attempts the second tangent equation and puts = 0 or uses vecrs find intercept e.g. y m 0 Way : M: Finds midpoint of PQ from symmetry. (This is at (0,6)) M: Uses this midpoint find second intercept or find difference between midpoint and first intercept. e.g = 9 then 6 9 = 3 so second intercept is at ( 3, 0) Ways and : A*: Obtain 58 correctly from a valid method. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 47

52 Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 48

53 Pearson Edecel Level 3 GCE Mathematics Advanced Subsidiary Paper : Statistics and Mechanics Sample assessment material for first teaching September 07 Time: hour 5 minutes You must have: Mathematical Formulae and Statistical Tables Calcular Paper Reference(s) 8MA0/0 Candidates may use any calcular permitted by Pearson regulations. Calculars must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae sred 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 boes at the p of this page with your name, centre number and candidate number. There are two sections in this question paper. Answer all the questions in Section A and all the questions in Section B. Answer the questions in the spaces provided there may be more space than you need. You should show sufficient working make your methods clear. Answers without working may not gain full credit. Ineact answers should be given three significant figures unless otherwise stated. Information A booklet Mathematical Formulae and Statistical Tables is provided. There are 9 questions in this question paper. The tal mark for this paper is 60. The marks for each question are shown in brackets use this as a guide as how much time spend on each question. Advice Read each question carefully before you start answer it. Try 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. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 49

54 SECTION A: STATISTICS Answer ALL questions. Write your answers in the spaces provided.. Sara is investigating the variation in daily maimum gust, t kn, for Camborne in June and July 987. She used the large data set select a sample of size 0 from the June and July data for 987. Sara selected the first value using a random number from 4 and then selected every third value after that. (a) State the sampling technique Sara used. (b) From your knowledge of the large data set, eplain why this process may not generate a sample of size 0. () () The data Sara collected are summarised as follows (c) Calculate the standard deviation. å å n= t= t = () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 50

55 Question Continued (Total for Question is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 5

56 . The partially completed hisgram and the partially completed table show the time, the nearest minute, that a random sample of morists were delayed by roadworks on a stretch of morway. Delay (minutes) Number of morists Estimate the percentage of these morists who were delayed by the roadworks for between 8.5 and 3.5 minutes. (5) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 5

57 Question Continued (Total for Question is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 53

58 3. The Venn diagram shows the probabilities for students at a college taking part in various sports. A represents the event that a student takes part in Athletics. T represents the event that a student takes part in Tennis. C represents the event that a student takes part in Cricket. p and q are probabilities. The probability that a student selected at random takes part in Athletics or Tennis is 0.75 (a) Find the value of p. (b) State, giving a reason, whether or not the events A and T are statistically independent. Show your working clearly. () (3) (c) Find the probability that a student selected at random does not take part in Athletics or Cricket. () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 54

59 Question 3 Continued (Total for Question 3 is 5 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 55

60 4. Sara was studying the relationship between rainfall, r mm, and humidity, h %, in the UK. She takes a random sample of days from May 987 for Leuchars from the large data set. She obtained the following results. Sara eamined the rainfall figures and found Q= 0. Q = 0.9 Q =.4 3 A value that is more than.5 times the interquartile range (IQR) above Q3 is called an outlier. (a) Show that r = 0.6 is an outlier. (b) Give a reason why Sara might (i) include this day s reading. (ii) eclude Sara decided eclude this day s reading and drew the following scatter diagram for the remaining 0 days values of r and h. () () (c) Give an interpretation of the correlation between rainfall and humidity. () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 56

61 Question 4 Continued The equation of the regression line of r on h for these 0 days is r.8 0.5h (d) Give an interpretation of the gradient of this regression line. () (e) (i) Comment on the suitability of Sara s sampling method for this study. (ii) Suggest how Sara could make better use of the large data set for her study. () Total for Question 4 is 7 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 57

