Level 2 Certificate Further Mathematics

Transcription

1 AQA Qualifications Level Certificate Further Mathematics Paper Mark scheme 86 June 6 Version:. Final

2 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation events which all associates participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every associate understands and applies it in the same crect way. As preparation f standardisation each associate analyses a number of students scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Assessment Writer. It must be stressed that a mark scheme is a wking document, in many cases further developed and expanded on the basis of students reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular examination paper. Further copies of this Mark Scheme are available from aqa.g.uk Copyright 6 AQA and its licenss. All rights reserved. AQA retains the copyright on all its publications. However, registered schools/colleges f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant exception: AQA cannot give permission to schools/colleges to photocopy any material that is acknowledged to a third party even f internal use within the centre.

3 MARK SCHEME LEVEL CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Glossary f Mark Schemes GCSE examinations are marked in such a way as to award positive achievement wherever possible. Thus, f GCSE Mathematics papers, marks are awarded under various categies. If a student uses a method which is not explicitly covered by the mark scheme the same principles of marking should be applied. Credit should be given to any valid methods. Examiners should seek advice from their seni examiner if in any doubt. M M dep A B B dep ft SC [a, b] Method marks are awarded f a crect method which could lead to a crect answer. A method mark dependent on a previous method mark being awarded. Accuracy marks are awarded when following on from a crect method. It is not necessary to always see the method. This can be implied. Marks awarded independent of method. A mark that can only be awarded if a previous independent mark has been awarded. Follow through marks. Marks awarded following a mistake in an earlier step. Special case. Marks awarded within the scheme f a common misinterpretation which has some mathematical wth. Or equivalent. Accept answers that are equivalent. eg, accept.5 as well as Accept values between a and b inclusive..4 Accept answers which begin.4 eg.4,.4,.46 of 54

4 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Examiners should consistently apply the following principles Diagrams Diagrams that have wking on them should be treated like nmal responses. If a diagram has been written on but the crect response is within the answer space, the wk within the answer space should be marked. Wking on diagrams that contradicts wk within the answer space is not to be considered as choice but as wking, and is not, therefe, penalised. Responses which appear to come from increct methods Whenever there is doubt as to whether a candidate has used an increct method to obtain an answer, as a general principle, the benefit of doubt must be given to the candidate. In cases where there is no doubt that the answer has come from increct wking then the candidate should be penalised. Questions which ask candidates to show wking Instructions on marking will be given but usually marks are not awarded to candidates who show no wking. Questions which do not ask candidates to show wking As a general principle, a crect response is awarded full marks. Misread miscopy Candidates often copy values from a question increctly. If the examiner thinks that the candidate has made a genuine misread, then only the accuracy marks (A B marks), up to a maximum of marks are penalised. The method marks can still be awarded. Further wk Once the crect answer has been seen, further wking may be igned unless it gs on to contradict the crect answer. Choice When a choice of answers and/ methods is given, mark each attempt. If both methods are valid then M marks can be awarded but any increct answer method would result in marks being lost. Wk not replaced Erased crossed out wk that is still legible should be marked. Wk replaced Erased crossed out wk that has been replaced is not awarded marks. Premature approximation Rounding off too early can lead to inaccuracy in the final answer. This should be penalised by mark unless instructed otherwise. Continental notation Accept a comma used instead of a decimal point (f example, in measurements currency), provided that it is clear to the examiner that the candidate intended it to be a decimal point. 4of 54

5 MARK SCHEME LEVEL CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method M Fully crect method Plots crect three points draws crect triangle 5 Alternative method 5 [6., 6.5] sin [7, 74] 5 [6.5, 6.9] sin [6, 65] M Fully crect method with tolerances on measurements Plots crect three points draws crect triangle [6., 6.5] [6.5, 6.9] sin [4, 47] 5 with no evidence that rounding has been applied eg 4.9 seen in wking Additional Guidance 5 from counting squares M Increct triangle drawn is zero unless recovered Answer only of 5 M 5of 54

6 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 x = B Additional Guidance (a) = x B y = B B.8 and 4.8 with no other answers B Both crect in either der B One crect and one increct missing (b) Additional Guidance The wd and is not needed If their answer has both.8 and 4.8 with no other solutions award B eg 4.8,.8 with no other solutions B 6of 54

