Mark Scheme (Results) Summer Pearson Edexcel GCE in Core Mathematics 1 (6663/01)

Transcription

1 Mark Scheme (Results) Summer 06 Pearson Edexcel GCE in Ce Mathematics (666/0)

2 Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding body. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes f employers. F further infmation visit our qualifications websites at Alternatively, you can get in touch with us using the details on our contact us page at Pearson: helping people progress, everywhere Pearson aspires to be the wld s leading learning company. Our aim is to help everyone progress in their lives through education. We believe in every kind of learning, f all kinds of people, wherever they are in the wld. We ve been involved in education f over 50 years, and by wking across 70 countries, in 00 languages, we have built an international reputation f our commitment to high standards and raising achievement through innovation in education. Find out me about how we can help you and your students at: Summer 06 Publications Code 666_0_606_MS All the material in this publication is copyright Pearson Education Ltd 06

3 General Marking Guidance All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last. Mark schemes should be applied positively. Candidates must be rewarded f what they have shown they can do rather than penalised f omissions. Examiners should mark accding to the mark scheme not accding to their perception of where the grade boundaries may lie. There is no ceiling on achievement. All marks on the mark scheme should be used appropriately. 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 wthy of credit accding to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited. Crossed out wk should be marked UNLESS the candidate has replaced it with an alternative response.

4 PEARSON EDEXCEL GCE MATHEMATICS General Instructions f Marking. The total number of marks f the paper is 75. The Edexcel Mathematics mark schemes use the following types of marks: M marks: Method marks are awarded f 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 will appear in the mark schemes. bod benefit of doubt ft follow through the symbol will be used f crect ft cao crect answer only cso - crect solution only. There must be no errs in this part of the question to obtain this mark isw igne subsequent wking awrt answers which round to SC: special case oe equivalent (and appropriate) d dep dependent indep independent dp decimal places sf significant figures The answer is printed on the paper ag- answer given d The second mark is dependent on gaining the first mark 4. All A marks are crect answer only (cao.), unless shown, f example, as A ft to indicate that previous wrong wking is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft. 5. F misreading which does not alter the character of a question materially simplify it, deduct two from any A B marks gained, in that part of the question affected.

5 6. If a candidate makes me than one attempt at any question: If all but one attempt is crossed out, mark the attempt which is NOT crossed out. If either all attempts are crossed out none are crossed out, mark all the attempts and sce the highest single attempt. 7. Igne wrong wking increct statements following a crect answer.

6 General Principles f Ce Mathematics Marking (But note that specific mark schemes may sometimes override these general principles). Method mark f solving term quadratic:. Factisation ( x bx c) ( x p)( x q), where pq c, leading to x = ( ax bx c) ( mx p)( nx q), where pq c and mn a, leading to x =. Fmula Attempt to use the crect fmula (with values f a, b and c).. Completing the square b Solving x bx c 0 : x q c 0, q 0, leading to x = Method marks f differentiation and integration:. Differentiation Power of at least one term decreased by. ( x n x n ). Integration Power of at least one term increased by. ( x n x n )

7 Use of a fmula Where a method involves using a fmula that has been learnt, the advice given in recent examiners repts is that the fmula should be quoted first. Nmal marking procedure is as follows: Method mark f quoting a crect fmula and attempting to use it, even if there are small errs in the substitution of values. Where the fmula is not quoted, the method mark can be gained by implication from crect wking with values, but may be lost if there is any mistake in the wking. Exact answers Examiners repts have emphasised that where, f example, an exact answer is asked f, wking with surds is clearly required, marks will nmally be lost if the candidate rests to using rounded decimals.