62 5. (a) The discrete random variable X ~ B(40, 0.7) Find (i) P(X = 9) (ii) P(X 6) Past records suggest that 30% of cusmers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of cusmers who buy baked beans in single tins. A random sample of 0 cusmers who had bought baked beans was taken. (b) Write down the hypotheses that should be used test the manager s suspicion. (c) Using a 0% level of significance, find the critical region for a two-tailed test answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05 (3) () (3) (d) Find the actual significance level of a test based on your critical region from part (c). Given that 9 of the 0 cusmers bought baked beans in single tins, () (e) comment on the manager s suspicion. () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 58

63 Question 5 Continued (Total for Question 5 is 9 marks) TOTAL FOR SECTION A IS 30 MARKS Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 59

64 SECTION B: MECHANICS Answer ALL questions. Write your answers in the spaces provided. Unless otherwise indicated, whenever a numerical value of g is required, take g = 9.8 m s - and give your answer either significant figures or 3 significant figures. 6. v V U 0 T t Figure A car moves along a straight horizontal road. At time t = 0, the velocity of the car is U m s -. The car then accelerates with constant acceleration a m s - for T seconds, reaching a velocity of V m s -. Figure above shows the velocity-time graph for the motion of the car for 0 t T. Using the graph, show that V = U + at. (No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.) (4) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 60

65 Question 6 Continued (Total for Question 6 is 4 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 6

66 7. A car is moving along a straight horizontal road with constant acceleration. There are three points A, B and C, in that order, on the road, where AB = m and BC = 04 m. The car takes s travel from A B and 4 s travel from B C. Find (i) the acceleration of the car, (ii) the speed of the car at the instant it passes A. (7) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 6

67 Question 7 Continued (Total for Question 7 is 7 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 63

68 8. A bird leaves its nest at time t = 0 for a short flight along a straight line. The bird then returns its nest. The bird is modelled as a particle moving in a straight horizontal line. The distance, s metres, of the bird from its nest at time t seconds is given by ( t s t t t ), where 0 0 (a) Eplain the restriction, 0 t 0 (b) Find the greatest distance of the bird from its nest. (3) (6) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 64

69 Question 8 Continued (Total for Question 8 is 9 marks) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 65

70 9. Figure A particle A of mass.5 kg is held at rest on a rough horizontal table. The particle is attached one end of a light inetensible string. The string passes over a small smooth pulley P which is fied at the edge of the table. The other end of the string is attached a particle B of mass.5 kg, which hangs freely, vertically below P and with B at a height of m above the horizontal floor. The system is released from rest, with the string taut, as shown in Figure. The resistance the motion of A from the rough table has constant magnitude.7 N. Particle B hits the floor before particle A reaches the pulley. (a) (i) Write down an equation of motion for A. (ii) Write down an equation of motion for B. (4) (b) Hence find the acceleration of B. () (c) Find the time it takes, from release, for B reach the floor. () The time taken for B reach the floor, in part (c), was calculated using the modelling assumption that the string is light and inetensible. (d) State how in your calculations you have used the assumption that the string is (i) inetensible (ii) light () Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 66

71 Question 9 Continued Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 67

72 Question 9 Continued Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 68

73 Question 9 Continued (Total for Question 9 is 0 marks) TOTAL FOR SECTION B IS 30 MARKS TOTAL FOR PAPER IS 60 MARKS Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 69

74 Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 70

75 Paper : Statistics and Mechanics Mark Scheme Question Scheme Marks AOs (a) Systematic (sample) Bcao. (b) In LDS some days have gaps because the data was not recorded B.4 (c) é 374 ù t = = 8.7 êë 0 úû 7600 s t = - t [= 30.3] 0 M.a (Accept use of t st = = ) 9 Part Notes (b) B a correct eplanation (c) M for a correct epression for t and t or A for t = awrt 5.5 or s t = awrt 5.65 = awrt 5.5 A.b (4 marks) s t. Ft an incorrect evaluation of t Question Scheme Marks AOs [ = 65] M.a (7 8 ) 4 or (6 0 ) 5 [Values may be seen in the table] M A 3.a.b "65" Percentage of morists is 6 "4" 7459 "5" 00 M 3.b = 67.7% A.b Part (5 marks) Notes st M for a fully correct epression for the number of morists in the interval nd M for clear use of frequency density in (4-6) or (3-5) cases establish the fd scale. Then use of area find frequency in one of the missing cases. st A for both correct values seen 3 rd M for realising that tal is required and attempting a correct epression for % nd A for awrt 67.7% Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 7

