maths 2020 past paper 1 mark scheme

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must mark the last candidate in exactly the same way as they mark the first. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded; exemplification/indicative content will not be exhaustive. When examiners are in doubt regarding the application of the mark scheme to a candidate's response, the response should be sent to review. All the marks on the mark scheme are designed to be awarded; mark schemes should be applied positively. Examiners should also be prepared to award zero marks if the candidate's response is not worthy of credit according to the mark scheme. If there is a wrong answer (or no answer) indicated on the answer line always check the working in the body of the script (and on any diagrams), and award any marks appropriate from the mark scheme. Questions where working is not required: In general, the correct answer should be given full marks. Questions that specifically require working: In general, candidates who do not show working on this type of question will get no marks full details will be given in the mark scheme for each individual question. Crossed out work This should be marked unless the candidate has replaced it with an alternative response. Choice of method If there is a choice of methods shown, mark the method that leads to the answer given on the answer line. If no answer appears on the answer line, mark both methods then award the lower number of marks. Incorrect method If it is clear from the working that the "correct" answer has been obtained from incorrect working, award 0 marks. Send the response to review for your Team Leader to check. Follow through marks Follow through marks which involve a single stage calculation can be awarded without working as you can check the answer, but if ambiguous do not award. Follow through marks which involve more than one stage of calculation can only be awarded on sight of the relevant working, even if it appears obvious that there is only one way you could get the answer given. 7 8 9 10 11 12 13 14 Ignoring subsequent work It is appropriate to ignore subsequent work when the additional work does not change the answer in a way that is inappropriate for the question or its context. (eg. an incorrectly cancelled fraction when the unsimplified fraction would gain full marks). It is not appropriate to ignore subsequent work when the additional work essentially makes the answer incorrect (eg incorrect algebraic simplification). Probability Probability answers must be given as a fraction, percentage or decimal. If a candidate gives a decimal equivalent to a probability, this should be written to at least 2 decimal places (unless tenths). Incorrect notation should lose the accuracy marks, but be awarded any implied method marks. If a probability fraction is given then cancelled incorrectly, ignore the incorrectly cancelled answer. Linear equations Unless indicated otherwise in the mark scheme, full marks can be gained if the solution alone is given on the answer line, or otherwise unambiguously identified in working (without contradiction elsewhere). Where the correct solution only is shown substituted, but not identified as the solution, the accuracy mark is lost but any method marks can be awarded (embedded answers). Range of answers Unless otherwise stated, when an answer is given as a range (eg 3.5-4.2) then this is inclusive of the end points (eg 3.5, 4.2) and all numbers within the range. Number in brackets after a calculation Where there is a number in brackets after a calculation eg 2 x 6 (12) then the mark can be awarded either for the correct method, implied by the calculation or for the correct answer to the calculation. Use of inverted commas Some numbers in the mark scheme will appear inside inverted commas eg "12" x 50; the number in inverted commas cannot be any number - it must come from a correct method or process but the candidate may make an arithmetic error in their working. Word