# Version 1.0. Level 2 Certificate in Further Mathematics June Paper /2. Final. Mark Scheme

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1 Version.0 Level Certificate in Further Mathematics June 0 Paper 80/ Final Mark Scheme

3 Mark Scheme Paper Level Certificate in Further Mathematics - 80/ June 0 Glossary f Mark Schemes These examinations are marked in such a way as to award positive achievement wherever possible. Thus, f these papers, marks are awarded under various categies. M A B M dep B dep ft SC Method marks are awarded f a crect method which could lead to a crect answer. 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 method mark dependent on a previous method mark being awarded. 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 0.5 as well as

4 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics Q Answer Mark Comments r = 5 r = 5 r = 5 B May be seen on diagram d = 0 ( their r) π their r [.45,.5] 00 5π ft ft from B0 Allow with wking (uses 5π = 79) Igne any units seen (a) 8 w < w... w < w < w ft ft A0 and inequality of fm a w < b a w b SC Answer with M0 SC Answer 9 5 with M0 8 SC < w (b) B their min from (a) (c) ft ft their min from (a) (a) (5, 0) B (5x, 0y) is B0 Check diagram f answer written next to P if answer line is blank (b) Crect elimination of a letter eg x = 5 x eg y = 5 y Crectly collects terms dep eg y y = 5 eg x x = 5 (, ) Allow x = and y = if not contradicted on answer line 4

5 Mark Scheme Level Certificate in Further Mathematics - 80/ - June 0 - Paper (c) their 5 their eg their 5 from (a) and their from (b) 5 ft ft their 5 from (a) and their from (b) 4 (a) B n 0.4n 5 (0m =) 0 their 5 n (= 4n) 0 (= 0) and 5 (= 5) 4 (b) 4 : ft numerical ratio of integers ft their 5 n if used 5x 5x 5x 9 4 terms with crect including a term in x 5 5x 5x 5x 9 Fully crect 5x 0x 9 Crectly differentiates their quadratic ft their 5x 5x 5x 9 50x x 0 0(5x ) 5(0x ) (5x 5) ft ft A0 if their 50x 0 factises to a(bx c) where a, b and c are integers > Alternative (5x ) 5 M 0(5x ) 5(0x ) A (5x 5) (c 4)(c ) (c ) Crect factisation (a) ( c 4)( c ) = ( c ) c 4 c 4 Must be a fraction and completed to Crectly converts to a common denominat eg ( c 4) c M c 8 c eg ( c 4) 8 ( c) 8 5

6 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics (b) Crectly expands their brackets (must have common denominat) c 8 c Allow if their first line of wking is c 4 c c 4 c c 8 c 5.8(..) Alternative method 5. A0 8 5 A0 8 A0 Crectly converts to a common denominat eg ( c 5c 4) ( c)(c ) (c ) (c ) May also expand the denominat Crectly expands their brackets (must have common denominat) c 0c 4 9c 9 c (c ) c May also expand the denominat c 0c 4 (c ) 9c 9 c c (c ) 5.8(.). 5. A0 8 5 A0 8 A0 Scale on the y-axis identified crectly eg Intercept of line A with y-axis identified as Scale on the x-axis identified crectly eg Intercept of line A with x-axis identified as B Must be unambiguous identification B Must be unambiguous identification 7 Crect attempt at gradient ft their scales eg their 5 their y = 5 x 5 y = 5x 0 ft ft B0 B B B0 5 x 5 is B A0

