Version 1.0. Level 2 Certificate in Further Mathematics Practice Paper Set 2. Paper /2. Mark Scheme

Transcription

1 Version 1.0 Level Certificate in Further Mathematics Practice Paper Set Paper 8360/ Mark Scheme

2 Mark Schemes Principal Examiners have prepared these mark schemes for practice papers. These mark schemes have not, therefore, been through the normal process of standardising that would take place for live papers. It is not possible to indicate all the possible approaches to questions that would gain credit in a live examination. The principles we work to are given in the glossary on page 3 of this mark scheme. Evidence of any method that would lead to a correct answer, if applied accurately, is generally worthy of credit. Accuracy marks are awarded for correct answers following on from a correct method. The correct method may be implied, but in this qualification there is a greater expectation that method will be appropriate and clearly shown. Further copies of this Mark Scheme are available to download from the AQA Website: Copyright 011 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres for AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within the centre. Set and published by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales (company number 36443) and a registered charity (registered charity number ). Registered address: AQA, Devas Street, Manchester 5 6EX

3 AQA Level Certificate in Further Mathematics / - Set 1 - Paper - Practice Paper Glossary for Mark Schemes These examinations are marked in such a way as to award positive achievement wherever possible. Thus, for these papers, marks are awarded under various categories. M A B M Dep BDep ft SC oe Method marks are awarded for a correct method which could lead to a correct answer. Accuracy marks are awarded when following on from a correct 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 for a common misinterpretation which has some mathematical worth. Or equivalent. Accept answers that are equivalent. eg, accept 0.5 as well as 1 3

4 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics Paper - Calculator 1(a) y = 3 1 x B oe B1 (gradient) 3 1 oe B1 y = mx (± 0) m 3 1 oe 1(b) 6 + (= = 40) 6 Dep 6.3(4 ) or 6.35 sf answer needed 6.3 B1ft ft Any answer seen sf that is rounded correctly to sf (a) m 10 B1 (b) m 16 B1 (c) m B1 (d) m 3 B B1 6 m or 1 3 m 3 (4, 0) B1 (0, 10) B1 1 their 4 their 10 0 ft 4 5(x 4) + 3x = 30 Allow one error 5x 0 + 3x = 30 ft Their one error if made 8x = ft oe ft From 4

5 AQA Level Certificate in Further Mathematics /- Set - Paper - Practice Paper 5(a) 3 B1 5(b) B1 5(c) x = 4 or 14 = 3x (x =) Do not allow if (x =) also included as a solution (x =) (d) f(0) (= 4) or f(3) (= 5) or f(5) ( = 1) 4 f(x) 5 A oe eg, [ 4, 5] Answer uses 4 and 5 but incorrect notation used eg, 4 x 5 6(a) and 5 B1 6(b) x = 3 B1 oe 6(c) x 5 Bft ft Their two values in (a) B1 x sin p (= 90sin p) sin p = Dep their Accept 163 with correct working SC 1.4 or 1.5 5

6 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics (= 1) or 5 16 (= 1) 3 11 (= 14) 11 3 (= 14) or or 4 their ( )1 (= ( ) 1) or 4 their ( ) 14 (= 8) 3 their ( ) 1 (= ( ) 9) or 3 their ( ) 14 (= ( ) 6) 16 their 1 or 16 + their 1 or 3 + their 8 or 3 their their 9 or 5 their 9 or 11 their 6 or 11 + their 6 (4, 5) cos40 (= ) their Dep 55(.0 ) or B1ft ft 85 + their (3x + 4) B1 x(9x 16) (3x 4x)(3x + 4) or (3x 4)(3x + 4x) x(3x 4)(3x + 4) x( 3 x 4) oe eg, 3 x x 6

7 AQA Level Certificate in Further Mathematics /- Set - Paper - Practice Paper 11(a) a 1 + b = a + b and b 1 + a = b + a B1 11(b) a + b or 3b + a a + b = 3b + a (leading to a = b) 11(c) bn b Allow for correct numerator or bn b denominator correct b(n + 1) or b(n + ) Factorises either their numerator or their denominator b( n 1) b( n ) and shows simplification 1(a) B1 1(b) (0, 16) B1 1(c) 3x 18x + 4 B B1 Any one correct term 1(d) Their 3x 18x + 4 = 0 (x )( x 4) or oe eg, (3x 6)( x 4) 6 ( 6 ) x = and x = 4 ft x = and y = 4 y = 4 and y = 0 ft x = 4 and y = 0 1(e) Curve passing through (1, 0) and their (0, 16) and one maximum point at their (, 4) and one minimum point at their (4, 0) Bft B1ft Curve with at least two of the four properties

8 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 13(a) πx + πx + πx (= 4πx ) oe πx + πx + 4πx (= 6πx ) oe Their 4πx = their 6πy Allow if equating curved surface areas Shows working leading to x = 3 y 13(b) x = 3 y oe B and x = 1. y which is less than y oe 14 (x + 1) or (x ) or (x 5) x x or x 4x 5 or x x + 10 Product of their remaining linear factor and their quadratic eg, x 3 x x 5x + 5x + 10 x 3 6x + 3x + 10 ft Allow one error but no omissions 15(a) a = 3 B1 b = 4 B1ft ft 5 (their 3) 15(b) (p 1) = 1 m oe eg, m 1 = (p 1) p 1 = ( ) their1 m p = their 1 m 1 Aft ft From one sign error ft p = their 1 m 1 8

9 AQA Level Certificate in Further Mathematics /- Set - Paper - Practice Paper 16 a 5 a b and a 3 b B3 B At least six entries correct a b B1 At least three entries correct their = their + a or their 5 = their 3 + b or their a + b = their b a = 0 and b = ft Equates any pair correctly 1(a) x x 5 = 8. ( 0) oe eg, x 5 x = x = 1.3(0)(x + 5) oe eg, (x + 5) = 3x 10 1(b) their 10 their (0) 10 8 their 10 their (= ) (= ) ft ft Their 10 if answer in standard form 18(a) s s + s 1 4 terms with at least 3 correct s s 1 18(b) (sin + 1) (sin 1) or B1 1 ( 1) and 10 and 330 Bft ft Their factorisation or formula B1ft At least one correct solution 9

10 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 19(a) 19(b) 1 B1 y = their 1 x + c oe Point where tangent intersects y-axis (0, 13) and (13 y) = 4 + (5 y) Substitutes (4, 5) ie, 5 = their c y 13y + y = y + y ft From their 13 Allow two errors in expansions but must not omit any terms c = 3 y = 3 (5 their 3) + 4 (= 0) x + (y their 3) = their 0 ft Alt 19(b) P M Diagram with only one of 4 and correct C 4 c = 3 y = 3 (5 their 3) + 4 (= 0) x + (y their 3) = their 0 ft 0 ( 1) 3 + 4( 1) 5( 1) 8 (= ) Allow one slip (4) 3 + 4(4) 5(4) 8 (= ) (x + 1) a factor (x 4) a factor (x + 1)(x. 8) (x 4)(x. + ) (x + 1)(x + 3x 8) (x 4)(x + 8x + ) (x + 1)(x + a)(x + b) ab = their 8 or a + b = their 3 (x 4)(x + c)(x + d) ab = their or c + d = their 8 (x + 1)(x 4)(x + ) SC Exactly one correct linear factor seen SC4 Exactly two correct linear factors seen 10