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More questions on this topic: IGCSE Algebra: Drawing & Using Graphs (144) Level 2 FM Algebra: Drawing & Using Graphs (29)
28 Sketch the graph of \(y = x^2 + 2\) (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph sketched | B2 | for fully correct graph with \(y\)-intercept labelled |
| (B1 | for the general shape in the correct orientation incorrectly positioned or graph with a \(y\)-intercept clearly labelled at 2) |
21
| \(x\) | −3 | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|
| \(y\) | 2 | −4 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \((2), -2, (-4), -4, -2, 2\) | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Curve must not have a horizontal segment between \((-1, -4)\) and \((0, -4)\)
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-0.5, -4.25\) | B1 | ft their graph with a single turning point or for \(x\) coordinate \(= -0.5\) and \(y\) coordinate \(= -4.25\) oe |
17 Here is a sketch of part of the graph of \(y = k^x\) where \(k\) is a positive constant.
The graph passes through the point with coordinates \((-1, 1.5)\)
Find the value of \(k\). (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{2}{3}\) | M1 | for correct substitution, eg \(1.5 = k^{-1}\) oe or \(\dfrac{1}{1.5}\) |
| A1 | oe but not \(\dfrac{1}{1.5}\) |
Accept rounded or truncated to 2dp or better
16 A train travels from Derby to Sheffield.
The travel graph shows information about the train’s journey.
The train stays at Sheffield for 15 minutes.
| Answer | Mark | Mark scheme |
|---|---|---|
| 70 | M1 | for a method to work with distance \(\div\) time, eg \(35 \div (30 \div 60)\) or \(35 \div 30 \times 60\) or \(35 \times (60 \div 30)\) or \(35 \div 30\ (= 1.16\ldots)\) or [distance] \(\div\) [time] |
| A1 | answer in the range 69.6 to 70.2 |
May use other readings from graph for [distance] and [time] but must be accurate
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | C1 | for a line from (30, 35) to (45, 35) |
13 Here are some graphs.
Each equation in the table is the equation of one of the graphs.
Complete the table. (3)
| Equation | Letter of graph |
|---|---|
| \(y = x^3\) | |
| \(y = \dfrac{2}{x}\) | |
| \(y = 5 - x^2\) | |
| \(y = 3^x\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| E C B H | B3 | for a fully correct table |
| (B2 | for 3 correct) | |
| (B1 | for 2 correct) |
12 The graph shows the velocity, \(v\) m/s, of a particle \(t\) seconds after it starts to move.
| Answer | Mark | Mark scheme |
|---|---|---|
| (i) 10 | M1 | for a tangent drawn at \(t = 3\) |
| M1 | for a complete method to find the gradient from tangent, eg \(\dfrac{30}{3}\) | |
| A1 | for answer in the range 8.5 to 11.5 or ft acceptable tangent at \(t = 3\) | |
| (ii) Acceleration or rate of change of velocity | C1 | for a correct explanation Acceptable examples acceleration rate of change of velocity increase in velocity each second how quickly the velocity increases increase in velocity over time the rate at which the particle is accelerating Not acceptable examples rate of change increase in velocity the velocity per second velocity ÷ time as time increases so does the velocity how steep the line is the acceleration of the particle and how far it got |
(i) A tangent must be seen to award any marks
This mark can be awarded if the tangent is drawn at \(t \neq 3\)
Accept answers in the form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers
Award 0 marks for a correct answer (in the range) with no (or incorrect) supportive working
(ii) Award if extra information is given provided not contradictory or incorrect
Accept ‘speed’ for ‘velocity’
| Answer | Mark | Mark scheme |
|---|---|---|
| 220 | M1 | for a method to find an appropriate area, eg \(\dfrac{1}{2} \times 30 \times 2\ (= 30)\) oe or \(\dfrac{1}{2}(30 + 50) \times (4 - 2)\ (= 80)\) oe or \(\dfrac{1}{2} \times (50 + 60) \times (6 - 4)\ (= 110)\) oe or for a method to find an estimate for the area of at least one rectangle with height at intersection of midpoint and curve, eg \(2 \times 16\ (= 32)\) oe or \(2 \times 42\ (= 84)\) oe or \(2 \times 56\ (= 112)\) oe |
| M1 | for a complete method, eg \(\dfrac{1}{2} \times 30 \times 2 + \dfrac{1}{2}(30 + 50) \times 2 + \dfrac{1}{2}(50 + 60) \times 2\) or \(\dfrac{1}{2} \times 2 \times (0 + 60 + 2(30 + 50))\) or \(2 \times 16 + 2 \times 42 + 2 \times 56\) | |
| A1 | for 220 or 228 |
Must have one correct expression for the award of this mark
May be seen as a rectangle added to a triangle
Allow 1 error in \(v\) values used
Allow 228 only if it comes from rectangle/midpoint method
9
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-2, 1\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| Point marked at (1, 2) | B1 | cao |
Accept without label provided unambiguous, ignore any lines drawn
| Answer | Mark | Mark scheme |
|---|---|---|
| Point marked at (5, 0) | B1 | cao |
Accept without label provided unambiguous, ignore any lines drawn
3
| \(x\) | −3 | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|
| \(y\) | 2 | −4 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \((2), -2, (-4), -4, -2, 2\) | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Curve must not have a horizontal segment between \((-1, -4)\) and \((0, -4)\)
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-0.5, -4.25\) | B1 | ft their graph with a single turning point or for \(x\) coordinate \(= -0.5\) and \(y\) coordinate \(= -4.25\) oe |
24
| \(x\) | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 0 | −3 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5, (0), -3, -4, (-3), 0, 5\) | B2 | for all 5 correct values |
| (B1 | for at least 2 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept freehand curves drawn that are not line segments
Ignore anything drawn outside the required range
22 Here is a velocity-time graph for an aeroplane.
Work out an estimate for the distance the aeroplane travelled in the first 30 seconds.
Use 3 strips of equal width. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| 1760 | M1 | for starting to find the area under the curve, eg \(0.5 \times 10 \times \mathbf{54}\ (= 270)\) oe or \(0.5 \times 10 \times (\mathbf{54} + \mathbf{76})\ (= 650)\) oe or \(0.5 \times 10 \times (\mathbf{76} + \mathbf{92})\ (= 840)\) or for a method to find an estimate for the area of at least 1 strip with heights at intersection of midpoint and curve eg \(10 \times [\mathbf{39}]\) oe or \(10 \times [\mathbf{67}]\) oe or \(10 \times [\mathbf{86}]\) oe |
| M1 | for a complete method to find the area under the curve, eg \(0.5 \times 10 \times 54 + 0.5 \times 10 \times (54 + 76) + 0.5 \times 10 \times (76 + 92)\) oe eg \(0.5 \times 10\,(92 + 2(54 + 76))\) or \(10 \times [39] + 10 \times [67] + 10 \times [86]\) oe | |
| A1 | for 1760 or 1890 to 1950 SCB2 for an answer in the range 1815 – 1855 |
Must have one correct expression for the award of this mark
May be seen as a rectangle added to a triangle
Where \(38 \leqslant [39] \leqslant 40\)
Where \(66 \leqslant [67] \leqslant 68\)
Where \(85 \leqslant [86] \leqslant 87\)
Allow 1 error in \(y\) values used
Allow 1890 to 1950 only if it comes from midpoint method
14 Amy walked from her home to the skate park.
The travel graph of Amy’s walk to the skate park is shown below.
On the way to the skate park Amy stopped at her friend’s house.
Amy stayed at the skate park for 2 hours.
Then she walked home at a steady speed.
She took 1 hour 30 minutes to walk home.
| Answer | Mark | Mark scheme |
|---|---|---|
| 2 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph completed | M1 | for straight line from (3, 5) to (5, 5) or for a straight line from (5, 5) to (6 30, 0) or line drawn from (3, 5) to (4 30, 0) or a line drawn from (\(x\), 5) to (\(x\) + 1 30, 0) where \(x \geqslant 3\) |
| A1 | cao |
Accept hand drawn, ruler not required but intention clear
8
| Answer | Mark | Mark scheme |
|---|---|---|
| (3, 1) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(B\) at \((3, -4)\) | B1 | cao |
Accept a cross without label as long as unambiguous.
Accept letter B or dot in place of cross.
| Answer | Mark | Mark scheme |
|---|---|---|
| (0, 2) | B1 | cao |
5
| \(x\) | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 0 | −3 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5, (0), -3, -4, (-3), 0, 5\) | B2 | for all 5 correct values |
| (B1 | for at least 2 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept freehand curves drawn that are not line segments
Ignore anything drawn outside the required range
24
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 6 | 0 | 2 |
| Answer | Mark | Mark scheme |
|---|---|---|
| (6) 2 (0) 0 (2) 6 | B2 | for all 3 values correct |
| (B1 | for 1 or 2 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-1.7\) to \(-1.5\) and 2.5 to 2.7 | M1 | for drawing the line \(y = 4\) or reading off intersections where \(y = 4\) or one correct solution or both solutions given as coordinates, eg \((-1.6, 2.6)\) or \((-1.6, 4)\) and \((2.6, 4)\) |
| A1 | for answers in the range \(-1.7\) to \(-1.5\) and 2.5 to 2.7 or ft their graph with at least 2 solutions |
ft their graph for this mark
Accept these coordinates reversed
Algebraic methods score 0 marks
16 You can use this graph to change between ounces and grams.
| Answer | Mark | Mark scheme |
|---|---|---|
| 170 | B1 | for answer in the range 167 to 173 |
| Answer | Mark | Mark scheme |
|---|---|---|
| 35 | M1 | for correctly using readings from the graph as a factor of 1000 from the grams scale, eg \(200 \times 5\) or \(100 \times 10\) or \(20 \times 50\) or \(250 \times 4\) or for method to use multiples of grams and corresponding ounces readings, eg \(1000 \div\) “answer to (a)” \(\times\, 6\) or \(1000 \div\) grams \(\times\) ounces oe |
| A1 | for an answer in the range 34 to 36 |
May be seen as a build-up method using multiple readings that can be read from the graph but must total 1000 grams
(ounces, grams) is a point on the line
14 The graph shows the velocity of a car, in metres per second, \(t\) seconds after it starts to slow down.
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-2.5\) | M1 | for drawing a tangent at \(t = 5\) |
| M1 | (dep on M1) for a complete method to find the gradient eg tangent at \(t = 5\) and \(\text{``}10\text{''} \div \text{``}4\text{''}\) or an answer in the range 2.0 to 2.8 | |
| A1 | for answer in the range \(-2.0\) to \(-2.8\) dependent on tangent drawn |
No tangent drawn score 0 marks
Working may be seen on the diagram
Accept answers in the form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers
| Answer | Mark | Mark scheme |
|---|---|---|
| 79 | M1 | for a method to find an estimate for the area of at least 1 trapezium under the curve, eg \(\dfrac{1}{2} \times 2 \times (25 + 16)\ (= 41)\) oe or \(\dfrac{1}{2} \times 2 \times (16 + 9)\ (= 25)\) oe or \(\dfrac{1}{2} \times 2 \times (9 + 4)\ (= 13)\) oe or for a method to find an estimate for the area of at least 1 rectangle with heights at intersection of midpoint and curve, eg \(2 \times 20.5\ (= 41)\) oe or \(2 \times 12.5\ (= 25)\) oe or \(2 \times 6\ (= 12)\) oe |
| M1 | for a complete method, eg \(\dfrac{1}{2} \times 2 \times (25 + 16) + \dfrac{1}{2} \times 2 \times (16 + 9) + \dfrac{1}{2} \times 2 \times (9 + 4)\) oe or \(\dfrac{1}{2} \times 2 \times (25 + 4 + 2(16 + 9))\) or \((2 \times 20.5) + (2 \times 12.5) + (2 \times 6)\) | |
| A1 | For 79 or 78 |
May be seen as a rectangle added to a triangle
Allow consistent use of incorrect width for both M marks
Allow 1 error in \(y\) values used
Allow 78 only if it comes from rectangle/midpoint method
12 Here are some graphs.
Write down the letter of the graph that could have the equation
| Answer | Mark | Mark scheme |
|---|---|---|
| H | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| F | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| J | B1 | cao |
8 The points \(A\) and \(B\) are shown on the grid.
| Answer | Mark | Mark scheme |
|---|---|---|
| 2, 1 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(0, -2\) | M1 | for an answer of \((0, y)\) where \(y \ne -2\) or \((x, -2)\) where \(x \ne 0\) or the correct midpoint identified on the grid or \((2 + -2) \div 2\) or \((1 + -5) \div 2\) |
| A1 | cao | |
| SCB1 if M0 scored for \(-2, 0\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(C\) shown at \((-4, 2)\) | C1 | cao |
5
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 6 | 0 | 2 |
| Answer | Mark | Mark scheme |
|---|---|---|
| (6) 2 (0) 0 (2) 6 | B2 | for all 3 values correct |
| (B1 | for 1 or 2 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-1.7\) to \(-1.5\) and 2.5 to 2.7 | M1 | for drawing the line \(y = 4\) or reading off intersections where \(y = 4\) or one correct solution or both solutions given as coordinates, eg \((-1.6, 2.6)\) or \((-1.6, 4)\) and \((2.6, 4)\) |
| A1 | for answers in the range \(-1.7\) to \(-1.5\) and 2.5 to 2.7 or ft their graph with at least 2 solutions |
ft their graph for this mark
Accept these coordinates reversed
Algebraic methods score 0 marks
21
| Answer | Mark | Mark scheme |
|---|---|---|
| Circle drawn | B2 | for drawing a circle centre \((0,0)\) and radius 13 |
| (B1 | for drawing a circle centre \((0,0)\) with radius \(\neq 13\) or a circle of radius 13 with a centre not \((0,0)\) or an incomplete correct circle drawn) |
Circle could be drawn freehand as long as it closely approximates to a circle
| Answer | Mark | Mark scheme |
|---|---|---|
| \(x = 7.2\), \(y = 10.8\) and \(x = -7.2\), \(y = -10.8\) | M1 | for drawing the line \(2y = 3x\) |
| A1 | for both \(x\) values correct or both \(y\) values correct or one pair of \(x\) and \(y\) values correct ft from (a) dep on B1 | |
| A1 | for both correct pairs of values correctly matched ft from (a) dep on B1 |
For both A marks accept answers in the ranges
\(x = 7.0\) to \(7.4\), \(y = 10.6\) to \(11.0\)
\(x = -7.0\) to \(-7.4\), \(y = -10.6\) to \(-11.0\)
Accept values given as coordinates
Check graph for answers
20
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 4 | \(-2\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| (4) 0 −2 (−2) 0 4 | B2 | for all 4 correct values |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | ft (dep B1) for plotting at least 4 points correctly |
| A1 | for a fully correct curve drawn |
Accept a freehand curve drawn that is not made of line segments.
15 An object falls from rest.
Here is the distance-time graph for the distance (\(d\) metres) fallen by the object \(t\) seconds after it starts to fall.
Work out an estimate for the gradient of the graph at \(t = 3\)
You must show how you get your answer. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| 25 | M1 | for a tangent drawn at \(t = 3\) |
| M1 | for a complete method to find the gradient from tangent eg \(50 \div (4 - 2)\) | |
| A1 | for answer in the range 22 to 28 or ft acceptable tangent |
A tangent must be seen to award any marks
This mark can be awarded if the tangent is drawn at \(t \neq 3\)
Award 0 marks for a correct answer (in the range) with no (or incorrect) supportive working
28 Here are five graphs.
The table shows the equations of these graphs.
| Equation | Graph |
|---|---|
| \(y = x^2 - 4x\) | |
| \(y = x + 3\) | |
| \(y = x^3 - 2\) | |
| \(y = \dfrac{1}{x}\) | |
| \(y = 5 - 2x\) |
Match the letter of each graph with its equation. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| B, D, E, C, A | B3 | for all correctly matched |
| (B2 | for 3 or 4 correctly matched) | |
| (B1 | for 2 correctly matched) |
26 Here is the graph of \(\; y = x^2 - 2x - 2\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \((1, -3)\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-0.7\) or 2.7 | B1 | for an answer in the range \(-0.8\) to \(-0.6\) or 2.6 to 2.8 |
19 Sana needs to draw the graph of \(y = 3^x\) for \(0 \leqslant x \leqslant 4\)
She draws the graph shown on the grid.
Write down one thing Sana has done wrong. (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| Statement | C1 | for explanation Acceptable examples should go through (0, 1) should not go through (0, 0) it has touched the \(x\)-axis \(3^0 = 1\) \(3^0 \neq 0\) Not acceptable examples the graph has been drawn wrong axes are not labelled |
17 Here is a distance-time graph.
| Answer | Mark | Mark scheme |
|---|---|---|
| 16.5 to 19.5 | M1 | for tangent drawn at \(t = 2.5\) |
| M1 | dep (M1) for a complete method to find the gradient from tangent eg \((48 - 0) \div (4 - 1.4)\ \ (= 18.4\ldots)\) | |
| A1 | for answer in the range 16.5 to 19.5 or ft acceptable tangent |
The tangent must be seen to award any marks
Award 0 marks for a correct answer (in the range) with no (or incorrect) supportive working
| Answer | Mark | Mark scheme |
|---|---|---|
| Description | C1 | eg speed or velocity |
“speed” may be expressed in a different way, eg distance per unit time (at that point)
Award if extra information is given provided not contradictory.
13 Rowena drove from her home to a beach.
Here is a travel graph for her journey.
Rowena stopped at a cafe on her way to the beach.
Rowena stayed at the beach for \(1\dfrac{1}{2}\) hours.
She then drove home without stopping.
Rowena arrived home at 16 00
| Answer | Mark | Mark scheme |
|---|---|---|
| (i) 30 | B1 | cao |
| (ii) 10 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| Drawn | M1 | for a line from (1330 , 35) to (1500 , 35) or a line to the \(x\) axis from a point on \(y = 35\) to 1600 on the \(x\) axis |
| A1 | fully correct graph |
| Answer | Mark | Mark scheme |
|---|---|---|
| 35 | B1 | for 35 or ft their graph |
ft must be \(35 \div\) time duration for their line
10 This graph can be used to find the cost of parking a car in a car park for up to 12 hours.
Justin drives into the car park at 08 00 in the morning.
When he drives out of the car park he has to pay £9
| Answer | Mark | Mark scheme |
|---|---|---|
| 6 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 14 00 | M1 | for use of graph to find the maximum time paid for, eg £9.00 = 6 hours |
| M1 | for intention to add times, eg 08 00 + “6” hrs | |
| A1 | for 14 00 or 2 pm |
May be seen on graph
8, 9, 10, 11, 12, 1 is enough to show a clear intention to add
For method marks condone use of incorrect time notation
Correct time notation required
9
| Answer | Mark | Mark scheme |
|---|---|---|
| (5, 2) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \((4, -2)\) marked | B1 | for the point \((4, -2)\) unambiguously marked on the grid |
Allow without label, provided unambiguous
| Answer | Mark | Mark scheme |
|---|---|---|
| (1, 3) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(y = -4\) shown | B1 | for correct single line unambiguously marked |
Need not be labelled if clear.
Accept a single line drawn freehand of any length.