8 Question Number 4 Scheme Notes Marks 5 5 x x x x x x c (x )dx x n n M: x x. One power increased by but not f just + c. n n This could be f x f x x on what they think is as a power of x. x A: One of these terms crect. 4 x Allow un-simplified e.g. 4, 4x, x 4 4 A: Two of these terms crect. 4 x Allow un-simplified e.g. 4, 4x, x Complete fully crect simplified expression appearing all on one line with constant. Allow 0.4 f 5. Do not allow x f x 0.5 Allow x x f x Igne any spurious integral signs and igne subsequent wking following a fully crect answer. MAA A [4] 4 marks

9 Question Number Scheme Notes Marks 9 x = f example x (x) (x) x x x x y y 9 9 y x Expresses 9 x crectly as a power of expresses y crectly as a power of 9 expresses y crectly in terms of x (This mark is not f just 9) 6 = x y = 6x + a = 6, b = Cao (isw if necessary) A Providing there is no increct wk, allow sight of 6x + to sce both marks Crect answer only implies both marks Special case: 6x only sces MA0 Alternative using logs x y x y 9 log9 log x log9 ylog Use power law crectly on both sides M y log 9 x log y 6x cao A marks M []

10 Question Number.(a) (b) WAY (b) WAY (b) WAY (b) WAY 4 Scheme Notes Marks and the other term 50 8 = 5 in the fm k. This mark may be implied M by the crect answer Or a = A = Uses part (a) by replacing denominat by their a where a is numeric. This is all that is 50 8 "" required f this mark. M Rationalises the denominat by a crect method e.g. multiplies numerat and denominat by k to obtain a multiple of 6 6.Note that multiplying numerat and "" 4 denominat by is quite dm common and is acceptable f this mark. May be implied by a crect answer. This is dependent on the first M. 6 b =, c = 6 Cao and cso A F rationalising the denominat by a crect method i.e. multiplying numerat and denominat by k 50 8 F replacing numerat by This is dependent on the first M and there is dm 50 8 no need to consider the denominat f this mark. 6 b =, c = 6 Cao and cso A 50 8 = "" Uses part (a) by replacing denominat by their a where a is numeric. This is all that is required f this mark. Cancels to obtain a multiple of dependent on the first M. 6. This is 6 Or b =, c = 6 Cao and cso A 50 8 = "" 4 "" Uses part (a) by replacing denominat by their a where a is numeric. This is all that is required f this mark. Obtains a multiple of on the first M. 6. This is dependent 6 Or b =, c = 6 Cao and cso (do not allow 6 ) A 5 marks M M dm M dm [] [] [] []

11 Question Number 4.(a) (b) (-, ) (-, 0) Scheme Notes Marks (0, -4) Note iginal points are A(-, 4) and B(, -8) Similar shape to given figure passing through the igin. A cubic shape with a maximum in the second quadrant and a minimum in the 4 th quadrant. There must be evidence of a change in at least one of the y-codinates (inconsistent changes in the y-codinates are acceptable) but not the x- codinates. Maximum at (-, ) and minimum at (, -4) with codinates written the right way round. Condone missing brackets. The codinates may appear on the sketch, separately in the text (not necessarily referenced as A and B). If they are on the sketch, the x and y codinates can be positioned crectly on the axes (, -4) rather than given as codinate pairs. In cases of ambiguity, the sketch has precedence. The igin does not need to be labelled. N do the x and y axes. (, -) A positive cubic which does not pass through the igin with a maximum to the left of the y-axis and a minimum to the right of the y-axis. Maximum at (-, 0) and minimum at (, -). Condone missing brackets. F the max allow just - (0, -) if marked in the crect place. If the codinates are in the text, they must appear as (-, 0) and must not contradict the sketch. The curve must touch the x-axis at (-, 0). F the min allow codinates as shown and - to be marked in the crect places on the axes. In cases of ambiguity, the sketch has precedence. Crosses y-axis at (0, -4). Allow just -4 (not +4) and allow (-4, 0) if marked in the crect place. If the codinates are in the text, they must appear as (0, -4) and must not contradict the sketch. In cases of ambiguity, the sketch has precedence. B B M A A [] [] 5 marks