7 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 f(x) 6 6 f(x) [, 6] B B f(x) f(x) 6 on their own embedded within an interval f f(x) only and 6 chosen Additional Guidance Allow as two inequalities f(x) f(x) 6 B to 6 inclusive B f(x) may be replaced by f y x 4x + f B and B (c) B may be seen with an increct inequality eg < f(x) 6 eg f(x) < 6 eg f(x) 5 F B igne increct notation if only and 6 chosen eg x 6 eg < x 6 eg to 6 eg4 to 6 = 9 (wking out a statistical range) eg5 f(x) 6 eg6, 6 B B B B B B B B B {,,,,,,, 4, 5, 6} B (a) ( a c, ) B (b) b a B 7of 54

8 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method Equates cfficients and eliminates a variable 4x + 6y = 6x + 9y = and and Equates cfficients of one variable 9x + 6y = 9 6x + 4y = 6 Allow one term err 4x 9x = 9y 4y = 9 6 Eliminates a variable 5x = 7 5y = 7 dep Must be crect method f their equations Unless crectly eliminated, intention to subtract ( add if appropriate) must be seen with the result of their subtraction (x =).4 (y =).4 7 eg x = 5 (.4,.4) 7 7 eg (, ) SC (.4,.4) Alternative method Makes a variable the subject in first equation y = x x = 5.5.5y y eg x = Makes y x the subject Allow one term err ( x) = x y = (5.5.5y) dep Eliminates a variable Must be crect method f their equations (x =).4 (y =).4 7 eg x = 5 (.4,.4) 7 7 eg (, ) 5 5 SC (.4,.4) MARK SCHEME CONTINUES ON THE NEXT PAGE 8of 54

9 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method Makes a variable the subject in second equation y = 6.5.5x x = y x eg y = Makes y x the subject Allow one term err x + (6.5.5x) = ( y) + y = dep Eliminates a variable Must be crect method f their equations (x =).4 (y =).4 7 eg x = 5 4 (.4,.4) Alternative method eg (, ) 5 5 SC (.4,.4) Makes same variable the subject in both equations y = x y = 6.5.5x x = 5.5.5y x eg y = x = y Makes y x the subject Allow one term err x = 6.5.5x 5.5.5y = dep y Eliminates a variable Must be crect method f their equations (x =).4 (y =).4 7 eg x = 5 (.4,.4) 7 7 eg (, ) 5 5 SC (.4,.4) ADDITIONAL GUIDANCE FOR Q4 IS ON THE NEXT PAGE 9of 54

10 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q4) Answer only (.4,.4) One value crect (possibly by drawing) with no increct wking seen f that variable M A M A If the same method is used f both x and y (eg equates cfficients and eliminates a variable), mark the attempt that favours the student 4 Alt 6x + 9y = 6x + 4y = 6 4y = 6 is M unless intention to subtract is seen (eg a subtraction symbol is seen the wd subtract is seen) which would then get Alts, and 4 Allow rounding truncating to dp better f up to eg (Alt ) y =.6.6x (.6.6x) = x (.4,.4) is SC M A if x =.4 and y =.4 seen in wking of 54

11 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Always true Sometimes true Never true Sometimes true B4 B f each crect answer 5 Additional Guidance Me than one box selected in a row is B f that row Allow any unambiguous indication of a selection in a row eg uses crosses instead of ticks Igne wking seen and mark the boxes 4 ( ) k ( ) + k 6k + k 5k Allow missing brackets even if not recovered eg k 4 + k + 6k + k + 7k 6 6k + k = 5k = 5k = 5 crect equation (brackets may be recovered) 4 ( ) and ( ) must be evaluated Implied by k = 5 SC Additional Guidance with no errs seen (recovered bracket is not an err) A Substituting x = M A of 54

12 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE x 9 y x y 7 7 B B Two of the three components crect and simplified Additional Guidance Allow multiplication signs f B and B..... Allow as a crect component.96x 9 y B x y followed by increct further wk (only penalise B responses) B 7 8x 9 y 7 ( ) x 9 y B B 8 9 x 7 B 8x 9 7y B 8 9 x + y 7 B of 54

13 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method ( 4.5, 5.875) eg ( 9, 8 47 ) SC (5.875, 4.5) 8 Alternative method ( 6).5 5 (9 4) eg ( 4.5, 5.875) eg ( 9, 8 47 ) SC (5.875, 4.5) MARK SCHEME CONTINUES ON THE NEXT PAGE of 54