in square brackets Where a word is used in square brackets eg [area] x 1.5 the value used for [area] does not have to come from a correct method or process but is the value that the candidate believes is the area. If there are any constraints on the value that can be used, details will be given in the mark scheme. Misread If a candidate misreads a number from the question. eg uses 252 instead of 255; method or process marks may be awarded provided the question has not been simplified. Examiners should send any instance of a suspected misread to review. Guidance on the use of abbreviations within this mark scheme M P A C B oe cao correct answer only follow through (when appropriate as per mark scheme) special case dep dependent (on a previous mark) indep independent awrt answer which rounds to isw ignore subsequent working ft method mark awarded for a correct method or partial method process mark awarded for a correct process as part of a problem solving question accuracy mark (awarded after a correct method or process; if no method or process is seen then full marks for the question are implied but see individual mark schemes for more details) SC communication mark awarded for a fully correct statement(s) with no contradiction or ambiguity unconditional accuracy mark (no method needed) or equivalent Paper: 1MA1/1H Question 1 2 3 4 Answer 3n-2 Shown BCDA A & D Mark B2 (B1 M1 M1 C1 B2 (B1 B1 for 3n - 2 oe for 3n+ k where k #-2 or is absent unambiguously shown) 7 15 for conversion to improper fractions eg. -or 3 4 (dep) for method to multiply fractions, 7x15 105 ·(=₁ -) or 3x4 12 28×45 1260 -(= -) oe 12×12 144 eg. Mark scheme 35 9 for complete working showing each stage as far as- or 8- 4 12 cao for two or three correct) cao Additional guidance Accept a different variable, eg. 3x-2 n = 3n-2 gets B1 only n+ 3 gets NO marks Need not be shown with operators Paper: 1MA1/1H Question 5 6 Answer 20 85 with working and reasons Mark P1 P1 A1 M1 M1 Al C2 (C1 Mark scheme for process to find SP of 24 chocolate bars, eg. 0.50 × 24 (= 12) oe or for process to find the overall profit eg (24 x 0.5) - 10 (2) or for process to find CP of one chocolate bar, eg. 1000 ÷ 24 (= 41.66...) oe (dep) for start to a process to find percentage profit, eg. using "12" 10 "12"-10 10 50-"41.66..." "41.66..." oe with consistent units or cao or for correct use of corresponding angles eg AEB = 63 or co-interior angles eg BCD = 180- 148 (32) or DEB= 180-63 (= 117) (dep) for a complete method to find angle EAB eg. 180 "63"-(180- 148) or 148 - "63" or "117" - (180-148) for EAB= 85 (identified) (dep on M2) all working correct with all appropriate reasons stated. Corresponding angles are equal Allied angles / Co-interior angles add up to 180 Angles on a straight line add up to 180 Angles in a triangle add up to 180 The exterior angle of a triangle is equal to the sum of the interior opposite angles. for one reason relating to parallel lines clearly used and stated or for any two reasons clearly stated for their fully correct method) Additional guidance Working can be carried out in either pounds or pence. Angles must be clearly labelled on the diagram or otherwise identified. Full solution must be seen. Correct method can be implied from angles on the diagram if no ambiguity or contradiction. When reasons are given the key words underlined must be present. Reasons need to be linked to their method; any reasons not linked, do not credit. There should be no incorrect reasons given. Paper: 1MA1/1H Question 7 8 9 Answer 20 or 24 or 168 Comparison 450 0.000 672, 67.2 x 10 6.72 105 672 × 104 Mark B1 C2 (C1 M1 M1 Al B2 (B1 Mark scheme for identification of the range of the girls (20) or the range (24) or the median (168) of the boys for a correct comparison of medians and a correct comparison of ranges supported Simply quoting values for median, by correct figures range is insufficient; they must be compared. eg the median height for girls (165) is less than the median height for boys (168) and the