Version 1.0. Level 2 Certificate in Further Mathematics Practice Paper Set 3. Paper /2. Mark Scheme

Transcription

1 Version 1.0 Level Certificate in Further Mathematics Practice Paper Set 3 Paper 8360/ Mark Scheme

2 Mark Schemes Principal Examiners have prepared these mark schemes f practice papers. These mark schemes have not, therefe, been through the nmal process of stardising that would take place f 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 wk to are given in the glossary on page 3 of this mark scheme. Evidence of any method that would lead to a crect answer, if applied accurately, is generally wthy of credit. Accuracy marks are awarded f crect answers following on from a crect method. The crect method may be implied, but in this qualification there is a greater expectation that method will be appropriate clearly shown. Further copies of this Mark Scheme are available to download from the AQA Website: Copyright 011 AQA its licenss. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres 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 centres to photocopy any material that is acknowledged to a third party even f internal use within the centre. Set published by the Assessment Qualifications Alliance. The Assessment Qualifications Alliance (AQA) is a company limited by guarantee registered in Engl Wales (company number ) a registered charity (registered charity number ). Registered address: AQA, Devas Street, Manchester 5 6EX

3 AQA Level Certificate in Further Mathematics / - Set 3 - Paper - Practice Paper 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 BDep 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 1 3

4 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics Paper - Calculat 1 Box 1 x + y = 4 B1 Box 3 y = 1 x B1 Box 4 y = 1 3x B1 Do not allow choice in any part Box 5 y = x + 1 B1 (a) h = 0.6m B1 60 eg, h = m 100 (b) their 0.6m 0.75m (= 0.8) 80 ft ft From their 0.6 3(a) ( 3) (b) x 7 = 1 eg, x = 8 (+) 4(a) x 4 B1 4(b) 14 4 (= ) Allow one sign err y 14 = their (x ) y = their (x 4) y = their x + c substitutes (, 14) (4, ) y 14 = (x + ) y = (x 4) Any crect fm eg, y + x = 10 Allow f(x) = 10 x t = 0 0 3t = 1 4t = 8 3t = SC seen 1 4

5 AQA Level Certificate in Further Mathematics /- Set 3 - Paper - Practice Paper 6 c + b = 5a c = a b 5 5 B1 c b 5 = a ft From their B1 if a the subject () c b 5 = a ft Only ft from B0 7(a) CBA = x base angles of isosceles triangle (are equal) DCE = x ACB = 180 x exteri angle = sum of interi opposite angles angle sum of triangle = 180 DCE = x (adjacent) angles on a straight line add up to 180 SC1 Crect solution without reasons 7(b) CDE = (180 x) base angles of isosceles triangle (are equal) CED = (180 x) base angles of isosceles triangle (are equal) 90 x AFD = 180 x (90 x) angle sum of triangle = 180 FEB = 90 x vertically opposite angles EFB = 180 x (90 x) angle sum of triangle = 180 SC Crect solution without reasons 5

6 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 8(a) True B1 False True B1 B1 8(b) x 3 x x x 4 B B1 One consecutive pair transposed eg x 3 x x 4 x 9(a) (= 35) Draws a Nth line at A marks 40 in crect place Accept any clear explanation 9(b) () cos 75 (= [ , 5043]) eg cos cos75 [ , 5043 ] [71, 71.01] Accept 70 with crect method seen 10 b = 4(a ) B1 b = 3a + k B1 Their 4(a ) = their 3a + k a = k + 8 ft ft From B0 B1 B1 B0 11(a) 48 B1 11(b) (= 65) 7 + their 48 (= 353) (= 5) 7 their (= [48.5, 48.51]) 7 their [48.5, 48.51] 5 [3.5, 3.51] ft ft Their 48 M3 1 q is odd (because f(0) = q) B1 f(1) = 1 + p + q 1 + q is even even + odd = odd eg, odd + odd + odd = odd 6

7 AQA Level Certificate in Further Mathematics /- Set 3 - Paper - Practice Paper 13 x 4 x B B1 F x 4 x 4x 3 1 ft Their x 4 their x 4x 3 1 ft Only ft from B0 14 : 3 3 : B1 ratio their their their 3 (11 1) their their their 3 (18 3) (5, 9) Aft ft f each Only ft from B0 15(a) (x 7)(x + ) B B1 x(x 7) + (x 7) (x + a)(x + b) where ab = 14 a +b = 3 15(b) 7 ft Their facts in (a) Their their ft 3 SC1 x = y 5 Alt 1 15(b) (y 5y 5y + 5) 3(y 5) 14 (= y 3y + 51) Allow one sign err in expansion (y 17)(y 3) 17 3 Alt Sub y 5 into (x 7)(x + ) ft Their facts in (a) 15(b) ie, ((y 5) 7)((y 5) + ) (y 17)(y 3)

