Method marks are awarded for a correct method which could lead to a correct answer.

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1 Pre Paper 3F Question Bank Answers November 2017 GCSE Mathematics (AQA style) Foundation Tier This set of answers is not a conventional marking scheme; while it gives a basic allocation of marks, its main purpose it to help students understand how to do each question and how they can avoid making mistakes. As such, its format is rather different from that of a normal mark scheme. Included with each answer is the statement from the specification to which it applies (where basic foundation content is in normal type, and additional foundation content is in underlined type); content in italics is taken from the notes sections of the specification. The "basic foundation content" and "additional foundation content" can be assessed on Foundation tier question papers. The following guidance is adapted from that issued by AQA Types of mark M A B 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. Working out Usually, if the question asks students to show working, marks are not awarded to students who show no working. As a general principle, where the questions does not ask students to show working, a correct answer is awarded full marks. However, if the answer is incorrect, students can still obtain method marks, assuming that they show some valid working out. An incorrect answer with no working out is always awarded zero. Premature approximation Rounding off too early can lead to inaccuracy in the final answer. This is normally penalised by 1 mark.

2 Multiple choice questions N13 use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate gram, tonne M2 N15 round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures) 7.60 M3 N15 round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures) use inequality notation to specify simple error intervals due to truncation or rounding 29.5 x < 30.5 M4 R6 express a multiplicative relationship between two quantities as a ratio or a fraction 3 : 1 M5 R8 relate ratios to fractions and to linear functions 4 7 M6 R9 interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively M7 G3 understand and use alternate and corresponding angles on parallel lines; colloquial terms such as Z angles are not acceptable and should not be used alternate M8 M9 G9 identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference tangent, arc, sector and segment segment G13 interpret plans and elevations of 3D shapes 5 0 G24 describe translations as 2D vectors 3 4

3 1 P5 understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size S4 interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers) S4 students should know and understand the terms: primary data, secondary data, discrete data and continuous data primary

4 Number 1 (a) N2 understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals) including questions set in context. Knowledge and understanding of terms used in household finance, for example profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate Be careful with the minus sign, and (for money questions) remember to use two decimal places, so here, not (b) R9 solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics Any method Remember that you have a calculator, so use it; don t spend your time doing 10%, 1%, 0.1%, etc. Methods include , or (a) N3 use conventional notation for priority of operations, including brackets, powers, roots and reciprocals Must see that the denominator is evaluated first. 2 2 (b) N3 use conventional notation for priority of operations, including brackets, powers, roots and reciprocals Trevor calculated , then added 5.56 afterwards. Order of operations was wrong. Any explanation must include some numerical calculation(s).

5 3 (a) N5 apply systematic listing strategies including using lists, tables and diagrams blue, green blue, red blue, white red, green red, red red, white white, green white, red white, white yellow, green yellow, red yellow, white All combinations for one shirt colour or one shorts colour listed (eg blue, green ; red, green ; white, green ; yellow, green ). At least nine correct pairs. If any repetitions are present, at least nine of the pairs given must be distinct. All correct, no repetitions (condone repetition of blue, green, but no others). 3 (b) P7 construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities 2 12 Denominator = 12. Complete fraction. Accept decimal ( ) or percentage ( %) equivalents, but not a ratio. May be simplified to 1 6, but not required here.

6 Algebra A2 substitute numerical values into formulae and expressions, including scientific formulae 4 (a) Choice of s = ut at2 is implied by calculation; to start picking up marks, some substitution must be seen. You would be allowed a minor mistake in the working for (b) substitute numerical values into formulae and expressions, including scientific formulae Substitution into v 2 = u 2 + 2as to get 121 = u or rearranges to get u 2 = v 2 2as Reaches u 2 = 49 u = 7 or u = 7 Either value accepted; only one needed. 5 A4 simplify and manipulate algebraic expressions (including those involving surds) by expanding products of two binomials 2x 2 + 7x 6x 21 At least three of the four terms must be correct for. 2x 2 + x 21 Remember that 1x should be simplified to x. A4 simplify and manipulate algebraic expressions (including those involving surds) by simplifying expressions involving sums, products and powers, including the laws of indices 6 4x y 2 4 or 4 2 4x y Simplify the fraction inside the square root first. Allow one error for. 2x 2 y or 2x2 y 1 A2 Take the square root of each term; A2 if all three terms, including the 2, are correct; if two of the three terms are correct. A5 rearrange formulae to change the subject 7 Moves 5 to right hand side Must see a 3 = b 5 a = 3(b 5) or a = 3b 15 Must have a =

