AQA Qualifications GCSE MATHEMATICS. Topic tests Higher tier - Mark schemes

Transcription

1 AQA Qualifications GCSE MATHEMATICS Topic tests Higher tier - Mark schemes

2 Our specification is published on our website ( We will let centres know in writing about any changes to the specification. We will also publish changes on our website. The definitive version of our specification will always be the one on our website, this may differ from printed versions. You can get further copies of this Teacher Resource from: The GCSE Mathematics Department AQA Devas Street Manchester 5 6EX Or, you can download a copy from our All About Maths website (

3 Contents Algebra 4 Number 7 Probability and statistics Problem solving 4 Real life 9 Shape Glossary for mark schemes 7 Version.0 3

4 Algebra Q Answer Mark Comments (a) 5x 0 (= 35) x = 7 5x = 45 x = A ft ft for M0 or M0 (b) 9y = 3y or 6y 9y (= 3y) 3 (= ) or 3 (= ) 4 A ft ft for M0 or M0 with only one rearrangement error (a) x 6 B (b),, 3, 4, 5, 6 B B For 5 correct and missing B For 6 correct and incorrect B For x 7 B0 For or more errors,, 3, 4, 5 B,, 3, 4, 5, 6, 7 B, 3, 4, 5, 6, 7 B0 3 y = 3x + 6 B3 oe B y = 3x c or 3x + 6 B Indication that gradient is 6 or 3 or y = mx Version.0

5 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments 4 (x + )(x + 7) B Either order B For (x + a)(x + b) where a + b = 9 or ab = 4 5 8x + 4y (= + 7y) x + y = 7y 4 8x = + 7y 4y 8x = + 3y x = 7y 4 y x = 3y A ft ft M0 or M0 and only one error 8 in expansion or rearrangement 3y SC 8 6 6x + y = 3 and 6x 0y = 4 or 0x + 0y = 5 and x 0y = 8 Either x =.5 or y = 0.5 Substituting their x or y into any of the linear equations and solving for the other variable, or balances again to eliminate and solve the other variable A dep Condone poor arithmetic if one coefficient is balanced 33, Condone poor arithmetic and rearrangement errors if the intention to solve is clear Either y = 0.5 or x =.5 A oe SC If T&I used and both answers correct Version.0 5

6 Q Answer Mark Comments 7 Alternative method abx + a x + b x + ab or ab = 0 Identifies and 0 or and 5 9 or 0 A 9 and 0 A Correct answer gets all 4 marks Alternative method (x + 0)(0x + ) (x + 5)(5x + ) 0x + 00x + x + 0 0x + 4x + 5x or 0 A 9 and 0 A Correct answer gets all 4 marks 6 Version.0

7 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Number Q Answer Mark Comments (a) A oe fraction 4 Bft Correctly changes their improper fraction to a mixed number (b) Yes with correct comparison and 0 0 B oe or ( ) or and % or 45.5% and 45% or < 0 oe implies Yes. or 0% B A Version.0 7

8 Q Answer Mark Comments 3 Alternative method (= ) oe or seen (litres) = their their 5 (= 9) Their 5 cannot be or 7 A SC 60 Alternative method 6 oe Diagram with and indicated identified between 8 and 8 6 Each section = 9 7 A SC 60 Alternative method 3 x 3x + 45 = oe 8 8 x = 360 = 6x oe 360 = 5x 7 A SC 60 4(a) (0).0046 B 4(b) ( ) 900 or 00 or or or other correct equivalent expression (.0) 0 A 8 Version.0

9 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments 5(a) 3 or 64 or ( ) 4 or 6 or ( ) 6 8 A 5(b) or 7 7 or A 5(c) Alternative method ( 6 ) ( ) oe Any expansion with 4 correct terms implied A oe eg A ft ft 8 + their (a) for term Alternative method ( 6 ) ( + ) 6( + + ) A 8 + A ft Version.0 9

10 Q Answer Mark Comments 6 9 = 3 or ( 7) = B or 3 8 = = 83 or or 3 8 or 3 8 or = or oe implies 3 8 = A All three numbers correct evaluated and in correct order ( 7) A Version.0

11 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Probability and statistics Q Answer Mark Comments (a) or 0 5 oe 4 A (b) or or 0 0 or or dep oe their 0 + their 5 45 A (c) 3 out of or out of 8 3 or or 8 oe 3 : or : 8 3 out of and out of 8 3 or and 8 or 4 or 5% or 0.5 A oe 3 : and : 8 All answers must be correct States the same Q ft Strand (iii) Must see a correct comparison from their relative frequencies dependent on SC For 0 3 and 0 and states boys larger oe Version.0