8 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 16(a) x 4 B1 16(b) (x + 1) (x + 1) (x 1) B B1 (x + 1)(x 1) B1 Shows f(1) = 0 f(1) = 0 17(a) Square with vertices (0, 0) (3, 0) (3, 3) (0, 3) B B1 At least one of (3, 0) (3, 3) 3 (0, 3) (b) 0 1 B 1 0 B SC ( 3x 7)( x 3) ( x 1)( 4x 11) ( x 1)( x 3) ( 5) 6x + 14x + 9x + 1 4x + 4x 11x 11 (5)(x + x + 3x + 3) 6x + 14x + 9x + 1 4x + 4x 11x 11 (5)(x + x + 3x + 3) Their 6x + 14x + 9x their 4x + 4x 11x 11 = 5 their (x + x + 3x + 3) 4 terms with any 3 crect All 4 terms crect 9 5 8

9 AQA Level Certificate in Further Mathematics /- Set 3 - Paper - Practice Paper 19 Crect shape that has min max points in crect quadrants one intersection with x-axis shown B3 B Crect shape that has min max points in crect quadrants but incomplete (eg intersection with x-axis not shown) B1 Crect shape 0 First differences attempted Allow one err Second difference of 4 (= ) n Subtracts their n from terms of sequence ( 3 6 9) n 3n Alt 0 3 equations in 3 variables obtained Allow one err in cfficients eg, a + b + c = 1 4a + b + c = 9a + 3b + c = 9 Eliminates one variable to obtain equations in two variables eg, 3a + b = 3 5a + b = 7 Eliminates one variable eg, a = 4 n 3n 9

10 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 1 A (0, 1) AC = 6 B1 Gradient BC is 1 B1 eg, equation BC y = 1 x + 7 x + 1 = their 1 x + 7 eg, x 1 7 x = their 1 1 x = 6 (x =.4) eg, 5x = 1 ft From their 1 x + 7 their 1 1 their 6 their.4 7. ft ft From B1 B0 M3 B0 B1 M3 Alt 1 Gradient BC is 1 B1 eg Equation BC y = 1 x + 7 x = their x eg x + 1 = their 1 x + 7 5x = 1 (x =.4) eg 1 x = 6 ft From their 1 their 1 x + 7 y = (their 5.8 7) their.4 (their 5.8 1) their.4 7. ft ft From B0 M B1 M A0 10

11 AQA Level Certificate in Further Mathematics /- Set 3 - Paper - Practice Paper x y + 3xy 5x y xy B1 10x 3 y ( ) 4x y (+) 15x y ( ) 6xy 3 4 terms with 3 crect ft Their x y + 3xy 5x y xy 10x 3 y 4x y + 15x y 6xy 3 ft Fully crect ft Their x y + 3xy 5x y xy 10x 3 y +11x y 6xy 3 B1ft Only ft if their 4 terms above require simplification Alt 10x ( ) 4xy (+) 15xy ( ) 6y 4 terms with 3 crect 10x 4xy + 15xy 6y Fully crect 10x + 11xy 6y B1ft Only ft if their 4 terms above require simplification 10x 3 y +11x y 6xy 3 B1ft ft xy(their 10x +11xy 6y ) 3 (3x) 3 + 3(3x) 7x 3 7x seen 7x 3 (+) 7x 7x (x + 1) k = 7 Alt 3 x (x + 3) (3x) (3x + 3) 9x (3x + 3) 7x (x + 1) k = 7 4(a) s(5s ) B1 4(b) sin x (5sin x ) ft Their factisation in (a) sin x = 0 sin x = Aft ft From their factisation in (a) M A F exactly 5 crect solutions F 5 crect with other increct solutions F any two crect solutions seen 11

12 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 5 Any facts of 150 except 1 150, 75 3, 50 5, 30 6, 5 10, 15 c = 5 d = 6 x (+) 5x (+) 5x (+) 5 ft Their c 4 terms with at least 3 crect (x + 10x + 5)(x + 6) = x 3 (+) 10x (+) 5x (+) 6x (+) 60x (+) 150 ft their c their d Allow one err one omission x x + 5x + 6x + 60x ft Fully crect f their c their d a = 16 b = 85 ft ft Their expansion Alt 5 x 3 + dx + cx + cdx + c x + c d Allow up to two errs omissions Their c d = 150 c = 5 d = 6 Their d + c = a their cd + c = b a = 16 ft ft Their d + c their c their d b = 85 ft ft Their cd + c their c their d 1