76 Question Scheme Marks AOs 3 (a) p = [ =] 0.0 B.b () (b) q = 0.5 Bft.b P(A) = 0.35 P(T) = 0.6 P(A and T) = 0.0 P(A) P(T) = 0. M. Since 0.0 ¹ 0. therefore A and T are not independent A.4 (3) (c) Part (a) (b) (c) P(not [A or C]) = 0.45 B.b () (5 marks) Notes Bcao for p = 0.0 Bft for use of their p and P(A or T) find q i.e p 0.40 or q = 0.5 M for the statement of all probabilities required for a suitable test and sight of any appropriate calculations required. A All probabilities correct, correct comparison and suitable comment. Bcao for 0.45 Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 7

77 Question Scheme Marks AOs 4(a) IQR =.3 and (= 5.85) (Compare correct values) (b)(i) (ii) (c) (d) (e) (i) (ii) e.g. it is a piece of data and we should consider all the data (o.e.) e.g. it is an etreme value and could unduly influence the analysis or it could be a mistake B.b () B.4 e.g. as humidity increases rainfall increases B.b e.g. a 0% increase in humidity gives rise a.5 mm increase in rainfall or represents 0.5mm of rainfall per percentage of humidity Not a good method since only uses days from one location in one month. e.g. She should use data from more of the UK locations and more of the months or using a spreadsheet or computer package she could use all of the available UK data Part Notes (a) B for sight of the correct calculation and suitable comparison with 0.6 (b)(i) B for a suitable reason for including the data point (ii) B for a suitable reason for ecluding the data point B () ().4 B 3.4 () B.4 B.4 () (7 marks) (c) B for a suitable interpretation of positive correlation mentioning humidity and rainfall (d) B for a suitable description of the rate: rainfall per percentage of humidity including reference values. (e)(i) B for a comment that supports the idea that her sampling method was not a good one (ii) B for some sensible suggestions that would give a better representation of the data across the UK. Must show some awareness of the fact that LDS has different locations and more months of data available but must be clear they are NOT using any overseas locations. NB B0 for a comment that says use more than one location without specifying that only UK locations are required. Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 73

78 Question Scheme Marks AOs 5(a) (i) (ii) (b) P(X = 9) = 0.08 = awrt 0. Bcao.b P(X 6) = -P( X 5) M.b 0 = = awrt A.b H : p= 0.3 H : p¹ 0.3 (Both correct in terms of p or ) B.5 (c) [Y ~ B(0, 0.3)] sight of P(Y ) = or P(Y 9) = (d) (3) () M. Critical region is {Y } or (o.e.) A.b { Y 0} (o.e.) A.b [ ( 0.950)] = or 8.35% Bft.b () (e) 9 is not in the CR so the manager s suspicion is not supported Bft 3.a () (9 marks) Part Notes (a)(ii) M for dealing with P(X 6) they ned use cumulative prob. function on calc. A awrt (from calcular) (b) B for both hypotheses in terms of p or and H must be -tail (c) M for correct use of tables find probability associated with critical value. st A for the correct lower limit of the CR. Do not award for P(Y ) nd A for the correct upper limit. (d) Bft ft on their and ( their 0.950) provided each probability is less than 0.05 (e) Bft for a comment that relates 9 their CR and makes a consistent comment relating this the manager s suspicion (3) Pearson Edecel Level 3 Advanced Subsidiary GCE in Mathematics Sample assessment materials (SAMs) Draft. January 07 Pearson Education Limited 07 74