7 8 (a) A6 know the difference between an equation and an identity 2a = (b) A6 know the difference between an equation and an identity your a 1 = b or your a x x = bx 5 Tolerate 5x 9 (a) A4 simplify and manipulate algebraic expressions involving sums, products and powers, including the laws of indices (i) x(x + 3) (ii) 7x Tolerate inclusion of multiplication signs here (iii) x(x + 3) + 7x Adds expressions from (i) and (ii). x x or x(x + 10) Must be simplified; no multiplication signs present. 9 (b) A4 simplify and manipulate algebraic expressions involving sums, products and powers, including the laws of indices (i) (x + 3)(x + 7) (ii) 3 7 (= 21) Tolerate inclusion of multiplication signs here (iii) (x + 3)(x + 7) 21 Adds expressions from (i) and (ii). x x or x(x + 10) Must be simplified; no multiplication signs present. 9 (c) 10 (a) A6 argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments Conclusion using correct answers to (a) and (b). A8 work with coordinates in all four quadrants ( 4, 1) If answers to (a) and (b) are not identical, one (or both) should be manipulated until equivalence is clear. 10 (b) A8 work with coordinates in all four quadrants G4 derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus Any point correctly drawn Any two points with y co-ordinate equal to 1 and x co-ordinate greater than 2, but excluding (0, 1) A2 for each set of co-ordinates. If no drawing is present on the diagram, award A2 if two correct pairs of coordinates are given, or if one correct pair of coordinates is given. Note that if D was located at (0, 1) the shape ABCD would be a rhombus.

8 A9 find the equation of the line through two given points, or through one point with a given gradient Gradient of PQ is (a) y = 1 2 x + 1 or 2y = x + 2 A2 y = mx + c (where m is the gradient found previously, c is any positive number) y = mx + 1 (where m is any positive number) y = 1 2 x + c (where c is any positive number) A9 use the form y = mx + c to identify parallel lines 11 (b) y = 1 2 x 3 y = mx + c (where m is the gradient used in (a), c is any negative number) y = mx 3 (where m is any positive number) or 2y = x 6 y = 1 2 x + c (where c is any negative number) Correct answer. 12 (a) 1 identify and interpret roots, intercepts and turning points of quadratic functions graphically including the symmetrical property of a quadratic x = 2 In fact this whole question is about the fact that quadratic curves have a line of symmetry. Don t worry that you aren t told the values of b and c, but use the top two points ( 1, 5) and (5, 5) to find where the line of symmetry is. 12 (b) 1 identify and interpret roots, intercepts and turning points of quadratic functions graphically including the symmetrical property of a quadratic a = 2 This follows from part (a) and you may still get it right if you got (a) wrong. 1 identify and interpret roots, intercepts and turning points of quadratic functions graphically 12 (c) x = 0 These are the x co-ordinates of the two points at which x = 4 the curve crosses the x axis.

9 13 (a) 2 recognise, sketch and interpret graphs of linear functions and quadratic functions including simple cubic functions and the reciprocal function y = 1 x with x = recognise, sketch and interpret graphs of linear functions and quadratic functions including simple cubic functions and the reciprocal function y = 1 x with x 0 13 (b) Any valid method Could be trial and improvement, but you must show every stage of your working out for. Best method, however, starts with 2x = 32 for. x Either x = 4 or y = 8 seen 2x = 32 x continues to 2x2 = 32, x 2 = 16, x = 4 (4, 8)

10 7 solve linear equations in one unknown algebraically 14 (a) Multiplies by 5 16 Must see 2x + 3 = 35 or better (2x + 3 = 7 5 is not enough) 14 (b) A4 simplify and manipulate algebraic expressions by multiplying a single term over a bracket 7 solve linear equations in one unknown algebraically including use of brackets Expands brackets Must see 4y + 8 = or solve linear equations in one unknown algebraically including those with the unknown on both sides of the equation 14 (c) Attempts to put terms in z together Reaches 6z = 12 3z + 3z = Allow one error (for example 17 5), but not if the result would cancel out terms in z (for example 3z 3z) Care with minus signs; this should follow completely correct working and award of first. 2 Do not award if this is fluked from incorrect working. 7 solve linear equations in one unknown algebraically including those with the unknown on both sides of the equation 14 (d) Attempts to put terms in x together Should see at least x 2 x 3 (may be reversed, for example if rearrangement puts x on right hand side). Reaches x 6 = 7 3 or equivalent (a) A4 factorising quadratic expressions of the form x 2 + bx + c, including the difference of two squares (x + 6)(x 7) 15 (b) 8 solve quadratic equations algebraically by factorising 6 Award A0 for two correct solutions obtained from 7 incorrect factorisation in part (a).