14 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 5 ( 6) (9 4).5 5 eg ( 4.5, 5.875) Alternative method 4 eg ( 9, 8 47 ) SC (5.875, 4.5) 8 x 6 6 = 5 y = 5 eg both fractions inverted x 5 = 6 5 x 6 = x 5 9 y 5 = y 4 = 9 y ( 4.5, 5.875) eg ( 9, 8 47 ) SC (5.875, 4.5) MARK SCHEME CONTINUES ON THE NEXT PAGE 4 of 54

15 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 5 5 (9 4) ( 6) sin(tan 9 4 ) tan 9 4 is the angle DE makes with 6 the hizontal (= 5...) (9 4) ( 6) is DE (= ) 8 5 (9 4) ( 6) cos(tan 9 4 ) ( 4.5, 5.875) eg ( 9, 8 47 ) SC (5.875, 4.5) MARK SCHEME CONTINUES ON THE NEXT PAGE 5 of 54

16 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method (9 4) ( 6) sin(tan 9 4 ) (9 4) ( 6) cos(tan 9 4 ).5 6 tan 9 4 is the angle DE makes with 6 the hizontal (= 5...) (9 4) ( 6) is DE (= ) ( 4.5, 5.875) eg ( 9, 8 47 ) SC (5.875, 4.5) Additional Guidance ( 4.5, 5.9) ( 4.5, 5.88) is A unless seen in wking (5.875, 4.5) is SC if x = 4.5 and y = seen in wking 4.5 in wking that becomes 4.5 on answer line should not be regarded as choice so gains at least marks if one codinate crect and marks if both crect (possibly from accurate drawing wking with midpoints) with no increct wking f that codinate Alts 5 and 6 also have equivalents where the angle DE makes with the vertical (= ) is used. Mark using the principles of alts 5 and 6 escalate 6 of 54

17 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method PBC in terms of x and in terms of y 9 8 x 6 (y + + y + 8) 6 (y + 8) 8 x and 6 (y + + y + 8) 6 (y + 8) 8 x and 8 y and x = y Both reasons given May be on diagram PBC (allow B) Must have seen crect wking f Must have (Co-)interi angles allied angles (add up to 8 ) and angles at a point (add up to 6 ) Alternative method PBC in terms of x + reflex PBC in terms of y = 6 8 x y + + y + 8 y x + y + + y + 8 = 6 8 x + y + 8 = 6 May be on diagram PBC (allow B) reflex PBC unsimplified crect equation Simplifies to x = y Must have seen crect wking f Both reasons given Must have (Co-)interi angles allied angles (add up to 8 ) and angles at a point (add up to 6 ) MARK SCHEME CONTINUES ON THE NEXT PAGE 7 of 54

18 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method x = 8 PBC in terms of y 6 (y + + y + 8) 6 (y + 8) x = 8 (6 (y + + y + 8)) x = 8 (6 (y + 8)) May be on diagram PBC (allow B) unsimplified crect equation Simplifies to x = y Must have seen crect wking f Both reasons given Must have (Co-)interi angles allied angles (add up to 8 ) and angles at a point (add up to 6 ) Alternative method 4 x + PBC = 8 and reflex PBC in terms of y + PBC = 6 9 x + PBC = 8 y + + y PBC = 6 PBC (allow B) y PBC = 6 x + PBC = 8 and y + + y PBC = 6 y PBC = 6 x + PBC = 8 Must have seen crect wking f and y + PBC = 8 and x = y Both reasons given Must have (Co-)interi angles allied angles (add up to 8 ) and angles at a point (add up to 6 ) MARK SCHEME CONTINUES ON THE NEXT PAGE 8 of 54

19 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 5 (Produces CB to X) PBX = reflex PBC in terms of y 8 y + + y + 8 y + 8 y + + y y May be on diagram reflex PBC PBX Simplifies to y and states x = y Must have seen crect wking f 9 Both reasons given Must have Angles on a (straight) line (add up to 8) and alternate angles (are equal) Alternative method 6 (Produces PB to Y) CBY = reflex PBC in terms of y 8 y + + y + 8 y + 8 y + + y y May be on diagram reflex PBC (allow reflex B) CBY Simplifies to y and states x = y Must have seen crect wking f Both reasons given Must have Angles on a (straight) line (add up to 8) and cresponding angles (are equal) MARK SCHEME CONTINUES ON THE NEXT PAGE 9 of 54