7 Mark Scheme Level Certificate in Further Mathematics - 80/ - June 0 - Paper 8 (a) y = y = 0 B Allow y = 0x 8 (b) x = x = 0 B 8 (c) < x < B Unambiguously selected (hizontal =) 8 cos 4 (= [5.9, ]) (hizontal =) 8 sin 48 (= [5.9, ]) M cos 4 = 8 x sin 48 = 8 x 9 (vertical =) 8 sin 4 (= [5.5, 5.4]) (vertical =) 8 cos 48 (= [5.5, 5.4]) M (x is the hizontal) y y sin 4 = cos 48 = 8 8 (vertical =) 8 their [5.9,] (= [5.5, 5.4]) (y is the vertical) 8 their [5.9, ] [5.4, 5.5] Alternative (vertical =) 8 sin 4 (= [5.5, 5.4]) (vertical =) 8 cos 48 (= [5.5, 5.4]) M sin 4 = 8 y cos 48 = 8 y (y is the vertical) (hizontal =) 8 cos 4 (= [5.9, ]) (hizontal=) 8 sin 48 (= [5.9, ]) (hizontal =) 8 their [5.5, 5.4] M x y cos 4 = sin 48 = 8 8 (x is the hizontal) (= [5.9, ]) 8 their [5.5, 5.4] [5.4, 5.5] SC [.8,.9] 0 Straight line through (, 0) and (0, ) B Lines must be ruled Straight line through (0, ) and (, ) Straight line through (, ) and (,) B B Only penalise (by mark) extended lines if B B B SC Any graph that passes through (, 0) and (0, ) and (, ) and (, ) (a) a 0 b c B B crect elements b (b) a = Bft ft their matrix in (a) if (a) crect ft their b when finding c b = Bft 7

8 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics c = Bft 5n 5n n 4 terms with crect including a term in n 5n 5n n Fully crect eg 5n n n (n ) states that both terms are multiples of Identifies (, ) (5, ) B May be implied by seen in a table of values on a graph as a mapping (eg ) their their their 5 their (= ) y their = their (x their ) y their = their (x their 5) y = their x c and substitutes their (, ) their (5, ) (y =) x Alternative Identifies (, ) (5, ) B May be implied by seen in a table of values on a graph as a mapping (eg ) their their their their 5 (= ) y their = their (x their ) y their = their (x their 5) y = their x c and substitutes their (, ) their (5, ) (y =) x Alternative m c = 5m c = B m c = 5m c = Eliminates a letter from their equations eg 5m m = Eliminates a letter from their equations eg 5m m = m = c = m = c = (y =) x (y =) x 8

9 Mark Scheme Level Certificate in Further Mathematics - 80/ - June 0 - Paper First and second differences crect ie 4 8 (0) () Crectly subtracts their n from given sequence ie 0 ( 4) ()n dep dep on M n n 9 eg n n 0 4 Alternative method Any three of a b c = 4a b c = 5 9a b c = a 4b c = 9 Allow one err but each of their three equations must have a, b and c 5a 5b c = 9 Eliminates one variable to obtain a pair of equations in two variables eg a b = 4 and Allow one err 5a b = Eliminates one variable crectly dep dep on M eg a = n n 9 eg n n 0 5 (a) 9 a ( ) b a ( ) b 0 a 9 ( ) b 0 4 b A a b 4 ( is implied) a ( ) b 4 a b (a ( ) b ) a SC a ( ) b 4 ( c) 9

10 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics 5 (b) q ( ) r q ( ) r B B q r 4 ( q ( ) r ) ( 4 q ( ) r ) q ( ) r 4 q ( ) r 4 (q ( ) r ) - p = q ( ) r = q ( ) r p p = q ( ) r 4 p = 4 q ( ) r Crect expressions value f any three of these angles B O is the centre of the circle angle PAC = x angle CAB = 90 angle PBA = x angle PCA = 80 x 90 x angle ACB = 90 x x angle COA = x 90 x angle PAO = 90 angle CAO = 90 x x angle BAD = x 90 x angle AOB = 80 x 90 x angle OAB = x Writes a crect equation that has solution 0 eg PAC CAB x PBA = 80 eg PCA ACB = 80 eg ACB CAB CBA = 80 eg 4 PAO APC POA = 80 0 D is the point at the end of PA extended B Any crect B Any crect 0