11 16 9 solve two simultaneous equations in two variables (linear/linear) algebraically Correct method to obtain x or y x = 3 Could use elimination or substitution. A likely first step is to double the second equation (to match terms in y), then find x from 10x 3x = 24 3 y = or y = (a) A22 solve linear inequalities in one variable; students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary. 3 x < 2 Note the link between the different circles and the symbols < and. A22 solve linear inequalities in one variable 17 (b) (i) Subtracts 7 from expression Must see either=r 5 2x or 2x < 8 (or, of course, 5 2x < 8) x < 15 7 isn t enough for x or x Could be 2 1 < 4 2 x < 4 with one of 21 2 or 4 incorrect. At this stage, tolerate a mistake with < and (but not > or unless other correct rearrangements are seen) x < 4 Must be fully correct, including and < symbols. 17 (b) (ii) A22 solve linear inequalities in one variable Either 2, at start or, 4 at end. 2, 1, 0, 1, 2, 3 Must be fully correct.

12 A23 generate terms of a sequence from either a term-to-term or a position-to-term rule 18 (a) 18 (b) A23 generate terms of a sequence from either a term-to-term or a position-to-term rule Any reference to increasing by 6 from term to term. 61 Best as a multiplication, eg or A6 use algebra to support and construct arguments A23 generate terms of a sequence from either a term-to-term or a position-to-term rule 18 (c) All terms in sequence are odd Could see 6 n + 1 = 200 at this stage; n = implies that there is not an integer that gives a pattern with 200 dots in it. Gemma is wrong box ticked Cannot award second mark without a valid reason. 18 (d) A25 deduce expressions to calculate the nth terms of linear sequences 6n + c (with any value for c) or just 6n seen anywhere. 6n (a) A24 recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions including Fibonacci-type sequences 8 cm 19 (b) A24 recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions including Fibonacci-type sequences Creates Fibonacci type sequence, starting with 2, 3, 5, or starting from 8 cm length of Rectangle 4 to give 8, 13, cm

13 Ratio 20 R1 change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed, rates of pay, prices) in numerical and algebraic contexts compound units (eg density, pressure) R11 use compound units such as density and pressure Volume of sphere = cm to find mass of sphere. Mass of sphere = g Care; diameter is 0.16 cm, so the radius is 0.08 cm Mass = volume density 1 kg = 500 g Any reasonable rounding of Alternatively, you could find the volume of 1 kg of 2 lead (= cm 3 ) and divide by volume of one sphere. 21 (a) R5 divide a given quantity into two parts in a given part : part or part : whole ratio 12 parts = 180 Must see (b) R5 divide a given quantity into two parts in a given part : part or part : whole ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations) = = 110 (may see 90 1 = 90) 1 part = 80 people 240 men 320 women 80 children A2 for either 320 women or 80 children; all three correct required for A2.

14 22 (a) R9 solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics Correct method to find 14% of or or (this last one will give you the total for Tuesday straight away). You have a calculator, so don t take lots of time working out 10%, 5%, 1%, etc 22 (b) R9 express one quantity as a percentage of another Correct method to find percentage of bricks % "your 741" 100 (then subtract from 100%) (650 + "your 741") or (c) R9 work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics Equates 104% to 572. No marks for any method in which 4% of 572 is found R6 apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations) including better value or best-buy problems R11 use compound units such as speed, rates of pay, unit pricing including making comparisons Either = = and = or = = and = (other variants possible) Large Either divide the price by the quantity (to find the cost of 1 litre or 1 ml) or divide the quantity by the price (to find the quantity per 1 or 1p). There are several alternatives (g or kg, 1 or 1p); two are given here. All divisions must be correct for the second mark. As well as ticking the box, write down your conclusion from the calculations. Of course, ticking a box (even the correct one) with no working out will get you no marks. 24 (a) R12 compare lengths, areas and volumes using ratio notation scale factors; make links to similarity (including trigonometric ratios) Finds scale factor = your cm 24 (b) R12 compare lengths, areas and volumes using ratio notation scale factors; make links to similarity (including trigonometric ratios) 10.8 your 2.4 from (a) 4.5 cm