12 Question Number 5. Scheme Notes Marks y 4x ( 4x) 5x x 0 x 0x 0 7xx 0 x, y 7, 7 WAY Attempts to makes y the subject of the linear equation and substitutes into the other equation. Allow slips e.g. substituting y = -4x + etc. Crect term quadratic (terms do not need to be all on the same side). The = 0 may be implied by subsequent wk. dm: Solves a term quadratic by the usual rules (see general guidance) to give at least one value f x. Dependent on the first method mark. A: x 7, (two separate crect exact answers). Allow exact equivalents e.g. 6 4 x 4, 4 M: Substitutes to find at least one y value (Allow substitution into their rearranged equation above but not into an equation that has not been seen earlier). You may need to check here if there is no wking and x values are increct. A: y 7, (two crect exact answers) Allow exact equivalents e.g. 8 4 y, 4 4 Codinates do not need to be paired Note that if the linear equation is explicitly rearranged to y = 4x +, this gives the crect answers f x and possibly f y. In these cases, if it is not already lost, deduct the final A. WAY x 4 y Attempts to makes x the subject of the linear 4 equation and substitutes into the other equation. y 5( 4 y 4) ( 4 y 4) 0 Allow slips in the rearrangement as above. Crect term quadratic (terms do not need to y y 0 y y 0 be all on the same side) The = 0 may be implied by subsequent wk. dm: Solves a term quadratic by the usual rules (see general guidance) to give at least one value f y. Dependent on the first method 7yy 0 y 7, mark. A: y 7, (two separate crect exact answers). Allow exact equivalents e.g. 8 4 y, x, M: Substitutes to find at least one x value (Allow substitution into their rearranged equation above but not into an equation that has not been seen earlier). You may need to check here if there is no wking and y values are increct. A: x 7, (two crect exact answers) Allow exact equivalents e.g. 6 4 x, 4 4 Codinates do not need to be paired Note that if the linear equation is explicitly rearranged to x = (y + )/4, this gives the crect answers f y and possibly f x. In these cases, if it is not already lost, deduct the final A. M A dm A M A M A [6] dm A M A [6] 6 marks

13 Question Number 6. (a) (b) r r Scheme Notes Marks a 5ka 5 4k a 5ka 5 k(5 4 k) r () a k k k 4 "5 4 " "5 5 4 " a 4, a n 5 kan, n M: Uses the recurrence relation crectly at least once. This may be implied by a 54k by the use of a 5 k their a A: Two crect expressions need not be simplified but must be seen in (a). Allow a 5k4anda 55k k 4 Isw if necessary f a. Finds ++ somewhere in their solution (may be implied by e.g k + 6 5k + 4k ). Note that k + 6 5k + 4k would sce B and the M below. Adds 4 to their a and their a where a and a are functions of k. The statement as shown is sufficient. ( ar) 7 9k4k Cao but condone = 0 after the expression A r MA Allow full marks in (b) f crect answer only [] (c) 500 cao B [] 6 marks B M []

14 Question Number Scheme Notes Marks 7. x 7 yx 6x x Attempts to split the fraction into terms 5 and obtains either x x. This may be implied by a crect power of x in their x 7 x differentiation of one of these terms. But x x x x x beware of x coming from x 7 x 7 x x x n n x Differentiates by reducing power by one f any of their powers of x M M dy 5 7 6x x x x dx 6 A: 6 x. Do not accept 6x. Depends on second M mark only. Award when first seen and isw. A: x. Must be simplified so do not accept e.g. x but allow. Depends x on second M mark only. Award when first seen and isw. 5 A: x. Must be simplified but allow e.g..5 x e.g. 5 x. Award when first seen and isw. 7 A:. 6 x Must be simplified but allow e.g. x 7 e.g.. Award when first 6 6 x seen and isw. In an otherwise fully crect solution, penalise the presence of + c by deducting the final A Use of Quotient Rule: First M and final AA (Other marks as above) x 7 d 6 7 x x x x x Uses crect quotient rule M dx x 5 A: Crect first term of numerat and 0x 7x crect denominat 6x A: All crect as simplified as shown 5 d So y x x x x sces full marks dx 6x AAAA AA [6] 6 marks