Accept a dotted (or dashed) line
7 Here is the graph of \(y = x^2 - 2x - 2\)
| Answer | Mark | Mark scheme |
|---|---|---|
| (1, −3) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| −0.7 or 2.7 | B1 | for an answer in the range −0.8 to −0.6 or 2.6 to 2.8 |
24 Here is the graph of \(\; y = x^2 - 6x + 4\)
| Answer | Mark | Mark scheme |
|---|---|---|
| 4 | B1 | for 4 |
Condone (0,4) or 0,4
| Answer | Mark | Mark scheme |
|---|---|---|
| \((3, -5)\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 5.1 to 5.3 and 0.7 to 0.9 | M1 | for a correct method, eg marking both intercepts with \(x\)-axis or one correct solution |
| A1 | for answers in the range 5.1 to 5.3 and 0.7 to 0.9 |
Accept both solutions given as a coordinate for M1
eg (5.2, 0.8) or (0.8, 5.2) or (5.2, 0) and (0.8, 0)
21 The graph below gives the volume, in litres, of water in a container \(t\) seconds after the water starts to fill the container.
| Answer | Mark | Mark scheme |
|---|---|---|
| 1.06 | M1 | for tangent drawn at \(t = 17.5\) |
| M1 | for a complete method to find the gradient eg tangent drawn at \(t = 17.5\), and \(18.5 \div 17.5\) | |
| A1 | answer in the range 0.9 to 1.2 |
No tangent drawn at \(t=17.5\) scores zero marks
Use of change in \(y\) over change in \(x\). Working may be seen on the diagram
Answer of \(\dfrac{10.5}{17.5}\) oe scores no marks.
Accept answers in the form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers
| Answer | Mark | Mark scheme |
|---|---|---|
| Explanation | C1 | suitable explanation, eg the rate of change of volume |
If units are given they must be correct
19 Carly cycles to her friend’s house.
She stays at her friend’s house for a number of minutes.
Then she cycles home.
Here is the travel graph for her journey.
| Answer | Mark | Mark scheme |
|---|---|---|
| 15 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 4.6 | B1 | for an answer in the range 4.4 to 4.8 |
| Answer | Mark | Mark scheme |
|---|---|---|
| 12 | M1 | for a method to calculate speed eg distance ÷ time (could be implied from figures used) eg \(4 \div 20\ (= 0.2)\) oe, \(4 \div 0.33(...)\) oe or \(4 \div \tfrac{1}{3}\) oe |
| A1 | cao |
Accept readings from the graph as an indication at this stage
15
Find the coordinates of the midpoint of \(PQ\). (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-0.5, 1\) | M1 | for one correct coordinate or midpoint shown on diagram or correct method, eg \(\dfrac{-3+2}{2}\) or \(\dfrac{-2+4}{2}\) or for the coordinates reversed, eg \(1, -0.5\) |
| A1 | for \(-0.5, 1\) oe |
14 You can use this graph to change between ounces and grams.
| Answer | Mark | Mark scheme |
|---|---|---|
| 30 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 2238 to 2296 | M1 | for a complete method eg attempts to read from the graph at a factor of 80 and scales up to 80 using a correct scale or attempts to read from the graph using numbers that sum to 80 and finds the sum of their readings or attempts to read from the graph a number that they then go on to scale up to 80 using a correct scaling factor |
| A1 | for an answer in the range 2238 to 2296 |
Condone some inaccuracy in reading from the graph, which should be given to within the nearest 50g
9
| \(x\) | \(-3\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \(y\) | 9 | 4 | \(-9\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| (9), −4, −5, 0, 5, (4), (−9) | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Ignore anything outside the required range
7 Here is the graph of \(y = x^2 - 6x + 4\)
| Answer | Mark | Mark scheme |
|---|---|---|
| 4 | B1 | for 4 |
Condone (0,4) or 0,4
| Answer | Mark | Mark scheme |
|---|---|---|
| (3, −5) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 5.1 to 5.3 and 0.7 to 0.9 | M1 | for a correct method, eg marking both intercepts with \(x\)-axis or one correct solution |
| A1 | for answers in the range 5.1 to 5.3 and 0.7 to 0.9 |
28
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | 1 | \(-1\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5, (1), (-1), -1, 1, 5\) | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand graph drawn that is not made of line segments
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| 0.3 to 0.5 and 2.5 to 2.7 | M1 | for a correct method, eg marking intercepts with \(x\)-axis or one correct solution or both solutions given as a coordinates, eg (0.4, 2.6) or (0.4, 0) and (2.6, 0) |
| A1 | for answers in the range 0.3 to 0.5 and 2.5 to 2.7 or ft their graph with at least 2 solutions |
ft their graph for this mark
Accept these coordinates reversed
14 Here is a speed-time graph.
| Answer | Mark | Mark scheme |
|---|---|---|
| 0.9 | M1 | for drawing a tangent at \(t = 2\) |
| M1 | for a complete method to find the gradient eg tangent at \(t = 2\) and \(\text{``}2.7\text{''} \div \text{``}3\text{''}\) | |
| A1 | for answer in the range 0.7 to 1.1 |
Use of change in \(y\) over change in \(x\)
Working may be seen on the diagram
Accept answers in the form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers
| Answer | Mark | Mark scheme |
|---|---|---|
| Statement | C1 | eg distance (travelled) |
If units are given they must be correct
6
| \(x\) | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | 1 | −1 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5, (1), (-1), -1, 1, 5\) | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | B2 | for a fully correct graph |
| (B1 | ft (dep on B1 in (a)) for plotting at least 5 of the points from their table correctly) |
Accept a freehand curve drawn that is not made of line segments
Ignore anything drawn outside the required range
| Answer | Mark | Mark scheme |
|---|---|---|
| 0.3 to 0.5 and 2.5 to 2.7 | M1 | for a correct method, eg marking intercepts with \(x\)-axis or one correct solution or both solutions given as a coordinates, eg \((0.4, 2.6)\) or \((0.4, 0)\) and \((2.6, 0)\) |
| A1 | for answers in the range 0.3 to 0.5 and 2.5 to 2.7 or ft their graph with at least 2 solutions |
ft their graph for this mark
Accept these coordinates reversed
28 Sketch the graph of \(y = \dfrac{1}{x}\) (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| Sketch | M1 | correct shape in one of the required quadrants or correct graph where the lines touch the axes |
| A1 | fully correct shape |
Lines do not need to extend to the ends of the axes if the intention is clear
24
| \(x\) | \({-2}\) | \({-1}\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 10 | 2 | 5 |
| Answer | Mark | Mark scheme |
|---|---|---|
| (10), 5, (2), 1, 2, (5), 10 | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | ft (dep on B1) for plotting at least 5 of their points correctly |
| A1 | for a fully correct curve drawn |
Accept a freehand curve drawn that is not made of line segments
| Answer | Mark | Mark scheme |
|---|---|---|
| \({-0.65}\) to \({-0.8}\) and 2.65 to 2.8 | M1 | for \(y = 4\) drawn or intersection with \(y = 4\) or \(y = x^2 - 2x - 2\) drawn or 1 correct value (ft a quadratic) |
| A1 | ft a quadratic graph or for answers in the range 2.65 to 2.8 and \({-0.65}\) to \({-0.8}\) |
If answers stated as coordinates, award M1 for both coordinates and M0 for one coordinate
21 Here is a speed-time graph showing the speed, in metres per second, of an object \(t\) seconds after it started to move from rest.
| Answer | Mark | Mark scheme |
|---|---|---|
| 19.1 | M1 | for a method to find an estimate for the area of at least 1 trapezium under the curve eg \(0.5 \times 1 \times (4 + 6)\) or \(0.5 \times 1 \times (6 + 7.2)\) or \(0.5 \times 1 \times (7.2 + 7.8)\) |
| M1 | for a complete method eg \(0.5 \times 1 \times (4 + 6) + 0.5 \times 1 \times (6 + 7.2) + 0.5 \times 1 \times (7.2 + 7.8)\) or \(0.5\{(4 + 7.8) + 2(6 + 7.2)\}\) | |
| A1 | cao |
Allow a maximum of 2 errors in \(y\) values used
Ignore any reference to units
| Answer | Mark | Mark scheme |
|---|---|---|
| Statement | C1 | eg distance (travelled) |
If units are given they must be correct
20 The diagram shows the graph of \(x^2 + y^2 = 30.25\)
Use the graph to find estimates for the solutions of the simultaneous equations
\(x^2 + y^2 = 30.25\)
\(y - 2x = 1\) (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(x = 2.1\), \(y = 5.1\) \(x = -2.9\), \(y = -4.7\) | M1 | for drawing the graph of \(y - 2x = 1\) |
| A1 | for one correct pair of values or for both correct \(x\) values, or for both correct \(y\) values | |
| A1 | for both correct pairs, correctly matched |
For both A marks accept answers in the ranges
\(x = 2.0\) to 2.2, \(y = 5.0\) to 5.2
\(x = -2.8\) to \(-3.0\), \(y = -4.6\) to \(-4.8\)
Accept values given as coordinates
15 The graph gives the volume of water, in litres, in a container at time \(t\) seconds after the water started to flow out of the container.
Using the graph, work out an estimate for the rate at which the water is flowing out of the container when \(t = 12\)
You must show your working. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| 0.7 to 1.1 | M1 | for tangent to the curve drawn at \(t = 12\) |
| M1 | for method to find the gradient of their tangent, eg \(28 \div 30\) | |
| A1 | for answer in the range 0.7 to 1.1 dependent upon tangent drawn |
Working may be seen on the diagram
Ignore negative signs
14 Nazima uses this graph to find out how much money she is paid for the number of hours she has worked.
Last week Nazima worked for 36 hours.
| Answer | Mark | Mark scheme |
|---|---|---|
| 15 | B1 | 14 to 16 |
| Answer | Mark | Mark scheme |
|---|---|---|
| 540 | M1 | for a complete method, eg \(30 \times (36 \div 2)\) or \(45 \times (36 \div 3)\) or \(60 \times (36 \div 4)\) or ft “hourly rate from (a)” × 36 |
| A1 | for 540 or ft (a) |
May be seen using a complete build up method
for “45” allow 44 to 46 ft for accuracy
Condone use of mixed rates
eg \(75 \times 7 + 16 = 541\)
4
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 10 | 2 | 5 |
| Answer | Mark | Mark scheme |
|---|---|---|
| (10), 5, (2), 1, 2, (5), 10 | B2 | for all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | ft (dep on B1) for plotting at least 5 of their points correctly |
| A1 | for a fully correct curve drawn |
Accept a freehand curve drawn that is not made of line segments
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-0.65\) to \(-0.8\) and 2.65 to 2.8 | M1 | for \(y = 4\) drawn or intersection with \(y = 4\) or \(y = x^2 - 2x - 2\) drawn or 1 correct value ft a quadratic graph |
| A1 | ft a quadratic graph or for answers in the range 2.65 to 2.8 and \(-0.65\) to \(-0.8\) |
If answers stated as coordinates, award M1 for both coordinates and M0 for one coordinate
25 Here are six graphs.
Write down the letter of the graph that could have the equation
| Answer | Mark | Mark scheme |
|---|---|---|
| F | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| D | B1 | cao |
21 Here is a speed-time graph for a train journey between two stations.
The journey took 100 seconds.
| Answer | Mark | Mark scheme |
|---|---|---|
| 52.5 | P1 | starts to find area under graph, eg \(\dfrac{30 \times 12}{2}\ (= 180)\) or \(50 \times 12\ (= 600)\) or \(\dfrac{20 \times 12}{2}\ (= 120)\) |
| P1 | complete process to find area under graph, eg \(\dfrac{30 \times 12}{2} + 50 \times 12 + \dfrac{20 \times 12}{2}\ (= 900)\) | |
| P1 | starts process to find half way time, eg \(((\text{``}900\text{''} \div 2) - 180) \div 12\ (= 22.5)\) | |
| A1 | 52.5 oe |
| Answer | Mark | Mark scheme |
|---|---|---|
| Comparison | C1 | acceptable comparison Acceptable (acceleration) during first part is positive but (acceleration) during last part is negative / deceleration (acceleration is) greater during the last part than during the first part gradient is steeper in the last part / longer to speed up than slow down speed / (acceleration) is increasing at start and decreasing at end (acceleration) is slower in the first part (acceleration) is ascending in the first part and descending in the last part 0.4 is the first part and \(-0.6\) in the last part Not acceptable goes down in the last part speed is greater in last part than first part |
19 The graph shows information about part of a cyclist’s journey.
Work out an estimate of the speed, in m/s, of the cyclist at time 6 seconds. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| 3.4 | M1 | for drawing a suitable tangent at \(t = 6\) |
| M1 | for a full method to find the gradient of the tangent at \(t = 6\), eg \(20 \div 5.8\) | |
| A1 | answer in the range 3.05 to 3.7 |
Use of change in y over change in x
Answers of \(\dfrac{10}{6}\) oe scores no marks
17 Here is the graph of \(y = x^2 - 3\)
Use the graph to find estimates for the solutions to the equation \(x^2 - 2x - 2 = 0\)
You must show how you get your solutions. (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| 2.7 and \(-0.7\) | M1 | for \(x^2 - 3 = 2x - 1\) oe or \(x^2 - 3 - 2x + 1\ (= 0)\) or completing the square eg \((y =)\ (x - 1)^2 - 1 - 2\) |
| M1 | (dep M1) draws graph of \(y = 2x - 1\) or drawing the translated graph or describing the translation in words or \(-1.7 + 1\ (= -0.7)\) or \(1.7 + 1\ (= 2.7)\) | |
| M1 | shows the points of intersection clearly for the given quadratic graph and linear graph or for one correct solution from appropriate supportive working | |
| A1 | for \(x\) in the range 2.6 to 2.8 and \(-0.6\) to \(-0.8\) | |
| SC B2 for plotting \(y = 2x + 1\) and values for \(x\) in the range \(-1.1\) to \(-1.3\) and 3.1 to 3.3 |
Line segments required
For 1.7 allow from 1.6 to 1.8
For \(-1.7\) allow from \(-1.8\) to \(-1.6\)
Points indicated or attempt to read off \(x\)-axis at the appropriate points – maybe indicated by dashes
No marks will be awarded for correct answers only
17
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | 10 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-10, -6, 2, 6\) | B2 | for 4 values correct \(-10, -6, (-2), 2, 6, (10)\) |
| (B1 | for 2 or 3 values correct) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph drawn | M1 | (ft from (a) if B1 awarded) for at least 5 points correctly plotted. |
| A1 | correct graph drawn from \(x = -1\) to 4 |
16 Steve drove from his home to his friend’s house.
He stayed at his friend’s house and then drove home.
Here is Steve’s travel graph.
| Answer | Mark | Mark scheme |
|---|---|---|
| 45 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 50 | M1 | for an attempt to find the gradient eg \(\text{``}25\text{''} \div \text{``}0.5\text{''}\) ft their readings from the travel graph; use of speed-time formula eg \(25 \div 30\) (ignore units if shown) |
| A1 | cao |
could be shown in working or on the graph using any acceptable triangle; could be shown by multiples of 25, 0.5 or multiples of ft figures
11 You can use this graph to change between stones and kilograms.
| Answer | Mark | Mark scheme |
|---|---|---|
| 19 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 12.4 to 12.8 | M1 | for a complete method, eg attempts to read from the graph at a factor of 80 and scales up to 80 or attempts to read from the graph at two numbers that sum to 80 and finds the sum of their readings or 1 stone = “6”kg and \(80 \div \text{``}6\text{''}\) |
| A1 | for an answer in the range 12.4 to 12.8 or ft correct reading from graph |
5 Here are six graphs.
Write down the letter of the graph that could have the equation
| Answer | Mark | Mark scheme |
|---|---|---|
| F | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| D | B1 | cao |
29 Here is the graph of \(y = x^2 - 2x - 3\)
| Answer | Mark | Mark scheme |
|---|---|---|
| 1, –4 | B1 | cao |
Brackets are given on the answer line, ignore any extra brackets seen
| Answer | Mark | Mark scheme |
|---|---|---|
| –1 and 3 | B2 | for both correct answers |
| (B1 | for one correct solution or \((x + 1)(x - 3)\) or \((-1, 3)\)) |
17 Here are four graphs.
The graphs represent four different types of function f.
Match each description of the function in the table to the letter of its graph.
| Description of function | Graph |
|---|---|
| \(\mathrm{f}(x)\) is inversely proportional to \(x\) | |
| \(\mathrm{f}(x)\) is a trigonometrical function | |
| \(\mathrm{f}(x)\) is an exponential function | |
| \(\mathrm{f}(x)\) is directly proportional to \(\sqrt{x}\) |
(2)
| Answer | Mark | Mark scheme |
|---|---|---|
| B, A, D, C | B2 | for all correct |
| (B1 | for two or three correct) |
14 A car moves from rest.
The graph gives information about the speed, \(v\) metres per second, of the car \(t\) seconds after it starts to move.
(a)
| Answer | Mark | Mark scheme |
|---|---|---|
| 0.83 | B1 | for a tangent drawn at \(t = 15\) |
| M1 | full method to use the tangent to find the gradient (eg \(7.5 \div 9\)) | |
| A1 | for answer in the range 0.6 to 1.0 |
Working: \(7.5 \div 9 = 0.83\ldots\)
No tangent scores 0 marks
This mark can be awarded if the tangent is drawn at \(t \neq 15\)
Working may be seen on the diagram
| Answer | Mark | Mark scheme |
|---|---|---|
| Statement | C1 | statement Acceptable examples acceleration rate of change of speed increase in speed over time Not acceptable examples rate of change m/s/s increase in speed |
| Answer | Mark | Mark scheme |
|---|---|---|
| 220 | P1 | for splitting the area into strips and correct process to find the area of one strip, eg \(\dfrac{5 \times 4}{2}\ (= 10)\) or \(\dfrac{(4 + 12)}{2} \times 5\ (= 40)\) or \(\dfrac{(12 + 18)}{2} \times 5\ (= 75)\) or \(\dfrac{(18 + 20)}{2} \times 5\ (= 95)\) |
| P1 | for a complete process using at least 4 strips to find the area under the curve eg, \(\text{``}10\text{''} + \text{``}40\text{''} + \text{``}75\text{''} + \text{``}95\text{''}\) | |
| A1 | for answer in the range 215 to 225 from correct working using at least 4 strips |
Working
4, 12, 18, 20
Allow one error in the reading of speeds
22
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | \(-4\) | \(-2\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(2, -4, 2, 8\) | B2 | all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | (dep B1) for at least 5 points plotted correctly ft from part a |
| A1 | for a fully correct curve drawn |
Accept freehand curves drawn that are not line segments; there must be some attempt to draw the minimum point below \(y = -4\).
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-2.6\) or 1.6 | B1 | for 1 correct value, ft a non linear graph |
Award for \(-2.6\) or 1.6 or both values but do not award the mark if a correct value is given with an incorrect value.
Accept 1.56 or \(-2.56\)
Note for ft to be applied the graph may be joined by line segments.