range for girls (20) is less than the range for boys (24) At least one comparison must be in context referring to height or quoting cm. for a correct comparison of medians or a correct comparison of ranges that could ft their incorrect figure(s)) for 18+ 3 (6) for substitution eg. 75=- cao F "6" cao or 75 x "6" Additional guidance for correct conversions to same format, condoning one error or for 3 numbers in the correct order (ignoring one) or for all 4 numbers listed in reverse order) Context not necessary for C1 Ignore units Accept correct numbers in any form Paper: 1MA1/1H Question 10 Answer 6:15:20 Mark P1 P1 A1 Mark scheme chooses a multiplier to equate the two fractions in terms of b 15 2 3 (=. ) or X 5 3 15 3 4 (= ) 20 or lists equivalent fractions to eg 555 up to at least 3 or lists equivalent fractions to or (a : b =) 2:5 and (b: c=) 3:4 or for 6 15 or 15 : 20 seen up to at least 6 15 15 20 3 3 eg. -, eg. 2 4 6 5 10 15 36 9 12 15 16 20 4 8 12 6 and 15 3 5 15 4 5 20 puts into related terms ready for ratio eg 5 or for (a : b =) 6:15 and (b: c =) 15:20 or lists equivalent ratios up to a common element for b, eg a : b = 2:5, 4: 10, 6 : 15 and b: c= 3:4, 6:8, 9 : 12, 12 : 16, 15 : 20 for 6 15 20 oe Additional guidance Need not be written in ratio form Accept equivalent ratios Accept a =6, b= 15 and c = 20 Paper: 1MA1/1H Question 11 (a) (b) O Answer 300 1 8 32-n Mark M1 Al M1 A1 M1 A1 Mark scheme for working out √81 as 3 or 108 as 10² or 100 for 300 or 3 × 10² or 3 × 100 for showing a square root of 64 as 8 1 or recognition of the reciprocal eg n or shows expressions that show an understanding of the ½ index and the minus index eg or other equivalent forms oe 64 for 32(n-1) or 32-2 or · (3²)¹-¹ for 32-" oe eg 3"-2(n-1) Additional guidance Mark may be awarded if operations are attempted on 8100000000 eg 300000000 Accept + oe 1 8 Paper: 1MA1/1H Question 12 (a) (b) 13 14 (c) Answer 5,15,35,55,70,80 Graph drawn Correct decision and correct figures 196 0.42 Mark B1 cao M1 A1 M1 M1 C1 P1 P1 Al P1 P1 Al for 5 or 6 of their points plotted correctly from a cf table for a fully correct graph SC B1 if 5 or 6 of their points plotted not at end but consistent within each interval and joined by a curve or line segments providing no gradient is negative for 60+ 100 x 80 (-48) oe reading value from graph at cf = 48 (=380) Mark scheme for density C = cao reading value from graph at wage = 360 (-40) or for 35+ × 20 (-39) 5 ft for correct decision and correct figures, eg No with 48 and "380" or with "40" and "50" (%) or with "40" and 48 for vol A = 1400 ÷ 70 (-20) or for mass B = 280 × 30 (-8400) 1400 + "8400" 9800 "20"+30 50 (= for "40" or for 60 oe 80 x 100 (-50(%)) 100 x 80 (-48) or answer with digits 196 for appropriate multiplication eg 0.3 x 0.7 (0.21) or 0.3 × 0.1 (-0.03) or 0.3 × 0.6 (-0.18) (dep) for complete process eg 0.3 × 0.7 +0.7 x 0.3 or 0.3 × 0.1 +0.3 × 0.6+0.6 × 0.3 +0.1 × 0.3 Additional guidance Ignore to the left of the first point and right of the last point If histograms drawn, points must be identified Accept a smooth curve or line segments ft from a cum freq graph An answer of 350 from 70+ 280 gets no marks Probabilities could also be given in fraction or percentage form Acceptable equivalents are 42% or oe 42 100 Paper: 1MA1/1H Question 15 16 (a) (b) Answer 1 y=-x+8 3 540 Explanation Mark M1 M1 A1 P1 P1 A1 C1 Mark scheme for a method for finding the gradient of L2 eg use of (dep) for substitution of (9, 5) into y=" for y=- -x+8 oe 120 for -(-6) or 20 for cao 20 90 120 n = for explanation 20 120 or 20 120 90 n ›="-1"x+c 3 90 (-0.16..) or (=4.5) or (-0.22..) 