12 Q Answer Mark Comments (a) No response section or No mention of websites or No mention of buying music B oe (b) Suitable question B eg Where do you buy music? Suitable response section B eg bookshops, websites, don t buy music Must include both shops and websites (c) Reason involving time or location B eg only Monday, only one morning, only customers asked, only in the shop (d) Complete description including correction of time and location B B Description correcting one problem Accept increased sample size as one of time / location 3 Fully correct labelled pie chart Spain 80 Portugal 90 Turkey 30 Other 60 Tolerance B4 B3 Two or three correct sectors and four sectors labelled correctly B3 Fully correct but incomplete or no labels B All angles calculated B Two or three sectors correct but incomplete or no labels B At least one angle calculated in table B One sector drawn an labelled correctly Version.0

13 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments B oe B At least one correct pair of probabilities or all top probabilities = or all bottom probabilities = oe or May be at the end of tree diagram 8 5 or 0.3 A ft oe ft their tree diagram (= 90) + 64 (= 354) (64 58) 6 (= ) Their dep 58 + their NB is M 6 59 A Version.0 3

14 Problem solving Q Answer Mark Comments Alternative method 3 5 (= 7) Check diagram y coordinate = 9 A (= 9) oe eg 6 7 x coordinate = 3 A (9, 3) is A max Alternative method Graph drawn with A at (5, 0) and B at (3, 36) Any rectangles drawn from A and B matching the diagram. x coordinate = 3 y coordinate = 9 A A 4 Version.0

15 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments Alternative method Any side chosen for square and squared, eg 0 = 00 Works out 75% of their square and a correct calculation of the circle area, or works out what percentage the circle area is of the square area A method mark gained and correct conclusion based on 75% of their square with their circle Alternative method r length of side of square giving 4r as area r as radius of circle giving r as area of circle 75% of their square (= 3r ) and correct expression for area of circle with their chosen radius A method mark gained and correct conclusion based on 75% of their square with circle eg 3 M A Q A Q M Is for both squares and circle areas attempted with correct numerical values (eg if 0 chosen for side of square, then 5 must be used as radius of circle, or if 4 chosen as radius then 8 is used as side of square) If both square and circle area attempted with one incorrect numerical value (eg if 0 chosen as side of square, then 0 used as radius of circle, or if 4 chosen as radius then 4 used as side of square) This can be awarded even if only awarded Allow used if a clear comparison, eg Strand (ii) Do not award if their circle area square area eg Strand (ii) Do not award if their circle area square area eg r r Version.0 5

16 Q Answer Mark Comments 3 Alternative method 3x (x 5) Condone omission of brackets x + 5 = 7 6 A SC Alternative method 3 x = 7 x 5 3x = + x 6 A SC Alternative method 3 Substitutes a value for x and evaluates correctly as a power of Substitutes a different value for x and evaluates correctly as a power of which is closer to 7 6 A SC 4 7 or [43.9, 44] oe or [0.9, ] or 7 3 or [3.9, 33] or [3.9, 3] dep dep oe 7 oe 4 oe [46.9, 47] A oe SC For [3.4, 3.5] or [30.4, 30.5] SC For [6.4, 6.5] 6 Version.0

17 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments 5 4 and and 00 B B For any correct value of n 0^n, where n > 00, 3000, , , etc 6 Alternative method n( n ) n( n ) B This mark is for combining fractions or if fractions dealt with separately, for combining n terms correctly n n n n is B0 as incorrect 4 combining of fractions n n n n = n B This is for eliminating n and n either by showing by crossing or writing on same line and writing next line without them n n n + n n + = n + n = n B This mark is for cancelling top and bottom n n + = n Alternative method n B This mark is for factorising out a ((n ) + (n + )) common factor n (n + n + ) is B0 4 n B This mark is for combining terms (n) inside bracket correctly n B n is OK Version.0 7

18 Q Answer Mark Comments 7.5 or 3 seen oe : 8 or seen as a scale factor 8 : 8 tan C = or 36 8 EC = or = or EC or or 8.5 dep oe CE = tan(their 36) 6.5 or A 6.5 ( ) or 5.5 ( ) (8 + ) their 5.5 or their 99 8 their their ED 8 + their ED 4 (8 + ) their ED 8 Version.0

19 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Real life Q Answer Mark Comments (a) 5 B (b) Any correct conversion between miles and km seen eg 5 miles = 8 km or mile =.6 km or km = 8 5 mile Slower as limit is 8 km A Slower as Alternative method 57 B Driving school A total ( 47) or 3.45 or 6.5 (3 their 3.45) 47 or 3 47 their (85) or 99 or 0.(36 ) A Total for B or Price per lesson for A (Driving school) B Q ft Strand (iii) ft conclusion based on two values if awarded Alternative method 57 B Driving school A total 47 3 or 08 Their (85) or 99 A Driving school B total (Driving school) B Q ft Strand (iii) ft conclusion based on two values if awarded Version.0 9