15 The equation of a circle is \(x^2 + y^2 = 42.25\)
Find the radius of the circle. (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| 6.5 | B1 | oe |
Accept (eg) \(6\frac{1}{2}\) and \(\sqrt{42.25}\)
14 On the grid, sketch the curve with equation \(y = 2^x\)
Give the coordinates of any points of intersection with the axes.
(2)
| Answer | Mark | Mark scheme |
|---|---|---|
| curve | C1 | sketch of graph which starts above \(x\)-axis for negative \(x\), and makes an increasing exponential rise into positive \(x\) |
| (0,1) labelled | C1 | for showing a label of (0,1) on the \(y\) axis |
Condone graph “touching” the \(x\) axis.
Do not award from a graph for positive \(x\) only.
Do not award if a point is given for crossing the \(x\)-axis. Accept the coordinates shown as a label of “1” written on the \(y\) axis at the intersection.
Question 14: sketch of \(y = 2^x\)
12 Tom uses his lorry to deliver bricks.
You can use this graph to find the delivery cost for different distances.
For each delivery, there is a fixed charge plus a charge for the distance.
Tom makes two deliveries of bricks.
The distance of one delivery is 20 miles more than the distance of the other delivery.
| Answer | Mark | Mark scheme |
|---|---|---|
| 10 | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 30 | M1 | for using the graph to take one correct reading |
| A1 | 30 or ft from correct use of graph |
May be shown on graph
3
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | \(-4\) | \(-2\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(2, -4, 2, 8\) | B2 | all 4 values correct |
| (B1 | for 2 or 3 correct values) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | (dep B1) for at least 5 points plotted correctly ft from part a |
| A1 | for a fully correct curve drawn |
Accept freehand curves drawn that are not line segments; there must be some attempt to draw the minimum point below \(y = -4\).
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-2.6\) or 1.6 | B1 | for 1 correct value, ft a non linear graph |
Award for \(-2.6\) or 1.6 or both values but do not award the mark if a correct value is given with an incorrect value.
Accept 1.56 or \(-2.56\)
Note for ft to be applied the graph may be joined by line segments.
24
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(6\) | \(-6\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(0, -4, -6, -4, 0\) | B2 | fully correct figures |
| (B1 | At least 2 correct figures) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | (dep B1) for at least 5 points correctly plotted ft from (a) |
| A1 | fully correct graph |
Must be a curve
| Answer | Mark | Mark scheme |
|---|---|---|
| 2.6 and \(-1.6\) | M1 | for \(y = -2\) drawn or intersections with \(y = -2\) or \(y = x^2 - x - 4\) drawn or 1 correct value |
| A1 | ft a quadratic graph or for answers in the range 2.5 to 2.7 and \(-1.5\) to \(-1.7\) |
If answers stated as coordinates, award M1 for both coordinates and M0 for one coordinate
16
| Answer | Mark | Mark scheme |
|---|---|---|
| Correct graph | B2 | for a circle radius 3.5, centre \((0, 0)\) |
| (B1 | for a circle centre \((0, 0)\) of a different radius, or for a circle drawn of radius 3.5 centre not \((0, 0)\) or incomplete correct circle) |
Circle could be drawn freehand as long as it approximates to a circle
| Answer | Mark | Mark scheme |
|---|---|---|
| \(x = 2.0\), \(y = -2.9\) \(x = -1.2\), \(y = 3.3\) | M1 | for \(2x + y = 1\) drawn, or for correctly eliminating one variable, eg \(x^2 + 1 - 4x + 4x^2 = 12.25\) or \(x^2 + (1 - 2x)^2 = 12.25\) |
| A1 | for the pair of \(x\) values, or the correct pair of \(y\) values, or one correct pair of \(x\)/\(y\) values ft from (a) (dep on B1) | |
| A1 | for both correct pair of \(x\)/\(y\) values, unambiguously matched ft from (a) (dep on B1) |
\(2x + y = 1\) does not have to be shown
Use professional judgment
Accept values given as coordinates.
Check graph for answers
15 The graph shows the speed of a car, in metres per second, during the first 20 seconds of a journey.
| Answer | Mark | Mark scheme |
|---|---|---|
| 488 to 507 | M1 | for method to find area of one strip using trapezia, eg \(\frac{1}{2} \times 5 \times 22\) (= 55) or \(\frac{1}{2} \times 5 \times (22 + 28)\) (= 125) or \(\frac{1}{2} \times 5 \times (28 + 32)\) (= 150) or \(\frac{1}{2} \times 5 \times (32 + 35)\) (= 167.5) OR for a method to find an estimate for the area using rectangles eg \(5 \times 22\) or \(5 \times 28\) or \(5 \times 32\) or \(5 \times 35\) |
| M1 | for complete and correct method to find the area using four strips, eg \(\frac{1}{2} \times 5 \times 22 + \frac{1}{2} \times 5 \times (22 + 28) + \frac{1}{2} \times 5 \times (28 + 32) + \frac{1}{2} \times 5 \times (32 + 35)\) or \(5 \times 22 + 5 \times 28 + 5 \times 32 + 5 \times 35\) | |
| A1 | for answer in the range 488 to 507 (SC B1 for using area under the curve) |
May use area of triangle + area of rectangle for the second, third and fourth strips – lengths must be correct.
May use triangle for first strip, \(\frac{1}{2} \times 5 \times 22\)
May use triangle for first strip, \(\frac{1}{2} \times 5 \times 22\)
| Answer | Mark | Mark scheme |
|---|---|---|
| Underestimate (supported) | C1 | (dep M1) for underestimate since parts not included below the graph OR ft their method |
14 The distance-time graph shows information about part of a car journey.
Use the graph to estimate the speed of the car at time 5 seconds. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| 11 – 19 | P1 | for drawing a tangent to the curve at time = 5 |
| P1 | for process to find the gradient, eg \(70 \div 5\) | |
| A1 | (dep on 1st P1) for answer in the range 11 - 19 m/s |
Using their drawn tangent, eg change in \(y\) ÷ change in \(x\)
Must come from gradient of a tangent.
5
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(6\) | \(-6\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(0, -4, -6, -4, 0\) | B2 | fully correct figures |
| (B1 | At least 2 correct figures) |
| Answer | Mark | Mark scheme |
|---|---|---|
| Graph | M1 | (dep B1) for at least 5 points correctly plotted ft from (a) |
| A1 | fully correct graph |
Must be a curve
| Answer | Mark | Mark scheme |
|---|---|---|
| 2.6 and \(-1.6\) | M1 | for \(y = -2\) drawn or intersections with \(y = -2\) or \(y = x^2 - x - 4\) drawn or 1 correct value |
| A1 | ft a quadratic graph or for answers in the range 2.5 to 2.7 and \(-1.5\) to \(-1.7\) |
If answers stated as coordinates, award M1 for both coordinates and M0 for one coordinate
29 Brogan needs to draw the graph of \(y = x^2 + 1\)
Here is her graph.
Write down one thing that is wrong with Brogan’s graph. (1)
| Answer | Mark | Notes |
|---|---|---|
| Comment | B1 | for correct mathematical comment eg line segments not a curve or should draw freehand or should not use a ruler, or should be a curve NB Do not accept statements about scale or plotting accuracy. |
18 Here is a speed-time graph for a train.
| Working | Answer | Mark | Notes |
|---|---|---|---|
| values 0, 2, 5, 10, 18 | 130 | M1 | for starting to find area under the curve, eg \(0.5 \times 5 \times 2\ (= 5)\) |
| M1 | for a complete method to find the area under the curve using 4 strips of equal width, eg \(\text{``}5\text{''} + 0.5 \times 5 \times (2 + 5)\ (= 17.5) + 0.5 \times 5 \times (5 + 10)\ (= 37.5) + 0.5 \times 5 \times (10 + 18)\ (= 70)\) | ||
| A1 | for 130 or answer in range 130.1 to 132 supported by accurate working |
| Answer | Mark | Notes |
|---|---|---|
| overestimate with reason | C1 | for “overestimate” and appropriate reason linked to method eg area between trapeziums and curve also included |
13
| \(x\) | \({-2}\) | \({-1}\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|
| \(y\) | \({-2}\) | \(0\) |
| Answer | Mark | Notes |
|---|---|---|
| (−2) −1.5 −1 −0.5 (0) 0.5 | B2 | for a fully correct table |
| [B1 | for 2 or 3 correct entries] |
| Answer | Mark | Notes |
|---|---|---|
| Correct line | M1 | for correctly plotting at least 5 of their points (provided B1 scored in part (a)) or for a straight line with gradient 0.5 or for a straight line through \((0, -1)\) with a positive gradient |
| A1 | for a correct line between \(x = -2\) and \(x = 3\) |
| Answer | Mark | Notes |
|---|---|---|
| 2.6 | B1 | for answer in the range 2.5 to 2.7 or ft a single straight line with positive gradient |
10 Here is part of a distance-time graph for a car’s journey.
| Answer | Mark | Notes |
|---|---|---|
| 0 to 20 seconds with reason | B1 | for between 0 seconds and 20 seconds |
| C1 | for reason given eg gradient is greatest oe |
| Answer | Mark | Notes |
|---|---|---|
| 18 | B1 | ft from (a) |
7 Brogan needs to draw the graph of \(y = x^2 + 1\)
Here is her graph.
Write down one thing that is wrong with Brogan’s graph. (1)
| Answer | Mark | Notes |
|---|---|---|
| Comment | B1 | for correct mathematical comment eg line segments not a curve or should draw freehand or should not use a ruler, or should be a curve NB Do not accept statements about scale or plotting accuracy. |
22
| \(x\) | 0.5 | 1 | 1.5 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 6 | 3 | 1.5 |
| Answer | Mark | Notes |
|---|---|---|
| 12, 4, 2, 1.2, 1 | B2 | for fully correct table (allow fractions or decimals) |
| (B1) | for 3 or 4 of 12, 4, 2, 1.2, 1 |
| Answer | Mark | Notes |
|---|---|---|
| Correct curve | M1 | ft (dep on B1 in (a)) for plotting at least 6 points from their table correctly |
| A1 | for a fully correct curve |
20 The diagram shows part of the graph of \(y = x^2 - 2x + 3\)
\(P\) is the point on the graph of \(y = x^2 - 2x + 3\) where \(x = 2\)
| Answer | Mark | Notes |
|---|---|---|
| \(-0.4\) to \(-0.2\) and 3.2 to 3.4 | M1 | for \((y =)\ x + 4\) |
| A1 | for answers in the range \(-0.4\) to \(-0.2\) and 3.2 to 3.4 |
| Answer | Mark | Notes |
|---|---|---|
| 1.6 to 2.5 | M1 | for drawing a tangent to the curve at \(x = 2\) |
| M1 | for method to find gradient of their tangent | |
| A1 | for answer in the range 1.6 to 2.5 |
20 The equation of a curve is \(y = a^x\)
\(A\) is the point where the curve intersects the \(y\)-axis.
The equation of circle C is \(x^2 + y^2 = 16\)
The circle C is translated by the vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give circle B.
| Answer | Mark | Notes |
|---|---|---|
| (0, 1) | B1 | (0, 1) |
| Answer | Mark | Notes |
|---|---|---|
| Circle radius 4 Centre (3, 0) and (−1, 0) and (7, 0) labelled | M1 | For centre (3, 0) implied by drawing or label or a circle of radius 4 or intersections on the \(x\)-axis at \(-1\) or 7 implied by drawing or labels |
| M1 | for 2 of centre (3, 0) implied by drawing or label intersections on the \(x\)-axis at \(-1\) and 7 implied by drawing or label circle drawn with radius 4 | |
| A1 | for a fully correct answer |
14 Here are some graphs.
In the table below, match each equation with the letter of its graph.
| Equation | Graph |
|---|---|
| \(y = \sin x\) | |
| \(y = x^3 + 4x\) | |
| \(y = 2^x\) | |
| \(y = \dfrac{4}{x}\) |
(3)
| Answer | Mark | Notes |
|---|---|---|
| C, F, A, H | B3 | for a fully correct table |
| [B2 | for 2 or 3 correct] | |
| [B1 | for 1 correct] |
11 The graph of \(y = \mathrm{f}(x)\) is drawn on the grid.
| Answer | Mark | Notes |
|---|---|---|
| (1, −3) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| \(-0.75\), 2.75 | B1 | accept \(-0.7\) to \(-0.8\), 2.7 to 2.8 |
| Answer | Mark | Notes |
|---|---|---|
| \(-2.8\) | B1 | cao |
11 You can use this graph to change between inches and centimetres.
Daniel’s height is 6 feet 3 inches.
1 foot = 12 inches
| Answer | Mark | Notes |
|---|---|---|
| 29 | B1 | answer in the range 29 to 30 |
| Answer | Mark | Notes |
|---|---|---|
| 186 to 195 | M1 | for changing 6ft 3 inches to inches e.g. \(6 \times 12 + 3\ (= 75)\) or changing 1ft to 30 cm |
| M1 | for a method to convert to cm, e.g. 25 → 63 then \(\times\, 3\), \(6 \times 30 + \frac{1}{4} \times 30\) | |
| A1 | for answer in the range 186 to 195 or ft from correct use of graph |
21
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | 3 | 0 |
| Answer | Mark | Notes |
|---|---|---|
| 5, 4, (3), 2, 1, (0) | 2 | M1 for 1 or 2 or 3 correct entries A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| Line drawn | 2 | M1 plots 5 of their points correctly or a straight line with gradient \(-1\) or a straight line through (0, 4) A1 correct line between \(x = -1\) and \(x = 4\) |
18 Amina cycled from her home to a shop.
She then cycled home.
The travel graph shows information about Amina’s journey.
At 11 20 Amina stopped to go into the shop.
Amina took more time to cycle home from the shop than she took to cycle to the shop.
| Answer | Mark | Notes |
|---|---|---|
| 10 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 7 | 2 | M1 for 11 20 – 11 00 (= 20) or 11 57 – 11 30 (= 27) or 57 – 20 (= 37) or 57 – 27 (= 30) A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 12 | 1 | B1 cao |
13
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 11 | 1 | \(-1\) | 1 |
| Answer | Mark | Notes |
|---|---|---|
| \(5, -1, 5\) | 2 | B2 for all 3 correct (B1 for 1 or 2 correct) |
| Answer | Mark | Notes |
|---|---|---|
| Correct graph | 2 | M1 ft for 5, 6 or 7 points plotted correctly, provided at least B1 awarded in (a) A1 for a fully correct graph (no line segments) |
| Answer | Mark | Notes |
|---|---|---|
| \(-0.6\), 3.6 | 2 | M1 for use of \(y = 3\) A1 for \(-0.5\) to \(-0.7\), 3.5 to 3.7 (ft quadratic graph) |
22
The graph of \(y = k^x\), where \(k\) is a positive constant, is shown above.
The graph of \(y = \sin x^\circ\) for values of \(x\) from \(-270\) to \(+270\) is shown above.
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(2 = k^{-1}\) | \(\tfrac{1}{2}\) | 2 | M1 for reading off and substituting a pair of values from the graph (excluding 0, 1) into the equation, eg \(x = -1\), \(y = 2\) A1 for \(\tfrac{1}{2}\) oe |
| Answer | Mark | Notes |
|---|---|---|
| correct graph | 2 | B2 cao (B1 for correct general shape, eg \(y = -\sin x^\circ\)) |
20
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | 6 | 0 |
| Answer | Mark | Notes |
|---|---|---|
| 10, 8, (6), 4, 2, (0) | 2 | B2 for fully correct table (B1 for 2 or 3 entries correct) |
| Answer | Mark | Notes |
|---|---|---|
| line drawn | 2 | B2 for correct straight line between \(x = -1\) and \(x = 4\) (B1 for a single straight line which passes through (0, 8) or for a single straight line with negative gradient \(-2\) or for at least 5 points from their table plotted correctly) |
17 Jane walked from her home to the ice rink and then walked back home.
The travel graph for Jane’s journey to the ice rink and back home is shown below.
On the way to the ice rink Jane stopped at her friend’s house.
| Answer | Mark | Notes |
|---|---|---|
| 30 minutes or \(\frac{1}{2}\) hour | 1 | B1 for 30 minutes or \(\frac{1}{2}\) hour |
| Answer | Mark | Notes |
|---|---|---|
| 4 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 5 30 pm | 1 | B1 for 5 30 (pm) or 17 30 |
17
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 0.25 | 2 |
| Working | Answer | Mark | Notes | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.5, 1, 4, 8 | 2 | B2 for all 4 correct (B1 for 2 or 3 correct) |
| Answer | Mark | Notes |
|---|---|---|
| Correct graph | 2 | M1 for at least 5 points plotted correctly from their table, provided at least B1 scored in part (a) A1 cao for a correct graph drawn from (−2, 0.25) to (3, 8) |
7 On the grid, draw the graph of \(y = 8 - 2x\) for values of \(x\) from \(-1\) to 4
(3)
| Working | Answer | Mark | Notes | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y = 8 - 2x\) drawn | 3 | B3 for a correct line between \(x = -1\) and \(x = 4\) OR B2 for a correct straight line segment through at least 3 of \((-1, 10)\), (0, 8), (1, 6), (2, 4), (3, 2), (4, 0) or for all of these points plotted but not joined or for a line drawn with a negative gradient through (0, 8) and clear intention to use of a gradient of \(-2\), eg line through (0,8) and (0.5, 6) OR B1 for at least 2 correct points stated or plotted or for a line drawn with a negative gradient through (0, 8) or a line with gradient \(-2\) |
4 Jane walked from her home to the ice rink.
The travel graph for Jane’s journey to the ice rink is shown below.
On the way to the ice rink Jane stopped at her friend’s house.
Jane was at the ice rink for 1 hour 30 minutes.
She then walked home at a steady speed.
Jane took 2 hours to walk home.
| Answer | Mark | Notes |
|---|---|---|
| 5 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| Lines added | 2 | B1 for a horizontal line from (5, 8) to (6.30, 8) B1 for a single straight line with the correct gradient and length, down to the time axis eg (6.30, 8) to (8.30, 0) |
19 The diagram shows the graph of \(y = x^2 - 4x - 2\)
| Answer | Mark | Notes |
|---|---|---|
| (i) \(-0.4\) to \(-0.5\) 4.4 to 4.5 | B1 for value in range \(-0.4\) to \(-0.5\) and value in range 4.4 to 4.5 NB: condone values given as part of coordinates. | |
| (ii) \(-1.0\) to \(-1.2\) 5.0 to 5.2 | 3 | M1 for \(x^2 - 4x - 2 = 4\) or line \(y = 4\) drawn on graph or points marked with a \(y\) coord. of 4 or a value in range \(-1.0\) to \(-1.2\) or a value in range 5.0 to 5.2 A1 for value in range \(-1.0\) to \(-1.2\) and value in range 5.0 to 5.2; do not accept coordinates. |
| Answer | Mark | Notes |
|---|---|---|
| \(-1.6\) to \(-1.8\) 4.6 to 4.8 | 3 | M1 for \(x + y = 6\) drawn on graph A2 for value in range \(-1.6\) to \(-1.8\) and value in range 4.6 to 4.8 (A1 for one correct value or both values given as coordinates) |
17
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \(y\) | 5 | 9 |
| Answer | Mark | Notes |
|---|---|---|
| 3 (5) 7 (9) 11, 13 | 2 | B2 for 3, 7, 11, 13 (B1 for 2 or 3 correct values) |
| Answer | Mark | Notes |
|---|---|---|
| Graph drawn | 2 | M1 (may ft from (a) if B1 awarded) for at least 5 points correctly plotted A1 for correct graph from \(x = 0\) to \(x = 5\) |
13
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 |
|---|---|---|---|---|---|
| \(y\) | 3 | 3 |
| Answer | Mark | Notes |
|---|---|---|
| \(-1\), 1, \(-1\) | 2 | B2 for all correct (B1 for 1 or 2 correct) |
| Answer | Mark | Notes |
|---|---|---|
| Correct graph | 2 | M1 ft for 4 or 5 points from their table plotted correctly, provided at least B1 awarded in part (a) A1 for a fully correct graph (no line segments) |
6 The graph gives information about the number of litres of petrol in the tank of Jim’s car during a journey.
At 3 30 pm Jim stopped and put some petrol into the tank.
| Answer | Mark | Notes |
|---|---|---|
| 35 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 30 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 50 | 2 | M1 for \(35 - 10\) (= 25) or \(40 - 15\) (= 25) or \(35 + \text{``}30\text{''} - 15\) oe A1 for 50 or f.t. from (b) + 20 |
14 You can use this graph to change between km and miles.
Chris drives 250 km.