20 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 7 (Produces CB to X) PBX = 8 PBC in terms of y 6 (y + + y + 8) 6 (y + 8) 8 (6 (y + + y + 8)) 8 (6 (y + 8)) May be on diagram PBC (allow B) PBX Simplifies to y and states x = y Must have seen crect wking f 9 All reasons given Must have Angles at a point and angles on a (straight) line (add up to 8) and alternate angles (are equal) Alternative method 8 (Produces PB to Y) CBY = 8 PBC in terms of y 6 (y + + y + 8) 6 (y + 8) 8 (6 (y + + y + 8)) 8 (6 (y + 8)) May be on diagram PBC (allow B) CBY Simplifies to y and states x = y Must have seen crect wking f All reasons given Must have Angles at a point and angles on a (straight) line (add up to 8) and cresponding angles (are equal) ADDITIONAL GUIDANCE FOR Q9 IS ON THE NEXT PAGE of 54

21 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q9) Recovery of brackets is not allowed as it is a proof Acceptable reasons must include the wd angles Angles at a point can be angles round a point These reasons are not allowed: 9 Alternating angles, alternative angles, angles in a circle, straight line, at a point, round a point, parallel lines Supplementary angles are not allowed unless accompanied by an acceptable reason eg Allied angles are supplementary is allowed Other variations on these methods will be seen. Escalate if necessary. Starting with x = y substituting values f x and y is zero unless M marks seen in wking ( x5)( x ) ( x5)( x ) factisation (5 x)( x) (5 x)( x ) x x x x numerat and denominat both linear (a) Additional Guidance Crect answer followed by increct further wk A x x x x from increct method M A Allow fraction with crect factisations and common facts crossed out Allow x / x + of 54

22 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 w x y (w + x y) B A crect partial factisation with at least one variable fully factised (b) B eg y (w 5 x + w x 6 y) eg w x(w x y + x 5 y ) a crect partial factisation with all three variables as facts eg wxy (w 4 x y + wx 5 y ) full common fact with one term in brackets crect w x y (w +...) w x y (... + x y) Must be two terms in each bracket Additional Guidance Allow multiplication signs and s and unnecessary brackets f B and B Use of negative fractional zero powers is a maximum of B w x y (w + x y) followed by increct further wk B Answer line increct, check wking lines f possible B of 54

23 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 4x px 4 + p May be seen in an expansion a grid Allow unsimplified eg x px their 4(x ) + their p(x ) = (x ) Crect ft their expansion ft is equating their terms in x to x dep Must be at least two terms with at least one linear term in p Allow unsimplified eg x px + 4x = x 9 Additional Guidance In this question, only consider terms in x If only one term in x the maximum mark is 4 + p = followed by 7p = A n + 5 B B n + k k 5 Additional Guidance (a) Allow any letter eg x + 5 Allow n = eg n = n + 5 n = x + 5 B B n B n + 5 = is B unless also seen with answer of.5 which then sces B of 54

24 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method (n ) ( ) their (n + 5) 6n + 5n n 5 dep ft their (a) Brackets needed but may be recovered 4 terms with at least crect Implied by an + n 5 (a ) 6n + n + b (b ) ft their two term linear expression in (a) 6n + n 5 ft ft their two term linear expression in (a) Alternative method (Second differences =) 6n a = 6 Seen at least once and not contradicted (b) (4 96) dep Subtracts 6n from the terms in sequence Z 8 4 (47) 6n + n 5 Alternative method Any two of a + b + c = 4 4a + b + c = 45 9a + b + c = 88 6a + 4b + c = 4 a + b = 45 4 and 5a + b = dep Obtains two crect equations in same two variables from their equations 6n + n 5 MARK SCHEME CONTINUES ON THE NEXT PAGE 4 of 54

25 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 4 (Second differences =) 6n a = 6 a + b = 45 4 and substitutes a = 6 6n + n 5 Alternative method 5 dep Seen at least once and not contradicted eg 5a + b = and substitutes a = 6 eg 7a + b = 4 88 and substitutes a = 6 (Second differences =) Seen at least once and not contradicted (b) 4 + (45 4)(n ) +.5 (n )(n ) dep Using p + q(n ) +.5r(n )(n ) p is st term q is nd term st term r is second differences 6n + n 5 Additional Guidance Allow any letter mixed letters eg 6x + x 5 6n + x 5 Allow n = eg n = 6n + x 5 6n + n 5 = is unless also seen with solutions which then sces A 5 of 54

26 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method a b = 7 49 a + b = a b = 7 49 and a + b = equation Allow an unsimplified equation eg a + b ( ) ( ) = Missing brackets can be recovered two equations Allow unsimplified equations 7 49 a + a = b b = and a their b = (, 7) dep dep on first Crect method to fm an equation in a crect method to fm an equation in b and substitutes to fm an equation in a Their two equations must both contain a and b SC4 Answer (, 7) from (x 7)( + x) x + x 7x 7 x 5x 7 MARK SCHEME CONTINUES ON THE NEXT PAGE 6 of 54