15 25 (a) 4 plot and interpret graphs, and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration 1.2 km 25 (b) R14 interpret the gradient of a straight-line graph as a rate of change Identifies BC. May be implied 0.7 km travelled in 30 sec km per second or 0.7 km kph

16 Geometry 26 G3 derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons) Total of interior angles in pentagon is 540, then for each interior angle or for each 5 interior angle. Other acute angle in right angled triangle with x is Seen or implied. Credit would be given for other correct methods. To ensure, must see correct value of 108 used for the interior angle of a pentagon. 27 G7 identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement Rotation Not turn. 90 anticlockwise or 270 clockwise and centre (1, 2) Angle/direction and centre needed for this mark. 28 G11 solve geometrical problems on coordinate axes Finds longer diagonal of a rhombus Finds shorter diagonal of a rhombus (9 5) 2 (16, 4) (11 (2 + your 8 )) 2 (if, following a miscalculation for first, this is negative, you won t get this. 29 G17 know the formulae: circumference of a circle = 2πr = πd; area of a circle = πr 2 ; calculate perimeters of 2D shapes, including circles, areas of circles and composite shapes Finds area of circle using π radius 2 Correct method to find area of grass m π = Must see radius = 1.5 m used your = Allow some mistakes (for example only taking away the area of one pond) if method/intention is clear. Eight bags needed (although rounds to 70, buying only seven bags would not be enough). No follow through here from incorrect area of grass.

17 30 (a) G20 know the formula for Pythagoras theorem, a 2 + b 2 = c 2 and apply to find angles and lengths in right-angled triangles in two dimensional figures Uses Pythagoras for DE Must see DE 2 = , or at least 12 2 = DE , with numbers substituted into the formula; just writing a 2 + b 2 = c 2 or similar isn t enough for a mark. You can use a symbol like x for DE if you prefer. DE = Uses square root to get DE; must see cm The final 0 must be present for 1 decimal place. 30 (b) opposite G20 know the trigonometric ratios sinx = hypotenuse, cosx = adjacent opposite and tanx = hypotenuse adjacent and apply them to find angles and lengths in right-angled triangles in two dimensional figures cosx = x = 34.6 Not enough just to identify trigonometry here; you must use the correct trigonometric ratio (sin, cos or tan) and make a fraction with the numbers.

18 Probability P1 record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees and 138 seen (a) Attempts to divide 288 in ratio 2 : 7 64 students wear glasses 224 students don t wear glasses (may be implied) and 92 seen 31 (b) R4 use ratio notation, including reduction to simplest form 46 : 92 1 : 2 32 (a) P8 calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions 0.3 seen on at least one branch for Steve wins All Dennis wins branches have 0.7 and all Steve wins branches have (b) P8 calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions including knowing when to add and when to multiply two or more probabilities or Both alternatives added and correct 32 (c) P8 calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions including knowing when to add and when to multiply two or more probabilities

19 Statistics 33 (a) S6 use and interpret scatter graphs of bivariate data; recognise correlation students should know and understand the terms: positive correlation, negative correlation, no correlation, weak correlation and strong correlation Greater age corresponds with greater mass 33 (b) S6 use and interpret scatter graphs of bivariate data; recognise correlation; know that it does not indicate causation draw estimated lines of best fit make predictions, interpolate and extrapolate apparent trends whilst knowing the dangers of so doing Line of best fit The question says Show clearly how you obtain your estimate ; you must draw the lines on the diagram that allow you to do this. 8 grams to 10 grams M0 if this is correct with no annotation on diagram 33 (c) S6 use and interpret scatter graphs of bivariate data; recognise correlation; know that it does not indicate causation draw estimated lines of best fit make predictions, interpolate and extrapolate apparent trends whilst knowing the dangers of so doing Any valid reason 365 days much bigger than 40 days, so line would be more inaccurate away from the original data; Lambs would not grow indefinitely. 34 S4 interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers) Correct method attempted = 42. One error allowed for if method used is correct. Divides answer by 40 Allow follow through from incorrect value for or equivalent (eg ). 35 S4 interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers) Mid values seen cm Use the middle of each interval. Should be 195, 205, 215 and 225. If your method is right, you will be let off a small mistake here. Whatever you get for the total height must be divided by the number of basketball players.