20 90 20 or Not acceptable examples My answer will be wrong It will increase the answer 90×120 20 m oe or 113 Acceptable examples If marks fall off Shirley will have over-estimated the number of bees There will be fewer bees Her amount will go down |y-5= Additional guidance 1 -=-70 3 (x-9) gets M2A1 Decimal values truncated or rounded to 2 dp or more Paper: 1MA1/1H Question 17 18 Answer f = 4d +3 d +3 20 Mark M1 M1 M1 Al P1 P1 Al for clearing the fraction eg df-4)=3(1-f) or df-4d-3-3f (dep M1) for isolating fterms in a correct equation eg df + 3f= 3 + 4d (dep on two terms in f) for factorising eg f(d+ 3) = 3 +4d oe Mark scheme for a statement of proportionality eg x= or 1.44 oe x=k√y cao for using √√1.44 as multiplier eg (x₂ =) k√1.44y or 1.2 oe Additional guidance Condone error in expansion of RHS for this mark Must be written in the form of an equation with a constant term, accept xxk√ √y Paper: 1MA1/1H Question 19 (a) (b) 20 (c) Answer 33 27 1 2 1+ 5 Mark B1 M1 A1 M1 A1 P1 P1 P1 A1 cao for f(9) = 12+√9 (=4) and a clear intention to find g("4") or for 3x (2x- 12 √9 12 or for stating gf eg 3(2x- √x+1) oe cao for g¹ as for ое 1 2 for writing √180 as 6-√√5 =+1) eg Mark scheme x-3 oe or for starting to solve 3(2x + 1) = 6 6 for process to rationalising the denominator √180-2√5 5√5 +5 4√5 5-√√5 +5 5√√5-5 5√5 +5 5√√5-5 5√√5 +5 eg X or ·X for 1+ (dep on previous P1) for expanding terms 5√5√180+5√180-50-10√5 100+20√5 125-25 100 √√5 5 oe or oe Accept Additional guidance y-3 6 This process mark can be awarded whenever this is seen, which might be later in the process. Accept written as a = 1, b = 5 Paper: 1MA1/1H Question 21 22 Answer Proof 0.5 Mark M1 M1 B1 C1 P1 P1 P1 P1 A1 for DQ = for PQ = or PQ = Mark scheme (b-a) oe or EQ=½ (a - b) oe for PQ = 1/2 b for complete proof with statement, eg FE = 2PQ or FE is a multiple of PQ or b = 2(b) a + DQ or 2 a + (b-a) oe a + b + EQ or -½ a+b+ (a - b) oe derive an algebraic expression for the area of A 1 eg [(5x-1)²-(3x - 1)²] 8 eg expand and simplify for either area A or area B 1 -π (16x²-4x) or (x²-2x+1) 8 (dep P2) equate and rearrange into a quadratic eqn of the form ax²+bx+c = 0 eg 2x² + 3x-2=0 oe (dep P3) factorise eg (2x - 1)(x+2) = 0 or use of formula eg -3+√√3²-4x2x-2 2x2 Additional guidance Vectors could be written on the diagram Accept only the single value of 0.5 oe but award 0 marks for a correct answer with no supportive working Paper: 1MA1/1H Question 23 Answer 27 56 Mark P1 P1 A1 3 for and 8 7 9 OR uses a total of 72 cards and shows a process to find the number of cards with a black shape or the number of cards with a triangle, eg 728 x 3 (27) or 72÷9 x 7 (=56) OR for for process shown to divide fractions Mark scheme for 39 27 (=. ) and 89 72 78 56 (= 98 72 37 39 + or -x 89 87 OR uses a total of 72 cards and shows a process to find the number of cards with a black shape and the number of cards with a triangle, eg 728 x 3 (27) and 72 ÷9 × 7 (=56) 27 56 or any other equivalent fraction Additional guidance 72 or any multiple of 72 Could be seen in a ratio, eg 27: 45 or 16:56 Accept the division shown as Could be seen in ratios, eg 27:45 and 16:56 Answer of 27:56 gets P2A0 3879 Modifications to the mark scheme for Modified Large Print (MLP) papers: 1 MA1 1H Only mark scheme amendments are shown where the enlargement or modification of the paper requires a change in the mark scheme. The following tolerances should be accepted on marking MLP papers, unless otherwise stated below: Angles: +5° Measurements of length: 15 mm PAPER: 1MA1/1H Question 3 4 Modification Wording added "Look at the diagram for Question 3 in the Diagram Book." The wording "The diagram shows four graphs" removed and replaced by "It shows four graphs labelled graph A, graph B, graph C and graph D." Diagrams enlarged. Graph lines made thicker. Open headed arrows. Headings moved above the graph. Wording added "Look at the diagram for Question 4 in the Diagram Book." The wording "The diagram shows four triangles" removed and replaced by "It shows four triangles." Diagram enlarged. Headings moved above the diagrams. Angles moved outside of the angle arcs and the angle arcs made smaller. Triangles straightened up so a 10 cm side lies horizontally. Braille only: Description added of the triangles. The wording "ADC is a triangle" removed. Diagram enlarged. Wording added "Look at the diagram for Question 6 in the Diagram Book. It shows