15 Question Scheme Notes Marks Number 8.(a) px 6px 4 p" "x 7 Either: Compares the given quadratic expression with the given linear expression using <, >, =, M y7 y7 (May be implied) y p 6p 4p Rearranges y = x 7 to make x the subject and substitutes into the given quadratic Examples px 6px 4p x 7 0, px 6px 4p x 7 0 (b) y7 y7 p 6p 4p y 0, py 0p9 y8p 0 y px 6px4px7 Moves all the terms to one side allowing sign errs only. Igne > 0, < 0, = 0 etc. The terms do not need to be collected. Dependent on the first method mark. Attempts to use b 4 ac with their a, b and c where a p, b ( 6p ) and c (4p7) f the quadratic in y, E.g. a p, b (0 p 9) and c 8 p. This b 4ac6p 4p4p 7 could be as part of the quadratic fmula as b 4ac0p9 4p8p b 4 ac as b 4ac as b 4ac etc. If it is part of the quadratic fmula only look f use of b 4 ac. There must be no x s y s. Dependent on both method marks. Obtains printed answer with no errs seen 4p 0p + 9 < 0 * (Allow 0 > 4p 0p + 9) but this < 0 must been seen at some stage befe the last line. p9 p 0 p... to obtain p = p 9, p 4 Allow equivalent values e.g. 6 8 f 4 Attempt to solve the given quadratic to find values f p. See general guidance. Both crect. May be implied by e.g. 9 p, p. Allow equivalent values e.g. 4.5, 6, 8 fmula allow 0.5 etc. If they use the quadratic f this mark but not 56 f 6 and allow e.g. 5 if they complete the square. M: Chooses inside region i.e. Lower Limit p Upper Limit e.g. Lower Limit p Upper Limit A: Allow p (,4 ) just (,4 ) and allow p and p 4 and 4 p but p, p 4 sces MA0 p 4 sces M0A0 dm ddm A* M A MA Allow wking in terms of x in (b) but the answer must be in terms of p f the final A mark. [4] [4] 8 marks

16 Question Number Scheme Notes Marks 9.(a) John; arithmetic series, a 60, d = 5* Finds and adds the first terms uses sum of terms of an AP and obtains the B * S 0 ( )(5) 5* printed answer, with no errs. Beware: The th term of the sequence is 5 also so look out f This is B0. [] (b) M: Uses 60 + (n )5 with n = 8 9 t 9 = 60 + (n )5 = ( )80 A: ( )80 M A Listing: M: Uses a = 60 and d = 5 to select the 8 th 9 th term (allow arithmetic slips) A: ( )80 (Special case ( )65 only sces MA0) [] n S n (c) 0 ( n )(5) Uses crect fmula f sum of n terms with a = 60 and d = 5 (must be a crect fmula but igne the value they use f n S n ( n )(5) n could be in terms of n) M S n Crect numerical expression A = ( )70 cao A Listing: M: Uses a = 60 and d = 5 and finds the sum of at least terms (allow arithmetic slips) A: ( )70 [] (d) n Uses crect fmula f sum of n terms 75 0 ( n )(5) with a = 60, d = 5 and puts = 75 M Crect three term quadratic. E.g n5 n, 75 n n n 8 ( n ) 5n 05n 6750 This may be implied by equations such as nn7 75 n 7n A (e) n 7n 5 8* Achieves the printed answer with no errs but must see the in factised fm e.g. 6750, 75 in factised fm i.e. an intermediate step. M: Attempts to solve the given quadratic n 8 Aged 7 states n = 8 A: Age = 7 just 7 Age = 7 only sces both marks (i.e. n = 8 need not be seen) Note that (e) is not hence so allow valid attempts to solve the given equation f M A* M A [] [] marks

17 n u n S n Age n u n S n Age

18 Question Number 0.(a) (b) (c) Scheme Notes Marks l : passes through (0, ) and (, 7) l : goes through (, 7) and is perpendicular to l Gradient of l is ml. Allow un-simplified. 0 0 May be implied. B 5 Crect application of perpendicular gradient ml their rule M M: Uses y 7 m( x) with their changed y 7 " "( x ) 5 gradient uses y mx c with (, 7) and their changed gradient to find a value f c MAft 44 y " " x c, 7 " " 5 5 c c 5 Aft: Crect ft equation f their perpendicular gradient (this is dependent on both M marks) x 5y 44 0 Any positive negative integer multiple. Must A be seen in (a) and must include = 0. [5] M: Puts y = 0 and finds a value f x from their equation 44 When y = 0 x A: 44 x exact M A equivalent. (y = 0 not needed) Condone x 5y 44 = 0 only leading to the crect answer and condone codinates written as (0, 44/) but allow recovery in (c) ` [] GENERAL APPROACH: Crect attempt at finding the area of any one of the triangles one of the trapezia but not just one rectangle. The crect pair of base and height must be used f a triangle and the crect fmula used f a trapezium. If Pythagas is required, then it must be used crectly with the crect end codinates. M Note that the first three marks apply to their calculated codinates e.g. their 44, 44 5, 6 5 etc. But the given codinates must be crect e.g. (0, ) and (, 7). A crect numerical expression f the area of one triangle one trapezium f their codinates. Aft Combines the crect areas together crectly f their chosen "way". Note that if crect numerical expressions f areas have been increctly simplified befe combining them, then this dm M may still be given. Dependent on the first method mark. Crect numerical expression f the area of ORQP. The expressions must be fully crect f this mark i.e. no follow through. A 6 6 Crect exact area e.g. 54,,, 54. any exact equivalent 6 A Shape Vertices Numerical Expression Exact Area Triangle TRQ Triangle SPO Triangle PWQ Triangle PVQ 7