11 Mark Scheme Level Certificate in Further Mathematics - 80/ - June 0 - Paper Q Answer Mark Comments 4(x ) x Must be crect 4( x ) ( x )( x ) x ( x )( x ) 7 4x x (= 5x 0) 4x ( x )( x ) x ( x )( x ) 5(x )(x ) Must have 5 and be crect Must be in an equation and not a denominat eg (5x 0)(x ) (5)(x x x ) 5 may be missing Must be in an equation and not a denominat 4 terms including term in x with crect eg x x eg 5x = 40 eg 5x 40 = 0 5x 5x 0x ( err) Must collect all terms and have an equation Crect attempt at solution of their quadratic eg x = 40 5 dep dep on M Quadratic fmula must have no errs in substitution If completing square must have no errs up to p(x q) = r p(x q) r = 0 [.8,.8] and [.8,.8] ft eg () 8 and 8 ± 8 ft their quadratic equation if M4 SC7 Both solutions crect (no valid method) SC One solution crect (no valid method) x b At least one term crect Substitutes into their equates to zero dy dx and dep Must have a term in x b = 0 8 ( ) b = 0 b = ( ) their b( ) c = 0 dep dep on M and having a value f b c = 4 ft ft their b and M A0 dep with no errs in their final

12 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics Q Answer Mark Comments 9 (a) ( ) (=.5) and 4 (=.5) 9 (b) 5.5 = 0.5 = 4.5 = 4.5 = 5.5 and = = 4.5 = 4.5 = = 4.5 tan MBN = sin MBN = cos MBN = 5.5 = Must be crect eg tan [.8,.804] 9 (c) 4.5 sin BNC = cos BNC = tan BNC = eg sin eg cos BNC = 4.5 [4, 4.004] SC [.89897, 7] Alternative BD = 4.5 BD = 4.5 and cos BND = their BD [4, 4.004] SC [.89897, 7] Alternative

13 Mark Scheme Level Certificate in Further Mathematics - 80/ - June 0 - Paper sin XNB = (= [5., 5.]) X is midpoint of AB 4. 5 cos XNB = (= [5., 5.]) tan XNB = 4.5 (= [5., 5.]) [4, 4.004] SC [5, 5.004] 0 ( ) π ( ) (p) ( ) 5p (= p ) B Missing brackets B0 unless recovered 0 their ( ) π ( ) (p) ( ) 5p = 500π May be implied by wking f eg 0 p = 500π π may already be cancelled value f π may be substituted in Must be equating two volumes Crectly rearranges to p = 0 eg p = 500π their dep eg p = 75 5 SC [8.8, 8.9] a 7a a Must be crect a 7a dep Must be crect May also see fact a (a )(a ) May also see fact a ft ft A0 Other solutions may be seen but must be selected as their answer Alternative method (x a)(x ax ) Must be crect (x) a (x) = 7a(x) dep Equating cfficients of x a 7a and (a )( a ) ft ft A0 Other solutions may be seen but must be selected as their answer

14 Mark Scheme Paper June 0-80/ - Level Certificate in Further Mathematics Q Answer Mark Comments tan Ɵ(tan Ɵ ) tan Ɵ = 0 sin Ɵ(sin Ɵ cos Ɵ) sin Ɵ = 0 eg t(t ) Must be crect 80 tan Ɵ = [08, 08.44] [88, 88.44] Bft ft 80 any angle (other than 0 and 90) if in range Appropriate and crect sine rule in triangle ABP eg BP sin x = AB sin 0 eg BP sin x = AB sin 0 Appropriate and crect sine rule in triangle ACP eg PC = sin x AC sin50 eg PC sin x = AC sin50 Eliminates sin x Must have eg PC BP sin 0 AB = AC sin50 eg BP PC sin 0 = sin 50 AB AC States sin 0 = sin 50 dep dep on eg Substitutes sin 0 = and sin 50 = Completes fully Must have all 4 previous marks eg PC BP = AC AB and AB BP = AC PC SC Sine rule in triangle ABP using angle 50 x Sine rule in triangle ACP using angle 0 x 4

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