He then drives 100 miles.
| Answer | Mark | Notes |
|---|---|---|
| 28 – 29 | 1 | B1 answer in range 28 – 29 |
| Answer | Mark | Notes |
|---|---|---|
| 8 – 8.5 | 1 | B1 answer in range 8 – 8.5 |
| Answer | Mark | Notes |
|---|---|---|
| 250 – 259 | 3 | M1 for use of conversion graph to change 250 km to miles (eg 140 – 160 miles) or 100 miles to km (eg 150 – 170 km) M1 (dep) for addition of 100 miles or 250 km in consistent units A1 for answer in the range 250 – 259 (miles) |
12
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| \(y\) | 6 | 2 | 12 |
| Answer | Mark | Notes |
|---|---|---|
| 2, 0, 0, 6 | 2 | B2 for 2, 0, 0, 6 (B1 for at least two of 2, 0, 0, 6); could be taken from graph |
| Answer | Mark | Notes |
|---|---|---|
| Correct curve | 2 | M1 (ft) for at least 5 points plotted correctly A1 for a fully correct curve |
| Answer | Mark | Notes |
|---|---|---|
| \(-0.6\), 3.6 | 2 | M1 (ft if M1 awarded in (b) and at least B1 in (a)) for indicating a point or line drawn at \(y = 4\), or one solution given A1 (ft) for both solutions |
10 A coach travels from Dronston to Luscoe.
The travel graph for this journey is shown below.
The coach stops in Luscoe for 15 minutes.
The coach then returns to Dronston at a constant speed of 42 km/h.
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(9 \times 6\) | 54 | 2 | M1 for a method to find the speed e.g \(9 \div 10\), \(9 \div 0.16\) A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| Graph completed | 3 | B1 horizontal line from (30,21) to (45,21) M1 for a complete method to show the return journey is 30 mins or \(\tfrac{1}{2}\) hour evidenced by the line on the graph or by calculation A1 Correct line drawn from Luscoe \((x, 21)\) to \((x + 30, 0)\) |
28
| Answer | Mark | Notes |
|---|---|---|
| Circle drawn | 2 | B2 fully correct circle drawn (B1 for circle drawn with centre (0,0) or circle drawn with radius 4) OR M1 at least 5 correct points calculated and plotted A1 fully correct circle drawn |
| Answer | Mark | Notes |
|---|---|---|
| \(x = 1.4\), \(y = 3.8\) \(x = -2.2\), \(y = -3.4\) | 3 | M1 for \(y = 2x + 1\) drawn or for elimination of one variable A1 for one correct pair of values given or for \(x = 1.4, -2.2\) (\(\pm 0.2\)) or ft from graph provided 2 marks in (a) A1 for second correct pair of values given (\(\pm 0.2\)) or ft from graph provided 2 marks in (a) |
10 You can use this graph to change between pounds and kilograms.
A trolley can carry a maximum weight of 200 pounds.
Jack has 4 bags of potatoes.
Each bag of potatoes weighs 25 kilograms.
| Answer | Mark | Notes |
|---|---|---|
| 5.8 to 6 | 1 | B1 for an answer in the range 5.8 to 6 |
| Answer | Mark | Notes |
|---|---|---|
| No (supported) | 3 | M1 for a correct conversion of any amount (lb to kg or kg to lb) excepting that in (a) M1 (dep M1) for a complete method to convert 100 kg (from \(25 \times 4\)) to lb (to compare with 200 lb) or to convert 50 lb (from \(200 \div 4\)) to kg (to compare with 25 kg) C1 for “no” and a comparison with a converted weight of 212 – 228 pounds or 88 – 94 kg |
25 On the grid, draw the graph of \(y = 2x - 3\) for values of \(x\) from \(-2\) to 3 (4)
| Answer | Mark | Notes |
|---|---|---|
| Straight line from \((-2, -7)\) to \((3, 3)\) | 4 | (Table of values) C1 for axes scaled and labelled M1 for at least 2 correct attempts to find points by substituting values of \(x\) M1 ft for plotting at least 2 of their points (any points plotted from their table must be plotted correctly) A1 for correct line between \(x = -2\) and \(x = 3\) (No table of values) C1 for axes scaled and labelled M1 for at least 2 correct points with no more than 2 incorrect points M1 for at least 2 correct points (and no incorrect points) plotted OR line segment of \(y = 2x - 3\) drawn A1 for correct line between \(x = -2\) and \(x = 3\) (Use of \(y = mx + c\)) C1 for axes scaled and labelled M1 for line drawn with gradient of 2 OR line drawn with a \(y\) intercept of \(-3\) M1 for line drawn with gradient of 2 AND with a \(y\) intercept of \(-3\) A1 for correct line between \(x = -2\) and \(x = 3\) [SC B2 (indep of C1) for the correct line between \(x = 0\) and \(x = 3\), ignore any additional incorrect line segment(s)] |
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | \(-7\) | \(-5\) | \(-3\) | \(-1\) | 1 | 3 |
19
| \(x\) | 0.5 | 1 | 2 | 4 | 5 | 8 |
|---|---|---|---|---|---|---|
| \(y\) | 4 | 2 |
| Answer | Mark | Notes |
|---|---|---|
| 8, (4), (2), 1, 0.8, 0.5 | 2 | B2 all 4 correct Accept \(\tfrac{4}{5}\) in place of 0.8 and \(\tfrac{1}{2}\) in place of 0.5 (B1 for 2 or 3 correct) |
| Answer | Mark | Notes |
|---|---|---|
| correct graph | 2 | M1 (ft dep on B1) for 5 or 6 points plotted correctly from their table (overlay) A1 cao for correct curve drawn from (0.5,8) to (8, 0.5) |
*13 You can use this graph to change between miles and kilometres.
The distance from Paris to London is 280 miles.
The distance from Paris to Amsterdam is 500 kilometres.
Is Paris further from London or further from Amsterdam?
You must show how you get your answer. (3)
| Answer | Mark | Notes |
|---|---|---|
| Amsterdam with figures | 3 | B1 for a correct conversion from miles to km or km to miles eg 8 km = 5 miles eg 28 miles = 44 km M1 for a correct method to convert 280 miles to km or 500 km to miles or 420 – 460 (km) or 300 – 320 (miles) C1 (dep on M1) for statement with correct conclusion and correct conversions (420 – 460 km or 300 – 320 miles) |
22 Anna drives 45 miles from her home to a meeting.
Here is the travel graph for Anna’s journey to the meeting.
Anna’s meeting lasts for 1 hour.
She then drives home at a steady speed of 30 miles per hour with no stops.
Complete the travel graph to show this information. (2)
| Answer | Mark | Notes |
|---|---|---|
| Graph completed | 2 | B1 for line from (2.5, 45) to (3.5, 45) B1 ft line of correct gradient to axis (after 1½ hour) |
18 Here is a conversion graph to change between UK pounds (£) and South African rand.
Jo changes £200 into rand.
Simon has £100 and 3700 rand.
He goes to a shop where he can spend both pounds and rand.
He wants to buy
a computer costing £360
or
a watch costing £400
or
a camera costing £375
| Answer | Mark | Notes |
|---|---|---|
| 2600 | 1 | B1 for 2600 |
| Working | Answer | Mark | Notes | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| computer, camera | 3 | M1 for method to convert 3700 rand into £ or for changing one amount in pounds into rand M1 for a complete method to compare total money Simon has with the cost of each item C1 (dep M2) for correct conclusion with correct figures e.g. £383 - £386 or 4950 rand to 5050 rand |
15
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 7 | \(-2\) | \(-1\) |
| Answer | Mark | Notes |
|---|---|---|
| \(2, -1, 2, 7\) | 2 | B2 for all correct (B1 for 2 or 3 correct) |
| Answer | Mark | Notes |
|---|---|---|
| Correct graph | 2 | M1 (dep on at least B1) for at least 6 points from their table plotted correctly A1 cao for fully correct graph |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(x^2 - 3x - 4 = 0\) \((x - 4)(x + 1) = 0\) | \(-1, 4\) | 2 | M1 for line \(y = x + 3\) drawn correctly or for reduction to correct 3 term quadratic (=0) and : \((x \pm 1)(x \pm 4)\) or formula using \(a = 1\), \(b = -3\) and \(c = -4\), allow one sign error in the formula, or \(\left(x - \dfrac{3}{2}\right)^2 = 4 + \left(\dfrac{3}{2}\right)^2\) A1 cao |
26 On the grid, draw the graph of \(y = 3x + 2\) for values of \(x\) from \(-2\) to 2
(4)
| Working | Answer | Mark | Notes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y = 3x + 2\) drawn | 4 | B1 for axes scaled and labelled (Table of values) M1 for at least 2 correct attempts to find points by substituting values of \(x\) M1 ft for plotting at least 2 of their points (any points from their table must be correctly plotted) A1 for correct line between \(x = -2\) and \(x = 2\) (No table of values) M1 for at least 2 correct points with no more than 2 incorrect points M1 for at least 2 correct points (and no incorrect points) plotted OR line segment of \(y = 3x + 2\) drawn A1 for correct line between \(x = -2\) and \(x = 2\) (Use of \(y = mx + c\)) M1 for line drawn with gradient of 3 OR line drawn with \(y\) intercept at 2 M1 for line drawn with gradient of 3 AND with \(y\) intercept at 2 A1 for correct line between \(x = -2\) and \(x = 2\) [SC B2 (indep of B1) for correct line segment between \(x = 0\) and \(x = 2\) – ignore any additional incorrect line segment(s)] |
18 You can use this graph to change between gallons and litres.
There are 80 litres of petrol in the petrol tank of car A.
There are 16 gallons of petrol in the petrol tank of car B.
Which car has the most petrol in its petrol tank?
You must show all your working. (3)
| Working | Answer | Mark | Notes |
|---|---|---|---|
| 80 litres ≈ 18 gallons or 16 gallons ≈ 72 litres | A with correct figures | 3 | M1 for reading from the graph eg. 8 gallons = 36 litres; 20 litres = 4.4 gallons M1 for a complete method to convert either 80 litres into gallons or 16 gallons into litres e.g. 80 litres = “4.4” × 4 gallons or 16 gallons = “36” × 2 litres A1 for car A with correct figures in range 17.5 – 18.5 gallons or 64 – 72 litres |
17
| \(x\) | \(-3\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \(y\) | 3 | 0 | 15 |
| Answer | Mark | Notes |
|---|---|---|
| \(-15, 0, 3, 0,\) \(-3, 0, 15\) | 2 | B2 for all correct (B1 for any 2 or 3 correct) |
| Answer | Mark | Notes |
|---|---|---|
| Correct graph | 2 | M1 for at least 5 points plotted correctly (ft from table if at least B1 awarded in (a)) A1 for a fully correct curve |
29
On the grid, draw the graph of \(y = \frac{1}{2}x + 5\) for values of \(x\) from \(-2\) to 4 (3)
| Working | Answer | Mark | Notes | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y = \frac{1}{2}x + 5\) drawn | 3 | (Table of values / calculation of values) M1 for at least 2 correct attempts to find points by substituting values of \(x\). M1 ft for plotting at least 2 of their points (any points plotted from their table must be plotted correctly) A1 for correct line between \(x = -2\) and \(x = 4\) (No table of values) M1 for at least 2 correct points with no more than 2 incorrect points M1 for at least 2 correct points (and no incorrect points) plotted OR line segment of \(y = \frac{1}{2}x + 5\) drawn A1 for correct line between \(x = -2\) and \(x = 4\) (Use of \(y = mx + c\)) M1 for line drawn with gradient of 0.5 OR line drawn with a \(y\) intercept of 5 M1 for line drawn with gradient of 0.5 AND with a \(y\) intercept of 5 A1 for correct line between \(x = -2\) and \(x = 4\) SC : B2 for the correct line from \(x = 0\) to \(x = 4\) |
22 On Monday, Holly walked from her home to school.
She stopped at her friend’s house on the way to school.
On Tuesday, Holly cycled from her home to school.
The travel graphs show Holly’s journey on Monday and on Tuesday.
Holly took less time to get to school on Tuesday than on Monday.
| Answer | Mark | Notes |
|---|---|---|
| 3 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 5 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 18 | 2 | M1 for “30” – “12” seen with at least one correct A1 cao (SC : B1 for 25 and 12 seen with an answer of 13) |
15
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | 3 | 0 | 3 |
| Answer | Mark | Notes |
|---|---|---|
| \(x\): \(-2\), \(-1\), 0, 1, 2, 3, 4 \(y\): 8, 3, 0, \(-1\), 0, 3, 8 | 2 | B2 for 8, \(-1\), 0, 8 (B1 for at least two of 8, \(-1\), 0, 8) |
| Answer | Mark | Notes |
|---|---|---|
| Correct curve | 2 | M1 (ft) for at least 5 points plotted correctly A1 for a fully correct curve |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(x^2 - 2x - 3 = 0\) OR \((x - 3)(x + 1) = 0\) | 3 and \(-1\) | 2 | M1 for the straight line \(y = 3\) drawn to intersect the “graph” from (a) A1 for both solutions OR M1 for identifying \(y = 3\) from the table A1 for both solutions OR M1 for \((x \pm 3)(x \pm 1)\) A1 for both solutions |
13 You can use this conversion graph to change between kilograms and pounds.
Daniel’s weight is 9 stone 6 pounds.
1 stone = 14 pounds.
What is Daniel’s weight in kilograms? (3)
| Answer | Mark | Notes |
|---|---|---|
| 60 | 3 | M1 for 9 × 14 + 6 (=132) M1 (dep) for full method to convert ‘132’ from lbs to kg using the graph or for ‘132’ ÷ 2.2 (=60) A1 for 59 – 62 OR M1 for reading off 14 lbs (= 6.2 − 6.5) and 6 lbs (= 2.4 − 2.9) M1 (dep) for 9 × ‘6.4’ + ‘2.75’ A1 for 59 – 62 [SC B2 for 66] |
21 Simon went for a cycle ride.
He left home at 2 pm.
The travel graph represents part of Simon’s cycle ride.
At 3 pm Simon stopped for a rest.
At 5 pm Simon stopped for 30 minutes.
Then he cycled home at a steady speed.
It took him 1 hour 30 minutes to get home.
| Answer | Mark | Notes |
|---|---|---|
| 30 | 1 | B1 for 30 minutes oe |
| Answer | Mark | Notes |
|---|---|---|
| 20 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| graph completed | 2 | B1 for horizontal line from (5, 20) to (5.30, 20) B1 for a single straight line with the correct gradient from ‘(5.30, 20)’ to the time axis |
14 You can use this conversion graph to change between pounds (£) and dollars ($).
Ella has $200 and £800
Her hotel bill is $600
Ella pays the bill with the $200 and some of the pounds.
| Answer | Mark | Notes |
|---|---|---|
| 8 | 1 | B1 for 8 (.00) |
| Answer | Mark | Notes |
|---|---|---|
| 550 | 4 | M1 for \(600 - 200\) (= 400) M1 for correct method to convert ‘$400’ to £ M1 (dep on the previous M1) for 800 − ‘$400’ in £s A1 for value in the range 540 – 560 OR M1 for correct method to convert $600 and $200 to pounds M1 for ‘375’ − ‘125’ M1 (dep on the previous M1) 800 − ‘250’ A1 for a value in the range 540-560 OR M1 for correct method to convert £800 to dollars M1 for ‘1280’ + 200 − 600 M1 (dep on the previous M1) for attempt to convert ‘$880’ back to £ A1 for value in the range 540 – 560 |
6 You can use this conversion graph to change between pounds (£) and dollars ($).
Ella has $200 and £800
Her hotel bill is $600
Ella pays the bill with the $200 and some of the pounds.
| Answer | Mark | Notes |
|---|---|---|
| 8 | 1 | B1 for 8 (.00) |
| Answer | Mark | Notes |
|---|---|---|
| 550 | 4 | M1 for 600 – 200 ( = 400) M1 for correct method to convert ‘$400’ to £ M1 (dep on the previous M1) for 800 – ‘$400’ in £s A1 for value in the range 540 – 560 OR M1 for correct method to convert $600 and $200 to pounds M1 for ‘375’ – ‘125’ M1 (dep on the previous M1) 800 – ‘250’ A1 for a value in the range 540 – 560 OR M1 for correct method to convert £800 to dollars M1 for ‘1280’ + 200 – 600 M1 (dep on the previous M1) for attempt to convert ‘$880’ back to £ A1 for value in the range 540 – 560 |
4
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 |
|---|---|---|---|---|---|
| \(y\) | 1 | 5 |
| Working | Answer | Mark | Notes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3, 7, 9 | 2 | B2 for all three values correct in the table (B1 for 2 values correct) |
| Working | Answer | Mark | Notes |
|---|---|---|---|
 | graph of \(y = 2x + 5\) | 2 | (From their table of values) M1 ft for plotting at least 2 of their points (any points from their table must be correctly plotted) A1 for correct line from \(x = -2\) to \(x = +2\) (Use of \(y = mx + c\)) M1 for line drawn with gradient of 2 or line drawn with a \(y\) intercept of 5 and a positive gradient) A1 for correct line from \(x = -2\) to \(x = +2\) |
3 Simon went for a cycle ride.
He left home at 2 pm.
The travel graph represents part of Simon’s cycle ride.
At 3 pm Simon stopped for a rest.
At 5 pm Simon stopped for 30 minutes.
Then he cycled home at a steady speed.