27 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method (7 x) ( + x) (x 7) ( x) (x.5) (.5 x) Brackets not needed x + x 7x 7 x 5x 7 (7 x)( + x) (x 7)( x) 7 + 7x x x 7 + 5x x eg (x 5x 7) y = not needed Expansion not needed Substitutes x = in their quadratic dep on first selects the constant term from their quadratic dep Expansion not needed f their quadratic but must be crect if attempted May be implied by the final answer (, 7) SC4 Answer (, 7) from (x 7)( + x) x + x 7x 7 x 5x 7 Alternative method b x b 4x Differentiates crectly b.5 = b 4.5 = b = 5 a + their b ( ) ( ) = 7 7 a + their b ( ) ( ) = (, 7) dep dep on first Must have substituted a value into b 4x and equated to Missing brackets can be recovered SC4 Answer (, 7) from (x 7)( + x) x + x 7x 7 x 5x 7 MARK SCHEME CONTINUES ON THE NEXT PAGE 7 of 54

28 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 4 {(x 4 b )... } completed square b =.5 b = 5 4 a + their b ( ) ( ) = a + their b ( 7 ) ( 7 ) = dep dep on first b Must have equated to a value 4 Missing brackets can be recovered (, 7) SC4 Answer (, 7) from (x 7)( + x) x + x 7x 7 x 5x 7 Additional Guidance Answer (7, ) Answer only (, 7) A M Alt rd Follow crect method rules as in Q4 Alt nd Answer (, 7) with wking may not sce full marks Check the wking each time eg (Alt ) x x + 7 (no method seen) Answer (, 7) Zero 8 of 54

29 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 8 (7 + 5) 5 (w =) 7 their 5 5 (y =) 5 their 5 75 dep May be seen on diagram M 5 : 75 8 their w their y dep dep on 7.5 SC Alternative method w + y = 8 and 5w = 7y w w = 8 y y = (w =) (y =) dep May be seen on diagram M 5 : 75 8 their w their y dep dep on 7.5 SC 5.5 MARK SCHEME CONTINUES ON THE NEXT PAGE 9 of 54

30 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method w 8 w 7 y = 5 8 y = (w =) (y =) dep May be seen on diagram M 5 : 75 8 their w their y dep dep on 7.5 SC Alternative method 4 y = x and w = 5 7 x x x = 8 4.8x = 8 dep 8 their dep dep on SC 5.5 Additional Guidance 75 : 5 implies M M unless recovered (award SC if answer 5.5) of 54

31 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 x = 5 x =.5 x = x = 4x = 5 Must have x =... have eliminated with no errs (a) Additional Guidance x = 5 (no further crect wk) M Allow unprocessed values f eg 5 4 x = x = 5 x may be seen as x.5 x x (x 5) (= ) x (5 x) (= ) (x =) (x =) 5 factisation eg (x )(x 5) eg x (x 5x) and 5 with no other solutions 5(b) Additional Guidance F, the wd and is not needed If their answer has both and 5 with no other solutions award eg, 5 with no other solutions, 5, 5 A Either both solutions seen embedded A of 54

32 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method yx = 8(w x) y = yx = 8w 8x 8w 8x x yx + 8x = 8w x(y + 8) = 8w dep eg yx 8w + 8x = Implies 8w y 8 dep dep on Implies x = 8w y 8 eg 8w y 8 Must have x = SC x = 8w y SC 8w y 6 Alternative method y = 8w x y + 8 = 8 y = 8w x 8w x 8x x dep eg y + 8 Implies 8 w = x yx + 8x = 8w x(y + 8) = 8w y 8 = x 8w 8w y 8 dep dep on Implies x = 8w y 8 eg 8w y 8 Must have x = SC x = 8w y SC 8w y MARK SCHEME CONTINUES ON THE NEXT PAGE of 54

33 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method yx = 8(w x) yx = w x 8 yx y + x = w x( + ) = w 8 8 w y 8 dep dep yx eg w + x = 8 Implies dep on Implies 6 x = w y 8 eg x = Must have x = w y 8 SC x = 8w y SC 8w y Additional Guidance x = 8w 8w in wking with on answer line M y 8 y 8 x = 8w 8w in wking with y y on answer line SC rd is f collecting terms in x ( x in numerat in Alt ) Allow multiplications signs and s throughout Crect answer followed by increct further wk M A of 54