20 Q Answer Mark Comments 3(a) Alternative method Midpoints seen or implied B 5, 5, 5, 35, 45 Their fx or or 970 This mark is for the sum of their midpoints frequencies but condone one error 5 5 = 5 5 = = = = 80 Their fx 80 dep Their ( ) A Accept 5 with working shown 3(b) Alternative method or or ( )(%) or 69 and No A 3. ( )(%) and No Alternative method or or or or and 56 and No or 56 is in the group so No A 4 and 5 and No 0 Version.0

21 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments (= 304) or AB + 64 = cos (C) = 80 their 304 (= 48) cos - 64 (= [36.8, 36 9]) their 48 (= 536) sin their [36.8, 36 9] (= 536) Their oe [46, 433.5] A Allow 430 with correct working seen 400 B ft ft value seen > 3sf rounded correctly to sf A Is implied by 400 if no incorrect working seen (= 6000) Answer of implies this ( %) Their 6000 = 90% Implied by division by 90 Their ( 00) or A Version.0

22 Shape Q Answer Mark Comments (a) Alternative method or A Attempts both ways obtaining two equal totals (35.5 if correct) Alternative method 3m or 4m Q Strand (iii) Applies full method in an attempt to show width = 0.5 Must score A0 Q possible A Q0 possible 3m = 4m A oe eg 3m + 34 = 4m Solves their equation correctly (m = 0.5 if correct) Q Strand (iii) Solves an equation in an attempt to show width = 0.5 Must score A0 Q possible A Q0 possible Version.0

23 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments (a) Alternative method 3 ( width =) (= 4.5) ( width =) A or Obtains a value for one width (0.5 if correct) Alternative method (= 34) or (= 33.5) Q Strand (iii) Attempts to show width = 0.5 using dimensions 8.5 and 4 Must score A0 Q possible A Q0 possible 33.5 and 34 A Subtracts their 34 and their 33.5 Q Strand (iii) Attempts to show width = 0.5 using dimensions 8.5 and 4 Must score A0 Q possible A Q0 possible (b) (= 3) (= 4.5) (= 9) Their 3 4 dep Their Their A SC 6.5 Version.0 3

24 Q Answer Mark Comments 3(0 x) or 30 3x Do not accept x = 3(0 x) Do not accept x = 30 x 8 5x x or x = 30 3x dep x = 0 x 5x + 3x = 30 8 Collecting their 4 terms ( stages) oe 5x + x = or 3 or A dep on 3rd 3 Alternative method (x =) cos 7 4 cos A.36 or.4 is A0 [8.385, 84] A Alternative method XB = and AX =.47 and XC = A X B C ( [8.385, 8.4] A 4 Version.0

25 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Q Answer Mark Comments 3 Alternative method 3 CX = and AX =.854 and XB = 6.45 A X B C ( [8.385, 8.4] A 4(a) 50 B 4(b) or 0 or 360 their or their 0 dep oe OCA = 8 seen or implied or or 87 oe OCB = 75 8 or 57 seen or implied or A or 6.8 Their 6.8 = π diameter dep oe Their 6.8 = π radius their 6.8 π dep Their 6.8 π radius = [9.95, 0] [9.9, 0] A SC For [4.9, 5] Version.0 5

26 Q Answer Mark Comments 6 w + 4e0 = 7 May be on diagram (w = ) 3 seen A w = 64 or w = their 3 or third angle = 7 or w + t + 7 = 80 oe or 80 7 their 3 oe A 6 Version.0

27 AQA GCSE MATHEMATICS TOPIC TESTS HIGHER TIER MARK SCHEMES Glossary for mark schemes GCSE examinations are marked in such a way as to award positive achievement wherever possible. Thus, for GCSE Mathematics papers, marks are awarded under various categories. M A B ft SC M dep B dep 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. Follow through marks. Marks awarded for correct working following a mistake in an earlier step. Special case. Marks awarded within the scheme for a common misinterpretation which has some mathematical worth. 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. Or equivalent. Accept answers that are equivalent. eg, accept 0.5 as well as [a, b] Accept values between a and b inclusive. 3.4 Allow answers which begin 3.4 eg 3.4, 3.4, Use of brackets It is not necessary to see the bracketed work to award the marks. Version.0 7

28 Acknowledgement of copyright holders and publishers Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright holders have been unsuccessful and AQA will be happy to rectify any omissions of acknowledgements in future papers if notified. Copyright 04 AQA Education and its licensors. All rights reserved. AQA Education (AQA) is a company limited by guarantee registered in England and Wales (company number ) and a registered charity (registered charity number ) Registered address: AQA, Devas Street, Manchester 5 6EX. SP/03/4