It took him 1 hour 30 minutes to get home.
| Answer | Mark | Notes |
|---|---|---|
| 30 | 1 | B1 for 30 minutes |
| Answer | Mark | Notes |
|---|---|---|
| 20 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| graph completed | 2 | B1 for horizontal line from (5, 20) to (5.30, 20) B1 for a single straight line with the correct gradient from ‘(5.30, 20)’ to the time axis |
27
| Answer | Mark | Notes |
|---|---|---|
| Circle, centre \(O\), radius 2 | 2 | B2 cao (B1 for a circle radius 2 any centre or for a circle or part of a circle centre (0, 0) any radius) |
| Answer | Mark | Notes |
|---|---|---|
| Cosine curve crossing at (0, 1), (90, 0), (270, 0) and (360, 1) | 2 | B2 cao (ignore if sketch outside region) (B1 for a curve with correct intercepts but incorrect amplitude OR for a curve starting at (0,1) with correct amplitude but incorrect intercepts; curves must have a shape that approximates to a cosine curve) |
21 Two companies, Barry’s Bricks and Bricks ArUs, deliver bricks.
The graph shows the delivery costs of bricks from both companies.
Prakash wants Bricks ArUs to deliver some bricks.
He lives 2 miles away from Bricks ArUs.
John needs to have some bricks delivered.
He lives 4 miles from Barry’s Bricks.
He lives 5 miles from Bricks ArUs.
| Answer | Mark | Notes |
|---|---|---|
| 56 | 1 | B1 for 56 (accept answer in the range 55 to 57) |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| Barry’s Bricks £50 Bricks ArUs £65 \(65 - 50\) | 15 | 3 | M1 for 50 or 65 (accept 64 – 66) M1 for 65 – 50 (accept 64-66 for 65) A1 for 15 (accept answer in range 14 to 16) |
18
| \(x\) | 0.5 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \(y\) | 6 | 3 | 1.5 | 1 |
| Working | Answer | Mark | Notes | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Correct table | 2 | B2 all 3 correct (B1 1 or 2 correct) |
| Working | Answer | Mark | Notes |
|---|---|---|---|
 | Correct graph | 2 | M1 at least 6 points plotted correctly from their table A1 cao for correct curve drawn from (0.5, 12) to (6, 1) |
15 You can use this graph to change between pounds (£) and dollars ($).
In London, Sano headphones cost £60
In New York, Sano headphones cost $100
Sano headphones cost more in New York than in London.
| Answer | Mark | Notes |
|---|---|---|
| 32 | 1 | B1 cao |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| e.g. $20 = £12.50 $100 = \(5 \times\) £12.50 = £62.50 £62.50 − 60 = £2.50 | £2.50 OR $4 | 3 | M1 for a correct method to convert $100 to £, e.g. \(5 \times\) ‘12.50’ (= 62.50) (‘12.50’ is their reading from the graph at $20) M1 (dep) for ‘62.50’ − 60 A1 for £2.5(0) (units must be stated) OR M1 for correct method to convert £60 to $, e.g. \(3 \times 32\) (=96) or ft their answer to part (a) M1 (dep) for 100 − ‘96’ A1 for $4 (units must be stated) |
21 On the grid, draw the graph of \(y = 3x - 2\) for values of \(x\) from \(-1\) to 3 (3)
| Working | Answer | Mark | Notes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Straight line from (\(-1\), \(-5\)) to (3, 7) | 3 | (Table of values) M1 for at least 2 correct attempts to find points by substituting values of \(x\). M1 ft for plotting at least 2 of their points (any points plotted from their table must be correctly plotted) A1 for correct line between \(-1\) and 3 (No table of values) M2 for at least 2 correct points (and no incorrect points) plotted OR line segment of \(y = 3x - 2\) drawn (ignore any additional incorrect segments) (M1 for at least 3 correct points plotted with no more than 2 incorrect points) A1 for correct line between \(-1\) and 3 (Use of \(y = mx + c\)) M2 for line segment of \(y = 3x - 2\) drawn (ignore any additional incorrect segments) (M1 for line drawn with gradient of 3 OR line drawn with a \(y\) intercept of \(-2\) and a positive gradient) A1 for correct line between \(-1\) and 3 |
*18 Bill uses his van to deliver parcels.
For each parcel Bill delivers there is a fixed charge plus £1.00 for each mile.
You can use the graph to find the total cost of having a parcel delivered by Bill.
Ed uses a van to deliver parcels.
For each parcel Ed delivers it costs £1.50 for each mile.
There is no fixed charge.
| Answer | Mark | Notes |
|---|---|---|
| 10 | 1 | B1 cao |
| Working | Answer | Mark | Notes | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ed is cheaper up to 20 miles, Bill is cheaper for more than 20 miles | 3 | M1 for correct line for Ed intersecting at (20,30) ±1 sq tolerance or \(10 + x = 1.5x\) oe C2 (dep on M1) for a correct full statement ft from graph eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles (C1 (dep on M1) for a correct conclusion ft from graph eg. cheaper at 10 miles with Ed; eg. cheaper at 50 miles with Bill eg. same cost at 20 miles; eg for £5 go further with Bill or A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances) OR M1 for correct method to work out Ed’s delivery cost for at least 2 values of \(n\) miles where \(0 \lt n \leqslant 50\) or for correct method to work out Ed and Bill’s delivery cost for \(n\) miles where \(0 \lt n \leqslant 50\) C2 (dep on M1) for 20 miles linked with £30 for Ed and Bill with correct full statement eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles (C1 (dep on M1) for a correct conclusion eg. cheaper at 10 miles with Ed; eg. cheaper at 50 miles with Bill eg. same cost at 20 miles; eg for £5 go further with Bill or A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances) SC: B1 for correct full statement seen with no working eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles QWC Decision and justification should be clear with working clearly presented and attributable |
12 You can use the graph to change between miles and kilometres.
| Answer | Mark | Notes |
|---|---|---|
| 4.8 | 1 | B1 for answer in range 4.6 – 5 |
| Answer | Mark | Notes |
|---|---|---|
| 37.5 | 2 | M1 for a valid method eg reading from graph for 6 km then \(\times 10\) A1 for answer in range 35 – 40 OR M1 for use of conversion factor \(60 \times 5/8\) oe A1 for answer in range 35 – 40 |
4 On the grid, draw the graph of \(y = 3x - 2\) for values of \(x\) from \(-1\) to 3 (3)
| Working | Answer | Mark | Notes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Using \(y = mx + c\) gradient = 3 \(y\) intercept = \(-2\) | Straight line from \((-1, -5)\) to \((3, 7)\) | 3 | (Table of values) M1 for at least 2 correct attempts to find points by substituting values of \(x\). M1 ft for plotting at least 2 of their points (any points plotted from their table must be correctly plotted) A1 for correct line between \(-1\) and 3 (No table of values) M2 for at least 2 correct points (and no incorrect points) plotted OR line segment of \(y = 3x - 2\) drawn (ignore any additional incorrect segments) (M1 for at least 3 correct points plotted with no more than 2 incorrect points) A1 for correct line between \(-1\) and 3 (Use of \(y = mx + c\)) M2 for line segment of \(y = 3x - 2\) drawn (ignore any additional incorrect segments) (M1 for line drawn with gradient of 3 OR line drawn with a \(y\) intercept of \(-2\) and a positive gradient) A1 for correct line between \(-1\) and 3 |
3 Bill uses his van to deliver parcels.
For each parcel Bill delivers there is a fixed charge plus £1.00 for each mile.
You can use the graph to find the total cost of having a parcel delivered by Bill.
Ed uses a van to deliver parcels.
For each parcel Ed delivers it costs £1.50 for each mile.
There is no fixed charge.
| Answer | Mark | Notes |
|---|---|---|
| 10 | 1 | B1 cao |
| Working | Answer | Mark | Notes | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ed is cheaper up to 20 miles, Bill is cheaper for more than 20 miles | 3 | M1 for correct line for Ed intersecting at (20,30) ±1 sq tolerance or \(10 + x = 1.5x\) oe C2 (dep on M1) for a correct full statement ft from graph eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles (C1 (dep on M1) for a correct conclusion ft from graph eg. cheaper at 10 miles with Ed ; eg. cheaper at 50 miles with Bill eg. same cost at 20 miles; eg for £5 go further with Bill OR A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances) OR M1 for correct method to work out Ed’s delivery cost for at least 2 values of \(n\) miles where \(0 \lt n \leqslant 50\) OR for correct method to work out Ed and Bill’s delivery cost for \(n\) miles where \(0 \lt n \leqslant 50\) C2 (dep on M1) for 20 miles linked with £30 for Ed and Bill with correct full statement eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles (C1 (dep on M1) for a correct conclusion eg. cheaper at 10 miles with Ed ; eg. cheaper at 50 miles with Bill eg. same cost at 20 miles; eg for £5 go further with Bill OR A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances) SC : B1 for correct full statement seen with no working eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles QWC: Decision and justification should be clear with working clearly presented and attributable |
23 A roller coaster starts at ground level.
The height above ground level, \(h\), in metres, of the roller coaster is given by
\[h = -(t - 7)^2 + 49\]where \(t\) is the time in seconds after the roller coaster starts.
Sam draws a graph of \(h\) against \(t\) for \(\quad 0 \leqslant t \leqslant 14\)
Make two criticisms of Sam’s graph. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two from valid criticism about the \(h\)-intercept or valid criticism about the maximum point or valid criticism about the non-linear scaling on the horizontal axis | B2 | eg graph should go through (0, 0) graph should intercept when \(h = 0\) \(y\)-intercept should be 0 eg the maximum point should be (7, 49) at \(t = 7\), \(h\) should be 49 eg 0, 7 and 14 should be evenly spaced B1 any one valid criticism |
Allow use of \(y\) for \(h\) or use of \(x\) for \(t\)
Each criticism may be implied by a correct, unambiguous graph drawn
Examples of valid criticisms
The intersection of the \(y\)-axis is wrong
The height at the beginning should not be 49
It should start at zero/ground level
The maximum/turning point is wrong
Goes higher than 49
Spacing of numbers on the \(x\)-axis is not equal
Examples of invalid criticisms
The \(y\)-axis is wrong
Wrong graph
Should be more symmetrical
The maximum point not labelled correctly
The coordinates are wrong
Needs more labels
The labels on the \(t\)-axis are wrong
Scales not easy to read
14
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| \(y\) | 1 | 4 |
| Answer | Mark | Comments | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B2 | oe B1 2 correct values |
Mark the table only
| Answer | Mark | Comments |
|---|---|---|
| 4 or 5 points plotted correctly | M1 | correct or ft their table in (a) \(\pm \dfrac{1}{2}\) square points may be implied by graph passing through them |
| Correct smooth curve | A1 | \(\pm \dfrac{1}{2}\) square for the five correct points |
| Ignore extra points plotted with different \(x\)-coordinates | |
| Two points with the same integer \(x\)-coordinate, with the curve going through neither, is choice for plotting of that point and therefore incorrect | |
| Ignore the curve drawn outside (−1, 0.5) and (3, 8) | |
| For tolerance, \(\pm \dfrac{1}{2}\) square can be vertically or horizontally | |
| Ruled straight lines | A0 |
4 Ola and Pip make orange paint by mixing red paint and yellow paint.
The graph shows how much of each colour paint to use.
Write down how much red paint Ola uses. [1 mark]
How much orange paint does Pip make? [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| [3.9, 4.1] | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: reads off at ‘10’ from graph | ||
| [17.9, 18.1] | B2 | B1 [7.9, 8.1] or their reading \(+\) 10 |
| Alternative method 2: uses their answer to part (a) | ||
| \(2 \times\) their (a) or (their (a) \(+\) 5) \(\times\) 2 | M1 | oe |
| [17.9, 18.1] | A1ft | ft \(2 \times\) their (a) \(+\) 10 |
2 Line \(AB\) is on a grid.
| Answer | Mark | Comments |
|---|---|---|
| (8, 1) | B1 |
Check the diagram if answer line is blank
| Answer | Mark | Comments |
|---|---|---|
| (8, 4) | B1 | SC1 (1, 8) in (a) and (4, 8) in (b) |
Check the diagram if answer line is blank
25 A triangle is drawn using the lines
\(y = x\)
\(x = -2\)
\(y = 4\)
Work out the coordinates of the three vertices of the triangle. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((-2, -2)\) and \((-2, 4)\) and \((4, 4)\) | B4 | B3 for 2 correct coordinates or the 3 correct lines drawn B2 for 1 correct coordinate or 2 correct lines drawn B1 for 1 correct line drawn |
Mark coordinates first
Ignore incorrect lines
23 The distance of a particle from a point is \(d\) metres after \(t\) seconds.
| \(d = a \times b^t \quad\) where \(a\) and \(b\) are constants |
Work out the values of \(a\) and \(b\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Constructs any equation using coordinates | M1 | eg \(3 = a \times b^0\) or \(3 = a \times 1\) \(6 = a \times b^1\) \(12 = a \times b^2\) \(24 = a \times b^3\) \(48 = a \times b^4\) |
| \(a = 3\) and \(b = 2\) | A2 | A1 \(a = 3\) |
| M1 may be awarded for correct work, with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(a = 3\) with no working | M1A1 |
| \(b = 2\) with no working and \(a \ne 3\) | M0A0 |
15 The graph represents the volume of water in a bath.
The bath is full after 10 minutes.
State the units of your answer. [2 marks]
the volume of water stays constant for 20 minutes
then
all the water empties out at a constant rate in 12 minutes.
Show this information on the graph. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(240 \div 10\) or \(120 \div 5\) or 24 or \(240 \times 6\) or \(120 \times 12\) or 1440 | M1 | |
| 24 litres per minute or 1440 litres per hour | A1 | oe eg 24 l/min or 0.4 litres per second |
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Units may be abbreviated but must be the correct units for their correct rate eg allow 0.4 l/s or 0.4 lps for 0.4 litres per second | M1A1 |
| Do not ignore further incorrect attempts to change units eg 24 litres per minute and 0.4 litres/hour on the answer line | M1A0 |
| Answer | Mark | Comments |
|---|---|---|
| Horizontal line from (10, 240) to (30, 240) | B1 | \(\pm\dfrac{1}{2}\) small square |
| Straight line from their (30, 240) to (12 + their 30, 0) | B1ft | \(\pm\dfrac{1}{2}\) small square for (12 + their 30, 0) |
| Mark intention | |
| No horizontal line and straight line from (10, 240) to (22, 0) | B0B1ft |
10 A car will travel 60 miles.
Draw a graph to show the average speed of the car for times taken between 1 hour and 6 hours.
You may use the table to help you. [3 marks]
| Time taken (hours) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Average speed (mph) | 60 | 30 | 10 |
| Answer | Mark | Comments |
|---|---|---|
| At least one of 20, 15 and 12 | M1 | may be seen in table or implied by correct point or curve passing through the correct point |
| Points plotted at (1, 60), (2, 30), (3, their 20), (4, their 15), (5, their 12) and (6, 10) | M1 | \(\pm\dfrac{1}{2}\) square implied by curve through these six points |
| Single smooth curve through the correct points | A1 |
| Ignore curve before \(x = 1\) and after \(x = 6\) if it is decreasing and does not touch either axis | |
| Curve touches at least one axis | A0 |
8 A triangle is drawn using the lines
\(y = x\)
\(x = -2\)
\(y = 4\)
Work out the coordinates of the three vertices of the triangle. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((-2, -2)\) and \((-2, 4)\) and \((4, 4)\) | B4 | B3 for 2 correct coordinates or the 3 correct lines drawn B2 for 1 correct coordinate or 2 correct lines drawn B1 for 1 correct line drawn |
Mark coordinates first
Ignore incorrect lines
4 Points \(A\), \(B\) and \(C\) are plotted on a grid.
| Answer | Mark | Comments |
|---|---|---|
| \((2, -2)\) | B1 |
Condone \(x\) and \(y\) written above the coordinates
Do not condone \((2x, -2y)\)
| Answer | Mark | Comments |
|---|---|---|
| (2, 1) | B1 |
Condone \(x\) and \(y\) written above the coordinates
Do not condone \((2x, 1y)\)
| Answer | Mark | Comments |
|---|---|---|
| Point plotted at (6, 1) | B1 | allow missing label |
Ignore point marked or working for part (b)
Mark intention
The correctly drawn rhombus implies the point has been plotted correctly
23
Complete the coordinates of \(C\) and \(D\). [2 marks]
\(C\) ( 0 , \(\ldots\ldots\ldots\) ) \(D\) ( \(\ldots\ldots\ldots\) , 0 )
Complete the following statements. [2 marks]
The value of the \(\boldsymbol{y}\)-intercept is \(\ldots\ldots\ldots\)
The \(\boldsymbol{x}\)-coordinate of the turning point, \(T\), is \(\ldots\ldots\ldots\)
| Answer | Mark | Comments |
|---|---|---|
| \(C\) (0, 6) | B1 | if answer space is blank, accept (0, 6) written at \(C\) on the diagram |
| \(D\) (3, 0) | B1 | if answer space is blank, accept (3, 0) written at \(D\) on the diagram |
For each part mark the answer space unless blank
Allow \(x\) and \(y\) written above the coordinates but do not allow eg \((0x, 6y)\)
| Answer | Mark | Comments |
|---|---|---|
| 5 | B1 | |
| 3 | B1 |
Mark the answer lines only
Do not allow eg (0, 5)
18 A circle has centre \(O\) and passes through \((0, 6)\)
Write down the equation of the circle. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + y^2 = 6^2\) or \(x^2 + y^2 = 36\) | B1 | oe equation |
| \(x^2 + y^2 = 6^2\) followed by an incorrect evaluation of \(6^2\) | B1 |
| Condone \(x^2 + y^2 = r^2\) and \(r = 6\) | B1 |
13 Here is a quadratic graph with equation \(\quad y = \mathrm{f}(x)\)
Write down the roots of the equation \(\quad \mathrm{f}(x) = 0\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((x =)\ -2.2,\ 1.5\) | B2 | B1 at least one of \(-2.2\) and 1.5 with at most one incorrect value or \((-2.2, 0)\) and \((1.5, 0)\) or \((-2.2, 1.5)\) |
| \((1.5, -2.2)\) | B0 |
| \((0, -2.2)\) and \((0, 1.5)\) | B0 |
11 Here is a table of values for the equation \(\quad y = 3x + 1\)
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(y\) | 4 | 7 | 10 | 13 |
| Answer | Mark | Comments |
|---|---|---|
| All 4 points plotted correctly with a straight line joining them | B2 | \(\pm\)\(\dfrac{1}{2}\) square B1 at least two correct points plotted mark intention for straight line |
Ignore additional or incorrect points for B2 or B1
Ignore any line or curve extended outside the range
The correct position of the line implies correctly plotted points
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: uses the graph | ||
| Vertical line from \(x = 2.5\) to their straight line | M1 | \(\pm\)\(\dfrac{1}{2}\) square implied by mark at correct point on graph or on vertical axis |
| their 8.5 | A1ft | \(\pm\)\(\dfrac{1}{2}\) square ft their straight line graph if at least B1 awarded in (a) |
| Alternative method 2: substitutes into the equation | ||
| \(3 \times 2.5 + 1\) | M1 | oe |
| 8.5 | A1 | |
| Alternative method 3: uses values from the table | ||
| \(\dfrac{7 + 10}{2}\) | M1 | oe eg \(\dfrac{4 + 7 + 10 + 13}{4}\) |
| 8.5 | A1 | |
Alternative method 1 – must have a line in part (a)
Alternative method 1
A vertical line from the \(x\)-axis does not need to be drawn if the reading from the graph is correct within tolerance for their graph
8 A lion is sprinting in a straight line away from its den.
The graph shows the lion’s distance from the den.