34 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 πx y 4 π(6y) π(6y) 44πy Only allow missing brackets if recovered in subsequent wking M (6y) = x y equation eg x y = 44 M 44 7 SC Additional Guidance and A Allow multiplication signs in M marks Answer only of Answers only of and Answer only of M M A M A 8(a) y > 45 B 4 of 54

35 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 (p + ) + (p ) May be within a square root p + p + p + + p p p + p + + p + p + (p + ) p May be within a square root Implies May be within a square root Must be simplified May be implied by final mark Allow a = b = c = d = p SC ( p ) p ( p ) 8(b) Additional Guidance p p and further increct wk M A Allow p f p Use of sin y = cos y p p can be marked by the scheme eg (p ) sin y = (p + ) cos y (p ) sin y = (p + ) ( sin y) ((p ) + (p + ) ) sin y = (p + ) First gained here ( may subsequently be gained) 5 of 54

36 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Continuous curve with point of inflection, labelled P (, ), in first quadrant and minimum point, labelled Q (a, b), in fourth quadrant, with x-codinate of Q > x-codinate of P eg y F B, allow the labelling of one codinate as sufficient f each point B As B but not sufficiently labelled B Curve with point of inflection, labelled P (, ), in first quadrant curve with minimum point, labelled Q (a, b), in fourth quadrant 9 B F B, allow labelling using one codinate as sufficient O P x SC As B but x-codinate of Q < x-codinate of P y eg Q P O x Q ADDITIONAL GUIDANCE FOR Q9 IS ON THE NEXT PAGE 6 of 54

37 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q9) F B, curve ds not have to cross the x-axis after Q and ds not have to cross the y-axis befe P F a stationary point, curve must not stop at the point At P, the curve must change from concave upward to concave downward f B B vice-versa f B SC 9 Note that other non-stationary points of inflection may also be seen (up to B possible) Curve may have hizontal asymptotes as x ± (up to B possible) Mark intention f stationary points, positioning of labels and smoothness of curve Me than stationary point of inflection and/ me than minimum point and/ maximum point(s) can sce a maximum of B Labelling using a codinate codinates may be seen by labelling on an axis on axes (axes may also show other numbers) 7 of 54

38 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 a 4 a 6 a 8 Allow ( a 6 a + 8) a 6 = a a + 8 = linear equation(s) (not a = ) Implies a 6 = a and a + 8 = equations (not a = ) Shows both equations have a common solution (a = ) and Yes ft ft A Must show that their two linear equations do not have a common solution and No SC4 4 = and Yes SC 4 = Additional Guidance a 4 is first M unless recovered In matrices, allow missing brackets inclusion of fraction lines Only one equation can sce a maximum of A A a = with no crect wking Zero a 6 a 8 a = with no further valid wk M A A The final A mark may be seen in various ways eg Solves both equations obtaining a = each time and Yes ( shows that both equations simplify to a = 6 and Yes) eg Solves one equation obtaining a = and shows by substitution that a = satisfies the other equation and Yes eg Adds the two equations to obtain a crect statement and Yes a 6 = a + 8 = = 8 of 54

39 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 (x ) x (x ) x 4 x and x 4 5x 7 Implies 5(x )(x ) 5(x x x + ) 5x 5x + (x )(x ) expanded and multiplied by 5 Allow one err in four term expansion of 5(x )(x ) Implied by 5(x x + k) 5(ax x + ) 5x x + 7 (= ) dep on rd dep -term quadratic equation eg 5x x = 7 Crectly collects terms in their expansion 5 5 ( ) (x ) 7 4 = 5 Crect use of quadratic fmula f their -term quadratic eg ( ) can be crect factisation of their -term quadratic attempt to complete the square f their -term quadratic 5[(x ) 4] = 7 Must be crect up to fm (x a) + b = c k[(x d) + e] = f. and.77 Must be significant figures Additional Guidance F, the wd and is not needed eg.,.77 (with method seen) M5 Brackets may be recovered throughout 5th may be implied by solutions of their quadratic equation seen M M M M A is possible if they have a -term quadratic equation Answers only Zero 9 of 54

40 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method Triangles VMB and VXM (M is the midpoint of BC) 7 (6 ) 5 7 = VM + (6 ) (VM =) 7 (6 ) 5 5 Implies May be seen on diagram cos x = their 5 x is required angle dep on sin x = tan x = 5 ( ) their 5 5 ( ) dep eg crect method using cosine rule sine rule simplified to cos x = sin x = 9 sin their Allow 4 with crect wking SC Answer MARK SCHEME CONTINUES ON THE NEXT PAGE 4 of 54