19 Triangle VWQ Triangle QUR Triangle PQR Triangle PNQ Triangle OPQ Triangle OQR Triangle OWR Triangle SQR Triangle OPR Trapezium OPQT ( 7) 7 Trapezium OPNR Trapezium OVQR EXAMPLES (c) WAY OPQT ( 7) 44 TRQ 7 44 ( 7) 7 M: Crect method f OPQT TRQ Aft: OPQT ( 7) 44 TRQ 7 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP Any exact equivalent e.g.,, MAft dma A (7 ) + "5"

20 WAY 7 PQR OPR M: Crect method f PQR OPR 7 Aft: PQR OPR dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP 6 6,, Any exact equivalent e.g. MAft dma A "44" 7 4 " 4 " WAY 8 SQR SPO M: Crect method f SQR SPO 8 Aft: SQR SPO 5 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP 6 6,, Any exact equivalent e.g. MAft dma A "8" "6"

21 WAY 4 PVQ 5 5 QUR OVUR M: Crect method f PVQ QUR Aft: PVQ 5 5 QUR 7 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP Any exact equivalent e.g.,, MAft dma A "44" "5" WAY OWR 5 44 PWQ M: Crect method f OWR PWQ Aft: OWR 5 44 PWQ 5 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP 6 6,, Any exact equivalent e.g. MAft dma A "44" "44"

22 WAY OPNR 4 PNQ M: Crect method f OPNR PNQ 4 44 Aft: OPNR 4 PNQ 5 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP 6 6,, Any exact equivalent e.g. MAft dma A "4" "44" "4" ( ) WAY 7 OPQ 44 OQR M: Crect method f OPQ OQR Aft: OPQ 44 OQR 7 dm: Crect numerical combination of areas that have been calculated crectly A: Fully Crect numerical expression f the area ORQP Any exact equivalent e.g.,, MAft dma A "44"

23 WAY 8 M: Uses the vertices of the quadrilateral to fm a determinant MAft Aft: dm: Fully crect determinant method with no errs A: Fully Crect numerical expression f the area ORQP 6 6,, Any exact equivalent e.g. There will be other ways but the same approach to marking should be applied. dma A W V (0, 7) Q (, 7) U P(0, ) S T (, 0) R V (0, 7) W Q (, 7) U P(0, ) N S R

24 Question Scheme Number. (a) y = x +k x +5 x + 6 Marks (b) dy dx 6x kx 5 Gradient of given line is 7 dy dx x k 6 5 n n M: x x f one of the terms including 6 0 M A A: Crect derivative [] Uses states 7 equivalent e.g Must be stated used in (b) and not just 7 seen as part of y x. Substitutes x = into their derivative (not the curve) B M (c) (d) 7 4 "44k 5" " " k 8 dm: Puts their expression = their 7 (Allow BOD f 7-7 but not the nmal gradient) and solves to obtain a value f k. Dependent on the previous method mark. A: Note: 7 6x kx5 x sces no marks on its own but may sce the first M mark if they substitute x = - into the lhs. If they rearrange this equation and then substitute x = -, this sces no marks. M: Substitutes x = and their numerical k into y = y 6 4k 0 6 4" k" 0 A: y Allow the marks f part (c) to be sced in part (b). 7 y" " " " x 7xy5 0 7 y" " xcc... 7xy5 0 y7x 4 7xy5 0 M: Crect attempt at linear equation with their 8.5 gradient (not the nmal gradient) using x = - and their A: cao (allow any integer multiple) dm A M A M A [4] [] [] 0 marks

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