Work out the speed of the lion in metres per second. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| A change in distance for an integer time interval or a change in distance for a non-integer time interval with the corresponding time interval | M1 | integer time intervals are [88, 92] [70, 74] [52, 56] [34, 38] [16, 20] may be seen on graph |
| \(\dfrac{\text{their change in distance}}{\text{corresponding time interval}}\) | M1dep | oe eg \(\dfrac{[88, 92]}{5}\) must see their change in distance and the corresponding time interval division by 1 may be implied |
| 18 | A1 | SC1 24 |
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(90 \times 5\) | M1M0 |
| (1 second) Answer [16, 20] is awarded at least M2 | |
| 18 from incorrect working cannot score A1 | |
| 18 followed by further work eg \(18 \div 5 = 3.6\) | M1M0 |
7 Draw the graph of \(\quad y = 1 - \dfrac{1}{2}x \quad\) for values of \(x\) from \(-2\) to 4 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two of \((-2, 2) \quad \left(-1, 1\dfrac{1}{2}\right) \quad (0, 1) \quad \left(1, \dfrac{1}{2}\right)\) \((2, 0) \quad \left(3, -\dfrac{1}{2}\right) \quad (4, -1)\) or other correct points | M1 | may be seen in a table with values assigned to \(x\) and \(y\) implied by points plotted |
| At least two of their points plotted | M1 | \(\pm\dfrac{1}{2}\) square implied by a line passing through two of their points |
| Single straight line from \((-2, 2)\) to \((4, -1)\) | A1 | \(\pm\dfrac{1}{2}\) square ignore line beyond \((-2, 2)\) and \((4, -1)\) |
| Ignore extra points listed or plotted if required line is drawn | |
| M marks can be awarded even if incorrect line drawn | |
| Correct line from \((-2, 2)\) to \((4, -1)\) within tolerance with no points plotted | M1M1A1 |
5
Complete the coordinates of \(C\) and \(D\). [2 marks]
\(C\,(\ 0\ ,\ \_\_\_\_\_\ ) \qquad D\,(\ \_\_\_\_\_\ ,\ 0\ )\)
Complete the following statements. [2 marks]
The value of the \(y\)-intercept is ______
The \(x\)-coordinate of the turning point, \(T\), is ______
| Answer | Mark | Comments |
|---|---|---|
| \(C\ (0, 6)\) | B1 | if answer space is blank, accept (0, 6) written at \(C\) on the diagram |
| \(D\ (3, 0)\) | B1 | if answer space is blank, accept (3, 0) written at \(D\) on the diagram |
For each part mark the answer space unless blank
Allow \(x\) and \(y\) written above the coordinates but do not allow eg \((0x, 6y)\)
| Answer | Mark | Comments |
|---|---|---|
| 5 | B1 | |
| 3 | B1 |
Mark the answer lines only
Do not allow eg (0, 5)
5
Which two lines are parallel? [1 mark]
line \(\ldots\ldots\ldots\) and line \(\ldots\ldots\ldots\)
There are four points on this grid that each have
both coordinates that are whole numbers
and
\(x\)-coordinate \(+\) \(y\)-coordinate \(= 3\)
Plot the four points on the grid. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| line Q and line S | B1 | either order, may be indicated on the diagram |
| Answer | Mark | Comments |
|---|---|---|
| (0, 3), (1, 2), (2, 1) and (3, 0) plotted with no other points plotted on the grid | B2 | B1 at least two of (0, 3), (1, 2), (2, 1) and (3, 0) plotted with up to two other points plotted on the grid or at least four points plotted that would lie on the line \(x + y = 3\) where each \(x\) and \(y\) are not all integers, with no other points plotted on the grid or all four correct coordinates given but not plotted, with no additional coordinates |
| Mark intention | |
| Line joining the four correct points with only the four correct points plotted | B2 |
| Line connecting the four correct points but without points plotted | B1 |
26 Here is the graph of \(\quad y = 0.5x^2 - 6x + 12\)
| Answer | Mark | Comments |
|---|---|---|
| \((x =)\ [2.25, 2.75]\) and \((x =)\ [9.25, 9.75]\) | B2 | B1 \((x =)\ [2.25, 2.75]\) or \((x =)\ [9.25, 9.75]\) or one or both values identified but not given in correct notation eg (2.5, 0) and/or (9.5, 0) or \(2.5 \lt x \lt 9.5\) or 2.5 and/or 9.5 written on the graph or in working |
| \(x =\) can be \(x \approx\) | |
| [2.25, 2.75] and/or [9.25, 9.75] with one extra value | B1 |
| [2.25, 2.75] and/or [9.25, 9.75] with more than one extra value | B0 |
| Answer from use of formula or completing the square | B0 |
23 Here is the velocity-time graph of a cyclist for 40 seconds.
State the units of your answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{2} \times 10 \times 3.2\) or 16 or \(\dfrac{1}{2} \times (3.2 + 5.8) \times 10\) or 45 or \(\dfrac{1}{2} \times (5.8 + 7.4) \times 10\) or 66 or \(\dfrac{1}{2} \times (7.4 + 6) \times 10\) or 67 | M1 | oe eg \(3.2 \times 10 + \dfrac{1}{2} \times (5.8 - 3.2) \times 10\) |
| At least three of \(\dfrac{1}{2} \times 10 \times 3.2\) or 16 and \(\dfrac{1}{2} \times (3.2 + 5.8) \times 10\) or 45 and \(\dfrac{1}{2} \times (5.8 + 7.4) \times 10\) or 66 and \(\dfrac{1}{2} \times (7.4 + 6) \times 10\) or 67 | M1dep | oe M2 \(\dfrac{1}{2} \times 10\,\{0 + 6 + 2(3.2 + 5.8 + 7.4)\}\) oe |
| 194 | A1 |
| The first M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Values may be seen on the diagram | |
| Answer 194 and extra strips used | M1M1A1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{6}{40}\) or \(\dfrac{3}{20}\) or 0.15 | B1 | |
| m/s\(^2\) or m/s/s or ms\(^{-2}\) | B1 | oe eg metres per second per second |
Ignore incorrect simplification or conversion after correct answer seen
21 Water flows from a tap at a constant rate.
A container is filled with water from the tap in 40 seconds.
The graph shows the height, \(h\) centimetres, of the water after time, \(t\) seconds.
Circle the letter of the correct shape. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| D | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Draws tangent at \(t = 10\) | M1 | |
| [0.3, 0.4] | A1 |
| For drawing of tangent mark intention | |
| No tangent drawn | M0A0 |
20 Salma is raising money for charity.
Her mum says,
“Whatever amount you raise, I will add double that amount.”
Show this information by drawing a graph on each grid below. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| First graph is a straight line from (0, 0) to (100, 200) and second graph is a straight line from (0, 0) to (100, 300) | B2 | B1 first graph is a straight line from (0, 0) to (100, 200) or second graph is a straight line from (0, 0) to (100, 300) or both graphs correct, but one or both does not reach to 0 or 100 on the horizontal axis or at least 3 correct points plotted on both graphs or B1ft first graph is an incorrect horizontal or increasing straight line to 100 on the horizontal axis, and second graph is a correct ft graph to 100 on the horizontal axis (must be joined) |
Ignore graphs to the right of 100 on the horizontal axes
B1ft can only be awarded if the graph fits onto the grid up to (100, 500)
17 Here is a sketch of the quadratic graph \(\quad y = \mathrm{f}(x)\)
The graph crosses the \(x\)-axis at \(\;x = a\;\) and \(\;x = b\)
Write an expression for the \(x\)-coordinate of the turning point. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{a + b}{2}\) | B1 | oe eg \(a + \left(\dfrac{b - a}{2}\right)\) |
14 On the grid, draw the graph of \(\quad y = 2x - 6 \quad\) for values of \(x\) from \(-2\) to 5 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two of \((-2, -10)\), \((-1, -8)\), \((0, -6)\), \((1, -4)\), \((2, -2)\), \((3, 0)\), \((4, 2)\), \((5, 4)\) | M1 | two correct pairs of coordinates may be in a table implied by points plotted \(\pm\) 2 mm |
| At least two correct points plotted or At least two of their points plotted correctly | M1 | implied by correct line of any length \(\pm\) 2 mm |
| Straight line from \((-2, -10)\) to \((5, 4)\) | A1 | ignore line outside the domain \([-2, 5]\) |
Ignore additional points listed or plotted
9 Here is the graph of \(\quad y = 0.5x^2 - 6x + 12\)
| Answer | Mark | Comments |
|---|---|---|
| \((x =)\) [2.25, 2.75] and \((x =)\) [9.25, 9.75] | B2 | B1 \((x =)\) [2.25, 2.75] or \((x =)\) [9.25, 9.75] or one or both values identified but not given in correct notation eg \((2.5, 0)\) and/or \((9.5, 0)\) or \(2.5 \lt x \lt 9.5\) or 2.5 and/or 9.5 written on the graph or in working |
| \(x =\) can be \(x \approx\) | |
| [2.25, 2.75] and/or [9.25, 9.75] with one extra value | B1 |
| [2.25, 2.75] and/or [9.25, 9.75] with more than one extra value | B0 |
| Answer from use of formula or completing the square | B0 |
8 The graph below is used to convert between pounds (£) and Norwegian krone (kr).
| Answer | Mark | Comments |
|---|---|---|
| 60 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Correct reading(s) taken from one or more number(s) of pounds and correct calculation to scale up | M1 | eg \(1200 \times 5\) or \(1200 \times 4 + 600 \times 2\) |
| 6000 | A1 | |
| Alternative method 2 | ||
| \(500 \times \dfrac{720}{\text{their } 60}\) or \(500 \times 12\) | M1 | oe |
| 6000 | A1ft | correct or ft their 60 |
| A correct ft answer implies both marks eg1 answer 62 in (a) and answer 5806(.45) or 5806.5 in (b) eg2 answer 72 in (a) and answer 5000 in (b) | M1A1ft M1A1ft | ||||||||
| Further work eg \(1200 \times 5 + 100 \times 5\) | M0A0 | ||||||||
For information:
| M1 |
4 At what point does the graph \(\quad y = x^3 - 1 \quad\) cross the \(y\) axis? [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \((0, -1)\) | B1 |
23 Erika tries to sketch the graph \(\quad y = \dfrac{1}{x} \quad\) with \(\;x \neq 0\)
Make two different criticisms of her sketch. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two from: Reference to graph passing through point where \(x = 0\) Reference to graph being incorrect for negative \(x\) values Reference to the graph stopping before the end of the axes/axis | B2 | B1 any one correct reference eg the graph touches the \(y\)-axis eg the graph to the left of the \(y\)-axis should be below the \(x\)-axis eg the graph should go to the ends of the axes |
| Ignore non-contradictory, irrelevant responses alongside a correct response | |
| Draws correct graph | B2 |
| Draws graph with one section correct for positive values of \(x\) or negative values of \(x\) | B1 for that section |
| ‘It isn’t the graph of \(y = \dfrac{1}{x}\)’ scores B0, but B1 may still be scored for the other criticism | |
| ‘There are no numbers on the axes’ scores B0, but B1 may still be scored for the other criticism | |
| Mark for graph touching \(y\)-axis | |
| You cannot have \(x = 0\) | B1 |
| The line in the top right should be moved to the right | B1 |
| It says \(x\) doesn’t \(= 0\) but it (the sketch) does | B1 |
| One line is touching the \(y\)-axis | B1 |
| The lines should be symmetrical | B0 |
| You cannot have \(y = 0\) | B0 |
| One line is touching the \(y\)-axis but the other isn’t | B0 |
| Mark for negative values being in the wrong quadrant | |
| There shouldn’t be anything in the top-left section | B1 |
| There should be something in the bottom-left section | B1 |
| It is the graph of \(y = \dfrac{1}{x^2}\) | B1 |
| It should have rotational symmetry | B1 |
| It should be symmetrical about \(y = x\) | B1 |
| It should be symmetrical about \(y = -x\) | B1 |
| It should be symmetrical | B0 |
| One should be negative | B0 |
| The bit on the left is wrong | B0 |
| The negative values are plotted incorrectly | B0 |
| Reference to the graph stopping before the end of the axes | |
| It stops before the end of the axes | B1 |
| The lines don’t go far enough | B1 |
| The lines need to be higher up | B0 |
20
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) |
|---|---|---|---|---|---|
| \(y\) | \(3\) | \(-1\) | \(0\) |
| Answer | Mark | Comments |
|---|---|---|
| 0 and 3 in the correct positions | B2 | B1 0 or 3 in the correct position |
| B2 |
| Answer | Mark | Comments |
|---|---|---|
| Plots at least three points correctly | M1 | correct or ft their table in (a) \(\pm\dfrac{1}{2}\) small square points may be implied by graph passing through them |
| Correct graph drawn through the five correct points | A1 | \(\pm\dfrac{1}{2}\) small square smooth quadratic curve |
| Correct graph drawn without plotting the correct points | M1A1 |
| Ignore any extra points plotted | |
| Ignore any part of graph drawn for \(x \lt -3\) or \(x \gt 1\) | |
| Ruled straight lines | A0 |
9 Erika tries to sketch the graph \(\quad y = \dfrac{1}{x} \quad\) with \(\ x \neq 0\)
Make two different criticisms of her sketch. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two from: Reference to graph passing through point where \(x = 0\) Reference to graph being incorrect for negative \(x\) values Reference to the graph stopping before the end of the axes/axis | B2 | B1 any one correct reference eg the graph touches the \(y\)-axis eg the graph to the left of the \(y\)-axis should be below the \(x\)-axis eg the graph should go to the ends of the axes |
| Ignore non-contradictory, irrelevant responses alongside a correct response | |
| Draws correct graph | B2 |
| Draws graph with one section correct for positive values of \(x\) or negative values of \(x\) | B1 for that section |
| ‘It isn’t the graph of \(y = \dfrac{1}{x}\)’ scores B0, but B1 may still be scored for the other criticism | |
| ‘There are no numbers on the axes’ scores B0, but B1 may still be scored for the other criticism | |
| Mark for graph touching \(y\)-axis | |
| You cannot have \(x = 0\) | B1 |
| The line in the top right should be moved to the right | B1 |
| It says \(x\) doesn’t \(= 0\) but it (the sketch) does | B1 |
| One line is touching the \(y\)-axis | B1 |
| The lines should be symmetrical | B0 |
| You cannot have \(y = 0\) | B0 |
| One line is touching the \(y\)-axis but the other isn’t | B0 |
| Mark for negative values being in the wrong quadrant | |
| There shouldn’t be anything in the top-left section | B1 |
| There should be something in the bottom-left section | B1 |
| It is the graph of \(y = \dfrac{1}{x^2}\) | B1 |
| It should have rotational symmetry | B1 |
| It should be symmetrical about \(y = x\) | B1 |
| It should be symmetrical about \(y = -x\) | B1 |
| It should be symmetrical | B0 |
| One should be negative | B0 |
| The bit on the left is wrong | B0 |
| The negative values are plotted incorrectly | B0 |
| Reference to the graph stopping before the end of the axes | |
| It stops before the end of the axes | B1 |
| The lines don’t go far enough | B1 |
| The lines need to be higher up | B0 |
7
| \(x\) | –3 | –2 | –1 | 0 | 1 |
|---|---|---|---|---|---|
| \(y\) | 3 | –1 | 0 |
| Answer | Mark | Comments |
|---|---|---|
| 0 and 3 in the correct positions | B2 | B1 0 or 3 in the correct position |
| B2 |
| Answer | Mark | Comments |
|---|---|---|
| Plots at least three points correctly | M1 | correct or ft their table in (a) \(\pm\dfrac{1}{2}\) small square points may be implied by graph passing through them |
| Correct graph drawn through the five correct points | A1 | \(\pm\dfrac{1}{2}\) small square smooth quadratic curve |
| Correct graph drawn without plotting the correct points | M1A1 |
| Ignore any extra points plotted | |
| Ignore any part of graph drawn for \(x \lt -3\) or \(x \gt 1\) | |
| Ruled straight lines | A0 |
6 Scarlett leaves home at 10.00 to cycle to the supermarket.
Here is part of a distance-time graph of her trip to the supermarket.
How far is the supermarket from her home? [1 mark]
How long does she stay at the supermarket? [1 mark]
It takes her 3 minutes longer than her journey to the supermarket.
Complete the distance-time graph. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| 7 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| 15 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(20 + 3\) or 23 or 10.58 | M1 | may be implied by a journey (lines or curves) ending at 10.58 on the graph |
| Straight line from (10.35, 7) to (10.58, 0) | A1 | \(\pm\dfrac{1}{2}\) small square ignore any other working lines on the graph |
| Fully correct graph | M1A1 |
| Accept unruled line if intention clear |
4 Points \(P\) and \(Q\) are shown on the grid.
Work out possible coordinates for \(R\). [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| (4, 3) | B1 | accept \((\overset{x}{4},\ \overset{y}{3})\) |
| Mark the answer line. If this is blank, the answer may be seen on the diagram but must be the coordinates for \(P\) | |
| Do not allow \(x\) and \(y\) within the coordinates eg \((4x, 3y)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \((x, -3)\) where \(x \neq 4\) | B1 | accept eg \((\overset{x}{5},\ \overset{y}{-3})\) |
| Do not allow \(x\) and \(y\) within the coordinates eg \((5x, -3y)\) | B0 |
26 Helena ran an 800-metre race in 140 seconds.
The speed-time graph represents the first 100 seconds of her run.
Helena ran the last 40 seconds with constant deceleration.
Work out her speed as she finished the race. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(6 \times 10 \div 2\) or 30 or \(6 \times 90\) or 540 or 570 | M1 | oe eg \(\dfrac{1}{2} \times \dfrac{6}{10} \times 10^2\) or \(\dfrac{1}{2} \times (100 + 90) \times 6\) may be on diagram |
| \(800 - 6 \times 10 \div 2 - 6 \times 90\) or \(800 -\) their \(30 -\) their 540 or \(800 -\) their 570 or 230 | M1dep | oe full method for remaining distance may be on diagram may be embedded eg \(230 \div 40\) |
| \(\dfrac{1}{2} \times (v + 6) \times 40 =\) their 230 \(2 \times\) their \(230 \div 40 - 6\) | M1dep | oe eg \(20v + 120 =\) their 230 any letter |
| 5.5 | A1 | oe value |
Up to M2 may be awarded for correct work with no answer, or incorrect answer, even if this is seen amongst multiple attempts
25 A tank contains 40 litres of water.
The leak is stopped after 20 minutes.
Show that, when the leak is stopped, the tank contains 16 litres of water. [1 mark]
The graph shows the amount of water in the tank after the leak is stopped.