41 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method Triangles BXM and VXB and VXM (M is the midpoint of BC) BX = (6 ) + ( ) 85 eg f BX and 7 their BX BX = ( BD) = 4 (6 + ) 7 = VX + their BX (VX =) 7 (their BX ) 4 6 [.9,.] Implies May be seen on diagram their [.9,.] tan x = sin x = their [.9,.] their VM dep x is required angle dep on eg crect method using cosine rule sine rule simplified to cos x = sin x = cos x = their VM 9 tan their [.9,.] 4.8 Allow 4 with crect wking SC Answer MARK SCHEME CONTINUES ON THE NEXT PAGE 4 of 54

42 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method Triangles VMB and VMN (M is the midpoint of BC, N is the midpoint of AD) 7 (6 ) 5 7 = VM + (6 ) (VM =) 7 (6 ) 5 5 Implies May be seen on diagram (8 cos their 5 their 5 their 5 their 5 ) dep dep on 4.8 Allow 4 with crect wking SC Answer Additional Guidance Alt rd their VM must be from crect method 4 of 54

43 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method B B their ( ) their dep Either der This mark cannot be implied Must have sced B B Crectly multiplies their pair of by matrices Must gain B B and scale fact MARK SCHEME CONTINUES ON THE NEXT PAGE 4 of 54

44 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 44 of 54 Alternative method Algebraic method B B their y x = y x their ( ) y x = y x This mark cannot be implied Must have sced B B Multiplications must be crectly wked out their ( ) their y x = y x their ( ) their y x = y x dep Multiplications must be crectly wked out y x and scale fact Must gain B B MARK SCHEME CONTINUES ON THE NEXT PAGE

45 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 45 of 54 ADDITIONAL GUIDANCE FOR Q IS ON THE NEXT PAGE Alternative method Unit square method B B their ( ) = their ( ) = This mark cannot be implied Must have sced B B Multiplications must be crectly wked out May be seen as three products their ( ) their = their ( ) their = dep Multiplications must be crectly wked out May be seen as three products and scale fact Must gain B B May be seen as three by matrices

46 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q) If both matrices are increct Zero Matrices must be used - igne diagrams In matrices, allow missing brackets inclusion of fraction lines Alt B gained then stated B M M A Allow enlargement f scale fact Do not allow f scale fact Scale fact with no valid wking Zero = sces B but ds not sce f the multiplication of two matrices with B sced Alt May also see wking f 46 of 54

47 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 x ( x ) x x (x + )(x + ) Crect common denominat f their two fractions Their two fractions must have different algebraic denominats (x + )(x + ) and x(x + ) dep Crect expressions f their numerats dep on nd 4 x + x + x + x x dep Subtracts their numerats Must expand all brackets Allow one sign err dep on nd (x )(x ) 4x 8x fraction in its simplest fm Additional Guidance Crect answer followed by increct further wk However, with an attempt to expand the denominat should (x )(x ) not be penalised even if their expansion is not crect M4 A nd and rd (x + ) must be used x x (x )(x ) x x (x )(x ) (no further crect wk) M A Allow = throughout, including in the answer 47 of 54

48 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 y-codinate of C is 5 (, 5) B May be on diagram implied by final answer x 6x Differentiates crectly 6 4 Substitutes x = in their dy dx their dy dx must be a function of x but cannot be x 5 y = their 4 (x ) dep on nd y = their 4x + c and substitutes (, ) dep y = 4x 7 y = their 4 ( ) Substitutes x = into their linear equation 5 y = 7 y = 7 (y-codinate of D =) 7 dep dep on rd May be seen on diagram their 4 sces rd and 4th ft ft B M4 with a positive length from their y-codinate of C 7 Additional Guidance A ( 5, ) is B unless recovered st Allow 6x + c (c an unknown constant) if c subsequently rejected Differentiates to 6x 5 gradient = 6 5 = 9 st M nd Could continue and gain rd and 4th rd Cannot be the equation of the nmal at (, ) 48 of 54