the triangle ADC." Angles moved outside of the angles arcs and the angle arcs made smaller. Mark scheme notes Standard mark scheme Standard mark scheme Standard mark scheme PAPER: 1MA1/1H Question 7 8 10 Modification Wording added "Look at the table and the diagram for Question 7 in the Diagram Book." Wording "This stem and leaf diagram shows..." removed and replaced with "The stem and leaf diagram shows..." The wording "...of a group of Year 9 girls" removed and replaced by "...of a group of girls in Year 9". The wording "...of a group of 15 Year 9 boys" removed and replaced by "...of a group of 15 boys in Year 9". Table enlarged and kept on the same page as the stem and leaf diagram. Diagram enlarged and a tracking line added. Key moved above the diagram. Model provided for all candidates with a base added to represent the horizontal floor. Wording added "Look at the diagram for Question 8 in the Diagram Book. You may be provided with a model." The wording "The diagram shows a prism..." removed and replaced by "The diagram and the model show a prism...". Diagram enlarged. Dashed lines to be made thicker and longer. The pressure formula to be kept in the Question Paper and moved to the left of the diagram in the Diagram Book. The wording "Write these numbers in order of size" removed and replaced by "Write these four numbers in order of size." The letter a changed to w. The letter b changed to x. The letter c changed to y. Mark scheme notes Standard mark scheme Standard mark scheme Standard mark scheme Standard mark scheme except for the letter changes indicated. PAPER: 1MA1/1H Question 12 17 21 (a) (b) (c) Modification The value of the 350-400 interval changed from '20' to '25'. The value of the 450-500 interval changed from '10' to '5'. Wording added "Look at the table for Question 12(a) in the Diagram Book." Table enlarged. The wording "The table gives information..." removed and replaced by "It gives information..." Wording added "Look at the diagram for Question 12(b) in the Diagram Book." The wording "On the grid opposite..." removed and replaced by "On the grid,..." Diagram enlarged. Right axis labelled. Small squares removed. The axes labels moved to the top of the vertical axis and to the left of the horizontal axis. The wording "60% of this group of people have a weekly wage of £360 or less" removed and replaced by "75% of this group of people have a weekly wage of £375 or less" to make the question accessible. The letter fchanged to p, d changed to m. The wording "DEF is a triangle" replaced by "Look at the diagram for Question 21 in the Diagram Book. It shows the triangle DEF." Diagram enlarged. Mark scheme notes Standard mark scheme Standard mark scheme M1 for 75 100x 80 (-60) oe M1 reading value from graph at wage = 375 (-48) M1 for "48" 80 × 100 (=60(%)) or for 75 100 × 80 (-60) Standard mark scheme (NO) with 60 and "400" or with "48" and 60(%) or with "48" and 60 M1 reading value from graph at cf = 60 (-400) Standard mark scheme but note the changes to the letters. Standard mark scheme PAPER: 1MA1/1H Question 22 23 Modification Wording added "Look at the diagram for Question 22 in the Diagram Book." The wording "The diagram shows..." removed and replaced by "It shows..." Diagram enlarged. Open headed arrows. Shading changed to dotty shading. Angle moved outside of the angle arc the angle arc made smaller. The shapes labelled 'shape A' and 'shape B.' The labels "(3x-1) cm" and "2x cm" added to the bottom of the diagram. The label "(5x-1) cm" added to the top of the diagram. Wording added: "Look at the information for Question 23 in the Diagram Book. It shows the four types of cards in a game." Diagram enlarged. Black cards changed to dotty shading. All reference of 'black' changed to 'shaded'. Headings added above the cards e.g. "shaded circle". Mark scheme notes Standard mark scheme Standard mark scheme Pearson Education Limited. Registered company number 872828 with its registered office at 80 Strand, London, WC2R ORL, United Kingdom