Complete this report by writing a number in each answer space. [3 marks]
| Report |
| \(\ldots\ldots\) minutes after the leak is stopped, the tap starts to refill the tank. |
| The rate at which the tank refills is \(\ldots\ldots\) litres per minute. |
| Answer | Mark | Comments |
|---|---|---|
| \(1.2 \times 20 = 24\) and \(40 - 24 = 16\) | B1 | oe eg \(1.2 \times 20 = 24\) and \(24 + 16 = 40\) or \(40 - 16 = 24\) and \(24 \div 20 = 1.2\) or \(24 + 16 = 40\) and \(24 \div 1.2 = 20\) may be seen as one calculation eg \(40 - 1.2 \times 20 = 16\) or \(16 + 1.2 \times 20 = 40\) or \(40 - 16 = 1.2 \times 20\) |
| \(40 - 24 = 16\) and \(40 - 16 = 24\) and \(24 + 16 = 40\) are equivalent | |
| \(1.2 \times 20 = 24\) and \(24 \div 1.2 = 20\) and \(24 \div 20 = 1.2\) are equivalent | |
| \(40 - 24 = 16\) or \(16 + 24 = 40\) or \(40 - 16 = 24\) | B0 |
| (20 minutes =) 24 litres leak out \(40 - 24 = 16\) | B0 |
| \(1.2 \times 20 = 24\) 16 litres left | B0 |
| Allow unambiguous working in ml and/or seconds | |
| For eg \(40 - 24 = 16\) condone \(24 - 40 = 16\) or \(24 - 40 = -16\) | |
| Condone incorrect use of equals sign eg \(1.2 \times 20 = 24 + 16 = 40\) or \(1.2 \times 20 = 24 - 40 = 16\) | B1 |
| Correct response with irrelevant work | B1 |
| 16 from two different ways with one way incorrect is choice eg \(1.2 \times 20 = 24\) and \(40 - 24 = 16\) and \(20 \div 1.2 = 16\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 3 | B1 | |
| Correct method for gradient eg \(\dfrac{40 - 16}{15 - \text{their } 3}\) or \(\dfrac{24}{12}\) | M1 | oe eg \(\dfrac{30 - 25}{10 - 7.5}\) or \(\dfrac{10}{5}\) or \(40 - 38\) |
| 2 | A1ft | correct or ft their 3 |
| Note that their 3 can be used to work out the rate but does not have to be | |
| Values seen on graph must be used correctly eg 24 and 12 seen on the graph is M0 unless subsequently used correctly in attempt to work out the gradient | |
| A1ft answers must be to 1 dp or better eg 3.5 \(\dfrac{40 - 16}{15 - 3.5}\) 2.1 (accept 2.08…) | B0 M1 A1ft |
| After B0 the method may be implied (use \(\dfrac{40 - 16}{15 - \text{their } 3}\) to check) eg 6 2.7 (accept 2.66…) | B0 M1A1ft |
| If the report is blank, 3 and 2 must be unambiguously identified in working to be acceptable | |
| Allow 2 to be written as \(\dfrac{2}{1}\) |
19 A circle has centre (0, 0) and passes through (0, 11)
Write down the equation of the circle. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + y^2 = 121\) or \(x^2 + y^2 = 11^2\) | B1 | oe equation |
| \((x - 0)^2 + (y - 0)^2 = 11^2\) | B1 |
| \(x^2 + y^2 = 11^2\) followed by incorrect working | B1 |
16 Here is a sketch of a graph.
Circle the possible equation of the graph. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(y = x^3 + 1\) | B1 |
15 The sketch shows
the line \(\;y = x\)
line A, which is vertical
line B, which is horizontal.
The point (3, 5) is on both line A and line B.
Write down the coordinates of \(P\) and \(Q\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(P\,(3, 0) \quad Q\,(5, 5)\) | B2 | B1 \(P\,(3, 0)\) or \(Q\,(5, 5)\) or both \(x\)-coordinates correct or both \(y\)-coordinates correct SC1 \(P\,(5, 5) \quad Q\,(3, 0)\) |
Accept eg \(P\,(\overset{x}{3}, \overset{y}{0})\)
Do not accept eg \(P\,(3x, 0y)\)
15 A mobile phone takes 2 hours to charge from empty.
When the phone is being charged, the current flow into the phone
This is shown on Graph A below.
Graph A
Graph B
Megan’s phone is empty of charge.
She starts to charge her phone at 10.00 am
estimate the time when the current flow starts to decrease. [2 marks]
estimate the percentage charge in the phone when the current flow is 40% [1 mark]
estimate the rate of increase in the percentage charge when the phone has 90% charge. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Vertical line drawn from correct point on graph to horizontal axis or [25, 27] | M1 | implied by mark on horizontal axis |
| [10.25, 10.27] (am) | A1 | SC1 10.23 (am) |
| Answer | Mark | Comments |
|---|---|---|
| 80 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Correct tangent drawn where charge is 90% | M1 | |
| Correct gradient for their tangent | A1 |
| No tangent drawn | M0A0 |
14 Craig wants to draw a graph, for values of \(x\) from \(-3\) to 3,
where the \(x\)-coordinate and \(y\)-coordinate are always in the ratio \(\quad 2 : 1\)
Here is his graph.
Make two criticisms of Craig’s graph. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Explanation that the ratio and graph do not match | B1 | eg This is the graph of \(y = 2x\), not \(y = \dfrac{1}{2}x\) This is the graph of \(x : y = 1 : 2\) It should go through (3, 1.5) |
| Explanation that the domain of the graph is incorrect | B1 | eg The graph goes from \(x = 0\), not \(x = -3\) |
| 3 : 6 is 1 : 2 | B1 |
| (3, 6) doesn’t work | B1 |
| The gradient is 2, not \(\dfrac{1}{2}\) | B1 |
| He got \(x\) and \(y\) mixed up | B1 |
| His graph is not going up in the ratio 2 : 1 | B0 |
| The gradient is 2 | B0 |
| He didn’t follow the ratio | B0 |
| The graph doesn’t have negative numbers | B1 |
| There are no minuses | B1 |
| It doesn’t go from \(-3\) to 3 | B1 |
| The axes should be the same length | B0 |
12 Here is the distance-time graph for a car between 1 pm and 3 pm
State the units of your answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| 30 or \(\dfrac{1}{2}\) and 15 or \(\dfrac{1}{4}\) or 45 or \(\dfrac{3}{4}\) | M1 | oe allow no units or incorrect units may be on graph |
| 45 minutes or \(\dfrac{3}{4}\) hour | A1 | oe |
| Allow abbreviated units eg 45 min(s) eg condone 45 m eg \(\dfrac{3}{4}\) h | M1A1 M1A1 M1A1 |
| 45 minutes in working with answer 45 | M1A1 |
| \(\dfrac{3}{4}\) hour in working with answer \(\dfrac{3}{4}\) | M1A1 |
| \(0.3 + 0.15\) is M0 unless recovered to 45 |
| Answer | Mark | Comments |
|---|---|---|
| 29 or \(4 + 25\) | M1 | oe may be embedded 29 may be on graph eg on \(y\)-axis |
| 58 | A1 | SC1 54 |
| \(29 \times 2\) with no or incorrect evaluation | M1A0 |
| Allow the first mark embedded in a calculation eg \(29 + 4\) or \(29 + 5 + 25\) or \(29 + 29 + 25 + 25\) or \(29 - 25\) | M1A0 |
11 A tank contains 40 litres of water.
The leak is stopped after 20 minutes.
Show that, when the leak is stopped, the tank contains 16 litres of water. [1 mark]
The graph shows the amount of water in the tank after the leak is stopped.
Complete this report by writing a number in each answer space. [3 marks]
Report __________ minutes after the leak is stopped, the tap starts to refill the tank. The rate at which the tank refills is __________ litres per minute. |
| Answer | Mark | Comments |
|---|---|---|
| \(1.2 \times 20 = 24\) and \(40 - 24 = 16\) | B1 | oe eg \(1.2 \times 20 = 24\) and \(24 + 16 = 40\) or \(40 - 16 = 24\) and \(24 \div 20 = 1.2\) or \(24 + 16 = 40\) and \(24 \div 1.2 = 20\) may be seen as one calculation eg \(40 - 1.2 \times 20 = 16\) or \(16 + 1.2 \times 20 = 40\) or \(40 - 16 = 1.2 \times 20\) |
| \(40 - 24 = 16\) and \(40 - 16 = 24\) and \(24 + 16 = 40\) are equivalent | |
| \(1.2 \times 20 = 24\) and \(24 \div 1.2 = 20\) and \(24 \div 20 = 1.2\) are equivalent | |
| \(40 - 24 = 16\) or \(16 + 24 = 40\) or \(40 - 16 = 24\) | B0 |
| (20 minutes =) 24 litres leak out \(40 - 24 = 16\) | B0 |
| \(1.2 \times 20 = 24\) 16 litres left | B0 |
| Allow unambiguous working in ml and/or seconds | |
| For eg \(40 - 24 = 16\) condone \(24 - 40 = 16\) or \(24 - 40 = -16\) | |
| Condone incorrect use of equals sign eg \(1.2 \times 20 = 24 + 16 = 40\) or \(1.2 \times 20 = 24 - 40 = 16\) | B1 |
| Correct response with irrelevant work | B1 |
| 16 from two different ways with one way incorrect is choice eg \(1.2 \times 20 = 24\) and \(40 - 24 = 16\) and \(20 \div 1.2 = 16\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 3 | B1 | |
| Correct method for gradient eg \(\dfrac{40 - 16}{15 - \text{their } 3}\) or \(\dfrac{24}{12}\) | M1 | oe eg \(\dfrac{30 - 25}{10 - 7.5}\) or \(\dfrac{10}{5}\) or \(40 - 38\) |
| 2 | A1ft | correct or ft their 3 |
| Note that their 3 can be used to work out the rate but does not have to be | |
| Values seen on graph must be used correctly eg 24 and 12 seen on the graph is M0 unless subsequently used correctly in attempt to work out the gradient | |
| A1ft answers must be to 1 dp or better eg 3.5 \(\dfrac{40 - 16}{15 - 3.5}\) 2.1 (accept 2.08…) | B0 M1 A1ft |
| After B0 the method may be implied (use \(\dfrac{40 - 16}{15 - \text{their } 3}\) to check) eg 6 2.7 (accept 2.66…) | B0 M1A1ft |
| If the report is blank, 3 and 2 must be unambiguously identified in working to be acceptable | |
| Allow 2 to be written as \(\dfrac{2}{1}\) |
9
| \(x\) | 0 | 2 | 4 | 6 | 8 | |
|---|---|---|---|---|---|---|
| \(y\) | 3 | 7 | 11 | 19 | 23 |
The \(x\)-values in the table make a linear sequence.
The \(y\)-values in the table make a different linear sequence.
| Answer | Mark | Comments |
|---|---|---|
| \((x =)\) 10 and \((y =)\) 15 | B2 | B1 \((x =)\) 10 or \((y =)\) 15 |
| B2 |
| Answer | Mark | Comments |
|---|---|---|
| Straight line from (0, 3) to (4, 11) | B2 | B1 at least two of (0, 3), (2, 7) and (4, 11) plotted or straight line from (0, 3) to (2, 7) or straight line from (2, 7) to (4, 11) \(\pm\dfrac{1}{2}\) square |
B2 or B1 may be awarded for a straight line without points plotted
Mark intention
Ignore line drawn after (4, 11)
Two points plotted with the same \(x\)-coordinate is choice unless the line is drawn through one of the points
| Answer | Mark | Comments |
|---|---|---|
| 9 | B1ft | correct or ft their line in (b) \(\pm\dfrac{1}{2}\) square |
25 Here is the speed-time graph for a 50-second journey.
| Answer | Mark | Comments |
|---|---|---|
| 0 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{2} \times (50 + 30) \times 20\) | M1 | oe complete method to work out the area of the trapezium eg \(\dfrac{1}{2} \times 10 \times 20 + 20 \times 30 + \dfrac{1}{2} \times 10 \times 20\) or \(50 \times 20 - \dfrac{1}{2} \times 10 \times 20 - \dfrac{1}{2} \times 10 \times 20\) or \(40 \times 20\) |
| 800 | A1 |
24 Here is a sketch of the graphs of \(\quad y = k^x \quad\) and \(\quad y = x^2 + c\)
\(k\) and \(c\) are positive constants.
Work out the value of \(r\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\sqrt[4]{81}\) or \(81^{\frac{1}{4}}\) or \(k = 3\) | M1 | may be seen on diagram and is implied by \(p = 9\) |
| (their value for \(k)^2 = 2^2 + c\) or \(9 = 4 + c\) or \(c = 5\) | M1 | does not need to be evaluated |
| \(r^2 +\) their \(5 = 43.44\) or \(\sqrt{43.44 - \text{their } 5}\) or \(\sqrt{38.44}\) | M1dep | oe equation dep on previous mark |
| 6.2 | A1 |
Coordinate (2, 9) implies \(p = 9\)
23 Here is a sketch of the curve \(\quad y = x^2 - 4x - 5\)
| Answer | Mark | Comments |
|---|---|---|
| \(-1\) and 5 | B1 | either order |
| Ignore \(x =\) written before answers | |
| \((-1, 0)\) or \((5, 0)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \((2, -9)\) | B2 | B1 \(x = 2\) or \((2, \ldots)\) or \(y = -9\) or \((\ldots, -9)\) or \((x - 2)^2 - 9\) B1ft correct \(y\)-coordinate for their \(x\)-coordinate with \(x \ne -1\), 0 or 5 SC1 \((-9, 2)\) |
| If answer line is blank, check diagram for indication of \(x\) or \(y\) values | |
| \((3, -9)\) | B1 |
| \((3, -8)\) | B1ft |
| \((1, -8)\) | B1ft |
| \((2.5, -8.75)\) | B1ft |
| \((0, -5)\) | B0ft |
21 Circle the letter of the possible sketch graph of \(y = x^3 - 4\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| A | B1 |
21 Which of these is the equation of a circle?
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + y^2 = 6\) | B1 |
16 Here is the graph of \(y = 7 - 3x\)
Draw the graph of \(y = 2x + 1\) on the grid
and then
work out an approximate solution to \(7 - 3x = 2x + 1\) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| At least two points from (0, 1) (1, 3) (2, 5) and (3, 7) | M1 | may be seen in a table of values or embedded in calculations may be implied by correct line \(\pm\dfrac{1}{2}\) square tolerance |
| Correct straight line between (1, 3) and (2, 5) | A1 | \(\pm\dfrac{1}{2}\) square tolerance |
| [1.15, 1.25] from using the graph or 1.2 | B1ft | oe ft \(x\)-coordinate of any line drawn that intersects the given line \(\pm\dfrac{1}{2}\) square tolerance |
| Ignore further work after B1 scored | |
| 1.2 with M0 scored 1.2 with two correct points seen but no or incorrect line | M0A0B1 M1A0B1 |
| For the A1, ignore incorrect lines unless used to read off for intersection and then only allow for the B1ft | |
| Answer given as coordinates eg (1.2, 3.4) | B0 |
16 Circle the letter of the possible sketch graph of \(\quad y = x^3 - 4\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| A | B1 | accept letter or graph circled |
10 Lines \(AB\) and \(BC\) are shown.
Work out the coordinates of \(D\). [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| (8, 1) | B1 | accept \((\overset{x}{8},\ \overset{y}{1})\) |
| \((8x, 1y)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| (7, 6) | B1 | accept \((\overset{x}{7},\ \overset{y}{6})\) |
| \((7x, 6y)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| (2, 1) | B1 | accept \((\overset{x}{2},\ \overset{y}{1})\) |
| \((2x, 1y)\) | B0 |
| If two or more parts have \((x, y)\) as \((y, x)\) then give the first 0 and condone the other(s) eg1 (a) (1, 8) (b) (6, 7) (c) (1, 2) eg2 (a) (1, 8) (b) (7, 6) (c) (1, 2) eg3 (a) (1, 8) (b) (6, 10) (c) (1, 2) eg4 (a) (8, 1) (b) (6, 7) (c) (1, 2) | B0 B1 B1 B0 B1 B1 B0 B0 B1 B1 B0 B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(y = 6\) or \(6 = y\) | B1 | accept \(y = 0x + 6\) |
| \(y = x + 6\) | B0 |
| \(x = 6\) | B0 |
| 6 | B0 |
6 Here is a sketch of the curve \(\quad y = x^2 - 4x - 5\)
| Answer | Mark | Comments |
|---|---|---|
| \(-1\) and 5 | B1 | either order |
| Ignore \(x =\) written before answers | |
| (\(-1\), 0) or (5, 0) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| (2, \(-9\)) | B2 | B1 \(x = 2\) or (2, …) or \(y = -9\) or (…, \(-9\)) or \((x - 2)^2 - 9\) B1ft correct \(y\)-coordinate for their \(x\)-coordinate with \(x \ne -1\), 0 or 5 SC1 (\(-9\), 2) |
| If answer line is blank, check diagram for indication of \(x\) or \(y\) values | |
| (3, \(-9\)) | B1 |
| (3, \(-8\)) | B1ft |
| (1, \(-8\)) | B1ft |
| (2.5, \(-8.75\)) | B1ft |
| (0, \(-5\)) | B0ft |
29 Here is a sketch of a graph.
Circle the equation of the graph.
\(k\) is a constant. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(y = \dfrac{k}{x}\) | B1 |
23 The equation of a curve is \(\qquad y = 16^x\)
Work out the \(x\)-coordinate. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| (2, 256) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(-1\) | B1 | accept \((-1,\) \(\dfrac{1}{16}\)\()\) |
| \(16^{-1} =\) \(\dfrac{1}{16}\) | B0 |
19 The graph shows the height above ground of a toy rocket for 10 seconds.
Circle your answer. [1 mark]
State the units of your answer. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| 6 seconds | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Correct tangent drawn at 6 seconds | B1 | |
| Correct gradient for their tangent | B1ft | ft their tangent, which must be an increasing straight line |
| m/s | B1 | oe eg metres per second or mps |
If no tangent is drawn the maximum mark possible is B0B0B1
Allow the units to be given in working lines if no units on the answer line
19 The equation of a circle is \(\qquad x^2 + y^2 = 11\)
Work out the length of the diameter.
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(2\sqrt{11}\) | B1 |
18 The cost of making a phone call is
a fixed charge
and
a charge per minute.
The costs of phone calls up to 5 minutes are represented by the graph.
| Answer | Mark | Comments |
|---|---|---|
| 20 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(28 - 20\) or \(\dfrac{36 - 20}{2}\) or \(\dfrac{44 - 20}{3}\) or \(\dfrac{52 - 20}{4}\) or \(\dfrac{60 - 20}{5}\) or correct calculation using any two points, eg \(\dfrac{60 - 44}{2}\) or \(2 \times 4\) | M1 | |
| 8 | A1 |
| \((60 \div 5 =)\ 12\) | M0A0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| their \(20 + 7 \times\) their 8 | M1 | oe |
| 76 | A1ft | correct answer or ft their values in (a) and (b) |
| Alternative method 2 | ||
| \(60 + 2 \times\) their 8 | M1 | oe |
| 76 | A1ft | correct answer or ft their values in (b) |
For Alt 2, they may read off any cost for \(n\) minutes (from 1 to 5) and add on \((7 - n) \times\) their (b) for M1. A1 or A1ft may follow from their working
15
| \(x\) | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
| \(y\) | 2 | −1 | −2 | −1 |
| Answer | Mark | Comments | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| B1 |
| Answer | Mark | Comments |
|---|---|---|
| Plots at least four points correctly | M1 | ft their points from part (a) \(\pm\,\dfrac{1}{2}\) small square |
| Correct graph drawn through the seven correct points | A1 | \(\pm\,\dfrac{1}{2}\) small square smooth quadratic curve |
| Correct graph drawn without plotting the correct points | M1A1 |
4 Here is a sketch of a graph.
Circle the equation of the graph.