49 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method (LHS RHS) sin x ( sin x) Must see ( sin x) sin x + sin x = 4 sin x Alternative method (LHS RHS) Must see crect expansion SC Crect rearrangement of given identity to sin x + cos x = and (sin x + cos x) = and sin x + cos x = 6(a) cos x cos x = 4 cos x = 4( sin x) sin x = 4 sin x Alternative method (RHS LHS) Must see ( cos x) and ( sin x) Must see crect expansion SC Crect rearrangement of given identity to sin x + cos x = and (sin x + cos x) = and sin x + cos x = 4 sin x (sin x + cos x) Must see (sin x + cos x) 4 sin x sin x cos x = sin x cos x Must see crect expansion SC Crect rearrangement of given identity to sin x + cos x = and (sin x + cos x) = and sin x + cos x = MARK SCHEME CONTINUES ON THE NEXT PAGE 49 of 54

50 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method 4 (RHS LHS) 4 ( cos x) = 4 4 cos x Must see ( cos x) and sin x + cos x = 4 cos x and crect expansion = sin x + cos x 4 cos x 6(a) = sin x cos x SC Crect rearrangement of given identity to sin x + cos x = and (sin x + cos x) = and sin x + cos x = Alternative method 5 (LHS and RHS common expression) cos x cos x = 4 cos x and 4( cos x) = 4 4 cos x = 4 cos x B Must see ( cos x) and crect expansion SC Crect rearrangement of given identity to sin x + cos x = and (sin x + cos x) = and sin x + cos x = ADDITIONAL GUIDANCE FOR Q6(a) IS ON THE NEXT PAGE 5 of 54

51 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q6(a)) As shown in the mark scheme, allow = signs but they may be seen (crectly) as the identity symbol = signs may be implied (eg wking down the page, line by line) To give the wking must not need any further identities applying The other side of the identity may be seen throughout wking in Alts to 4 However, full wking on one side of the identity is needed f eg (Alt ) cos x cos x = 4 sin x 4 cos x = 4 sin x 4( sin x) = 4 sin x sin x = 4 sin x A (with 4 sin x = 4 sin x it would be ) 6(a) Other examples may be seen, escalate if necessary Allow any variable mixed variables no variables Allow (sin x) f sin x and (cos x) f cos x Allow s f sin x and c f cos x Do not allow sin x f sin x (but could still gain ) eg Alt sin x ( sin x) = sin x + sin x = 4 sin x eg Alt sin x ( sin x) = sin x + sin x = 4 sin x A M A Do not allow recovery of missing brackets as this is a proof SC Instead of factisation, they can divide by Other examples of SC may be seen where the identity is assumed to be crect and crect wking with use of sin x + cos x = is seen 5 of 54

52 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method sin x = sin x = 4 eg (sin x) = 4 sin x = 4 Allow f 6 Must have sin x = sin x = sin Allow s f sin x Do not allow sin x f sin x but may be recovered sin x = sin x = 4 Allow f 4 6 6(b) 6 and and 4 and with no other angles in range A 6 and 4 and Alternative method tan x = tan x = eg (tan x) = 6 4 Allow.7... f Must have tan x = tan x = tan Allow t f tan x Do not allow tan x f tan x but may be recovered tan x = 6 Allow.7... f 6 and and 4 and with no other angles in range A 6 and 4 and MARK SCHEME CONTINUES ON THE NEXT PAGE 5 of 54

53 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Alternative method cos x = 4 cos x = eg (cos x) = 4 6(b) cos x = 4 6 Must have cos x = cos x = cos Allow c f cos x Do not allow cos x f cos x but may be recovered cos x = cos x = and and 4 and with no other angles in range A 6 and and 4 ADDITIONAL GUIDANCE FOR Q6b IS ON THE NEXT PAGE 5 of 54

54 MARK SCHEME LEVEL TWO CERTIFICATE FURTHER MATHEMATICS 86 JUNE 6 Additional Guidance (Q6(b)) Igne any solutions outside of < x < 6 ie and 6 are outside the range and can be igned All four solutions with extra solutions in range, < x < 6,are penalised one accuracy mark eg Only penalise extra solutions in range when all four crect solutions are given Answer line blank, award any marks gained from wking lines If angles are found in wking lines but only some are listed on answer line award any method marks gained from the wking lines award any accuracy marks gained from the answer line eg Wking lines sin x = ± 6 and and 4 and Answer line 6 and and 4 6(b) eg Wking lines tan x = 6 4 Answer line 6 eg Wking lines sin x = Answer line 6 sin x = M A A Answers only can sce up to 4 marks All 4 crect 4 marks crect marks crect marks crect mark M M are possible eg sin x = 6 M eg sin x = 4 M Embedded answers can sce up to A Wking in rads grads can sce M marks if method seen 54 of 54