\(k\) is a constant. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(y = \dfrac{k}{x}\) | B1 |
28 A horse runs in a field.
The speed-time graph represents the first 12 seconds of the run.
After how many seconds had the horse run a distance of 75 metres? [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(0.5 \times 4 \times 10\) or 20 | M1 | oe may be seen on graph |
| \(\dfrac{75 - 0.5 \times 4 \times 10}{10}\) or \(\dfrac{55}{10}\) or 5.5 | M1dep | oe may be embedded eg \(5.5 \times 10 = 55\) |
| 9.5 | A1 | oe |
| Alternative method 2 | ||
| Correct method or value for distance travelled in the first \(t\) seconds where \(t > 4\) | M1 | eg distance for 12s = 100 or distance for 9s \(= 0.5 \times (9 + 5) \times 10\) or 70 may be seen on graph |
| \(\dfrac{\text{their distance} - 75}{10}\) or \(\dfrac{75 - \text{their distance}}{10}\) | M1dep | eg \(\dfrac{100 - 75}{10}\) or \(\dfrac{75 - 70}{10}\) |
| 9.5 | A1 | oe |
1st M can be awarded even if not subsequently used
25 The equation of a circle is \(\quad x^2 + y^2 = 9\)
Work out the length of the diameter.
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| 6 | B1 |
24 A and B are points on a curve.
A is (2, 7) B is (12, 0)
Which statement is correct?
Tick one box. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Draws a tangent at (2, 7) | M1 | Must see a tangent on the graph |
| Their gradient at (2, 7) | A1ft | ft their tangent ± 0.2 tolerance on their readings |
| Mark intention for drawing of tangent | |
| No tangent drawn | M0A0 |
| Answer | Mark | Comments |
|---|---|---|
| It is negative | B1 |
22 Here is the graph of \(\quad y = x^2 - 7x + 10 \quad\) for values of \(x\) from 0 to 7
| Answer | Mark | Comments |
|---|---|---|
| 2 and 5 with no other roots | B2 | either order B1 at least one correct root with up to one incorrect root SC1 (2, 0) or (5, 0) or (2, 5) or (5, 2) |
| \(x = 2\) and \(x = 5\) | B2 |
| 2, 5 or 5, 2 | B2 |
| (2, 0) and (5, 0) and 2 and 5 | SC1 |
| (2, 0) and (5, 0) and \(-2\) and \(-5\) | B0 |
| 2, 0 and 5, 0 (both pairs imply coordinates) | SC1 |
| 2, 0 or 5, 0 (one pair implies roots) | B1 |
| (0, 2) and (0, 5) | B0 |
| 0, 2 and 0, 5 (both pairs imply coordinates) | B0 |
| 0, 2 or 0, 5 (one pair implies roots) | B1 |
| Both answers embedded \(2^2 - 7 \times 2 + 10 = 0\) and \(5^2 - 7 \times 5 + 10 = 0\) | B1 |
| \((x - 2)(x - 5)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 3.5 | B1 | oe |
| \(x = 3.5\) | B1 |
| \(3.5x\) | B0 |
| Ignore any \(y\)-coordinate even with brackets omitted eg \((3.5, -2.25)\) or \(3.5, -2\) or \(x = 3.5\;\; y = -2.25\) or \(x = 3.5\;\; y = 2\) | B1 |
| \((-2.25, 3.5)\) | B0 |
21
Here is his graph.
Make one criticism of his graph. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| (2.5, 0.4) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Valid criticism | B1 | eg the graph should go through (4, 16) |
| (4, 15) should be (4, 16) | B1 |
| It should be (4, 16) | B1 |
| Graph should end at \((y =)\) 16 | B1 |
| (4, 15) is not on the graph | B1 |
| The point at \(x = 4\) is wrong | B1 |
| The point at 4 is wrong | B0 |
| \(2^4\) is 16 | B1 |
| \(4^2\) is 16 | B0 |
| The last point is wrong | B1 |
| One of the points is wrong | B0 |
| Graph isn’t high enough | B0 |
7 Here is the graph of \(\quad y = x^2 - 7x + 10 \quad\) for values of \(x\) from 0 to 7
| Answer | Mark | Comments |
|---|---|---|
| 2 and 5 with no other roots | B2 | either order B1 at least one correct root with up to one incorrect root SC1 (2, 0) or (5, 0) or (2, 5) or (5, 2) |
| \(x = 2\) and \(x = 5\) | B2 |
| 2, 5 or 5, 2 | B2 |
| (2, 0) and (5, 0) and 2 and 5 | SC1 |
| (2, 0) and (5, 0) and \(-2\) and \(-5\) | B0 |
| 2, 0 and 5, 0 (both pairs imply coordinates) | SC1 |
| 2, 0 or 5, 0 (one pair implies roots) | B1 |
| (0, 2) and (0, 5) | B0 |
| 0, 2 and 0, 5 (both pairs imply coordinates) | B0 |
| 0, 2 or 0, 5 (one pair implies roots) | B1 |
| Both answers embedded \(2^2 - 7 \times 2 + 10 = 0\) and \(5^2 - 7 \times 5 + 10 = 0\) | B1 |
| \((x - 2)(x - 5)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 3.5 | B1 | oe |
| \(x = 3.5\) | B1 |
| \(3.5x\) | B0 |
| Ignore any \(y\)-coordinate even with brackets omitted eg (3.5, \(-2.25\)) or 3.5, \(-2\) or \(x = 3.5\) \(y = -2.25\) or \(x = 3.5\) \(y = 2\) | B1 |
| (\(-2.25\), 3.5) | B0 |
7 Line \(AB\) is shown where \(A\) is the point (1, 0) and \(B\) is the point (5, 8)
The distance \(AP\) is half the distance \(AB\).
Work out the coordinates of \(P\). [1 mark]
parallel to the \(x\)-axis
meets the \(y\)-axis at point \(Q\).
Work out the coordinates of \(Q\). [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| (3, 4) | B1 |
| \((3x, 4y)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| (0, 8) | B1 | SC1 (4, 3) in (a) and (8, 0) in (b) |
| \((0x, 8y)\) | B0 |
29 The graph shows how much Molly is paid for working for up to 40 hours.
She receives
a basic rate of pay for the first 35 hours worked
a higher rate of pay for the next 5 hours worked.
Work out the difference between the higher rate of pay and the basic rate of pay.
Give your answer in £ per hour. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(280 \div 35\) or 8 | M1 | oe eg \(80 \div 10\) |
| \((350 - 280) \div (40 - 35)\) or \(70 \div 5\) or 14 | M1 | oe |
| 6 | A1 | |
| Alternative method 2 | ||
| 320 or \(350 - 320\) or 30 or \(350 - 280\) and \(320 - 280\) or 70 and 40 | M1 | oe |
| \((350 - 320) \div 5\) or \((70 - 40) \div 5\) or \(30 \div 5\) | M1dep | oe |
| 6 | A1 | |
Do not allow a misread from the graph
Alt 2 40 must come from \(320 - 280\) and not 40 hours worked
25 Here is a sketch of a speed-time graph for the first part of a journey.
The total distance for the journey is 130 kilometres.
How far is left to travel? [4 marks]
Answer in km
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Correct method to work out any viable distance, eg \(\dfrac{1}{2} \times \dfrac{5}{60} \times 102\) or 4.25 (first section) or \(102 \times \dfrac{40}{60}\) or 68 (second section) or \(\dfrac{1}{2}(102 + 96) \times \dfrac{15}{60}\) or \(96 \times \dfrac{15}{60}\) and \(\dfrac{1}{2} \times 6 \times \dfrac{15}{60}\) or 24 and 0.75 or 24.75 (third section) or \(\dfrac{1}{2}\left(\dfrac{40}{60} + \dfrac{45}{60}\right) \times 102\) or 72.25 (first and second sections) | M1 | |
| Correct method to work out all parts of distance, eg \(\dfrac{1}{2} \times \dfrac{5}{60} \times 102\) or 4.25 and \(102 \times \dfrac{40}{60}\) or 68 and \(\dfrac{1}{2}(102 + 96) \times \dfrac{15}{60}\) or 24.75 | M1dep | 97 scores M1M1 |
| \(130 -\) their whole distance or \(130 - 97\) | M1dep | eg \(130 -\) their \(4.25 -\) their \(68 -\) their 24.75 dep on M2 |
| 33 | A1 | |
| Alternative method 2 | ||
| Correct method to work out \(60 \times\) any viable distance, eg \(\dfrac{1}{2} \times 5 \times 102\) or 255 (first section) or \(102 \times 40\) or 4080 (second section) or \(\dfrac{1}{2}(102 + 96) \times 15\) or \(96 \times 15\) and \(\dfrac{1}{2} \times 6 \times 15\) or 1440 and 45 or 1485 (third section) or \(\dfrac{1}{2}(40 + 45) \times 102\) or 4335 (first and second sections) | M1 | |
| Correct method to work out \(60 \times\) all parts of distance, eg \(\dfrac{1}{2} \times 5 \times 102\) or 255 and \(102 \times 40\) or 4080 and \(\dfrac{1}{2}(102 + 96) \times 15\) or 1485 | M1dep | 5820 implies M1M1 |
| \(130 -\) their whole distance or \(130 - \dfrac{5820}{60}\) or \(130 - 97\) | M1dep | eg \(130 - \dfrac{\text{their } 255 + \text{their } 4080 + \text{their } 1485}{60}\) dep on M2 |
| 33 | A1 | |
Accept fractions used as decimals correct to 2 dp or better
22 Here is a sketch of a distance-time graph.
Which of these represents the average speed between A and B?
Tick one box. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| The gradient of the chord from A to B | B1 |
18 Jenny leaves home at 06.00
She runs for 3 hours.
Here is a distance-time graph of her run.
She then gets a bus home.
She arrives home at 11.30
Complete the distance-time graph.
Assume the bus travels at a constant speed. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| 20 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Horizontal line from (09.00, 20) to (10.00, 20) | B1 | |
| Straight line with a negative gradient from their (10.00, 20) to (11.30, 0) | B1ft | ft their (10.00, 20) |
Tolerance within one small square
Accept unruled line if intention for straight line is clear
Their (10.00, 20) can be (09.00, 20)
Their 10.00 cannot be earlier than 09.00
17 Draw the graph of \(\quad y = 3x - 1 \quad\) for values of \(x\) from \(-1\) to 3 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any two of \((-1, -4)\), \((0, -1)\), \((1, 2)\), \((2, 5)\) and \((3, 8)\) or other correct points | M1 | may be seen in a table may be implied by points plotted |
| At least two correct points plotted correctly or at least two of their points plotted correctly | M1 | implied by correct line which does not have to extend from \((-1, -4)\) to \((3, 8)\) \(\pm\dfrac{1}{2}\) small square |
| Straight, ruled line from \((-1, -4)\) to \((3, 8)\) | A1 | \(\pm\dfrac{1}{2}\) small square ignore line beyond \((-1, -4)\) and \((3, 8)\) |
Ignore extra points listed or plotted
M marks can be scored even if wrong line drawn
M marks are independent, the second mark can be awarded for correct plotting of two of their points
14 A graph has equation \(\quad y = x^3 + a \quad\) where \(a\) is an integer.
The graph passes through the point (3, 29)
Draw the graph for values of \(x\) from −3 to 3 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(y = x^3 + 2\) or \(a = 2\) | M1 | implied by at least two correct points identified or plotted from (−3, −25), (−2, −6), (−1, 1), (0, 2), (1, 3), (2, 10) |
| At least five correct points identified or plotted for their value of \(a\) | M1 | correct points are (−3, −25), (−2, −6), (−1, 1), (0, 2), (1, 3), (2, 10) may be seen in a table or in working |
| Seven correct points plotted and joined with a smooth curve | A1 | \(\pm\dfrac{1}{2}\) square SC1 fully correct curve for \(y = x^3\) for \(-3 \le x \le 2\) |
12 The graph shows how much Molly is paid for working for up to 40 hours.
She receives
a basic rate of pay for the first 35 hours worked
a higher rate of pay for the next 5 hours worked.
Work out the difference between the higher rate of pay and the basic rate of pay.
Give your answer in £ per hour. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(280 \div 35\) or 8 | M1 | oe eg \(80 \div 10\) |
| \((350 - 280) \div (40 - 35)\) or \(70 \div 5\) or 14 | M1 | oe |
| 6 | A1 | |
| Alternative method 2 | ||
| 320 or \(350 - 320\) or 30 or \(350 - 280\) and \(320 - 280\) or 70 and 40 | M1 | oe |
| \((350 - 320) \div 5\) or \((70 - 40) \div 5\) or \(30 \div 5\) | M1dep | oe |
| 6 | A1 | |
Do not allow a misread from the graph
Alt 2 40 must come from \(320 - 280\) and not 40 hours worked
4 Here is the sketch of a graph.
Circle the equation of the graph. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(y = \dfrac{1}{x}\) | B1 |
28 Leo runs for 12 seconds.
The graph shows his speed.
State the units of your answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{2} \times 5 \times 8\) or 20 or \(\dfrac{1}{2} \times (8 + 9) \times (9 - 5)\) or 34 | M1 | oe eg \(\dfrac{1}{2} \times 4\ (\times 1)\) and \(4 \times 8\) or 2 and 32 |
| \(\dfrac{1}{2} \times 5 \times 8\) or 20 and \(\dfrac{1}{2} \times (8 + 9) \times (9 - 5)\) or 34 | M1dep | \(\dfrac{1}{2} \times 4\ (\times 1)\) and \(4 \times 8\) or 2 and 32 |
| \(\dfrac{1}{2} \times (9 + 4.6) \times 1\) \(+\ \dfrac{1}{2} \times (4.6 + 2) \times 1\) \(+\ \dfrac{1}{2} \times 1 \times 2\) or 6.8 + 3.3 + 1 or 11.1 or \(\dfrac{1}{2} \times (9 + 4.6) \times 1 + \dfrac{1}{2} \times 2 \times 4.6\) or 6.8 + 4.6 or 11.4 or \(\dfrac{1}{2} \times (9 + 2) \times 2 + \dfrac{1}{2} \times 1 \times 2\) or 11 + 1 or 12 or \(\dfrac{1}{2} \times 3 \times 9\) or 13.5 | M1 | correct attempt to estimate the full area below curve using trapezia, a trapezium and a triangle or a triangle |
| Correctly evaluates 20 + 34 + their correct estimate for the full area below curve, which must sum to an answer which is less than or equal to 67.5 | A1 | M3 must be awarded |
If first two marks are awarded, the third area must not come from 67.5 minus their two areas
If a concluding statement is made do not award A mark if it contains an error
| Answer | Mark | Comments |
|---|---|---|
| 1 | B1 | |
| m/s2 or ms–2 or metres per second per second | B1 | oe allow mps2 or m/s/s |
| Do not accept fractions | |
| m/s–2 | B0 |
26 The number of items, \(n\), made in 1 hour by a machine is given by \(\quad n = \dfrac{60}{t}\)
\(t\) is the time in minutes the machine takes to make one item.
The value of \(t\) changes for different types of item.
Use your graph to estimate the value of \(n\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Plots the points (1, 60), (2, 30), (3, 20) and (4, 15) | M1 | \(\pm \dfrac{1}{2}\) small square |
| Correct smooth curve through correct four points | A1 | \(\pm \dfrac{1}{2}\) small square |
Ignore any calculations and mark the graph only
Points cannot be implied by a bar chart or vertical line graph, but condone crosses at the top of a vertical line graph for M1 and the correct curve superimposed for M1A1
For M1, ignore the curve outside the domain \(1 \leqslant t \leqslant 4\)
For A1, whether or not the curve extends outside the domain \(1 \leqslant t \leqslant 4\) it must not have a positive gradient at any point
If there is no curve, for M1 there must be no other points with \(x\)-coordinate 1, 2, 3 or 4
The curve should be a single line with no feathering
Unless it affects the shape of the curve (in which case A1 cannot be awarded), ignore incorrect evaluations of \(60 \div\) a non-integer value
eg \(60 \div 1.5 = \ldots\)
| Answer | Mark | Comments |
|---|---|---|
| Vertical line from \(3\dfrac{1}{2}\) minutes to their graph | M1 | \(\pm \dfrac{1}{2}\) small square implied by mark at correct place on the graph or on the vertical axis (but not on the horizontal axis) or by correct reading from their graph |
| Correct reading from their graph for \(t = 3.5\) | A1ft | ft their graph \(\pm \dfrac{1}{2}\) small square |
| Correct reading for their graph, with or without evidence of using graph | M1A1 |
| No graph in (a) | M0A0 |
| To score any marks, their graph must be decreasing in the domain \(1 \leqslant t \leqslant 4\), but may be a straight line or series of connected straight lines | |
| Answer from \(60 \div 3.5\) with no graph, or which does not match graph | M0A0 |
| Reading from 3.3 | M0A0 |
23 Here is a sketch of the curve \(\quad y = 2^x\)
On the axes above, sketch the curve \(\quad y = 3^x\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Correct curve | B2 | B2 correct curve must be correct shape and pass through (0, 1) and be in correct position relative to \(y = 2^x\) B1 correct shape and pass through (0, 1) |
Correct curve must be an exponential graph
Correct position must be
above \(\ y = 2^x\ \) for \(x > 0\)
below \(\ y = 2^x\ \) for \(x < 0\)
16 The graph below is used to convert between
temperature in degrees Fahrenheit (\(F\))
and
temperature in degrees Celsius (\(C\)).
At one temperature, \(T\),
the number of degrees Celsius is double the number of degrees Fahrenheit.
The graph of \(\quad C = 2F \quad\) can be drawn to help find this temperature.
You may use the table to help you. [2 marks]
| \(F\) | \(-25\) | ||
| \(C\) | \(-50\) |
| Answer | Mark | Comments |
|---|---|---|
| [4, 5] | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Correct ruled straight line from \((-25, -50)\) to \((25, 50)\) | B2 | \(\pm \dfrac{1}{2}\) small square ignore ends of line outside \([-25, 25]\) B1 two correct points added to the table or at least two correct points plotted or correct line too short but crosses 2 horizontal centimetre squares |
| The correct points in the table or on the graph may be outside \([-25, 25]\) eg \((100, 200)\) and \((-100, -200)\) in the table | B1 |
| For B1, do not count a point as correct if another point has the same \(x\)-coordinate, otherwise ignore extra points that are incorrect | |
| The B1 for points plotted cannot be implied by a line – you must see eg crosses or dots | |
| Ignore incorrect points in the table if B1 or B2 gained elsewhere |
| Answer | Mark | Comments |
|---|---|---|
| Correct reading of \(C\) coordinate of intersection of their graph with the given graph | B2ft | ft their intersection from any line or curve \(\pm \dfrac{1}{2}\) small square B1 line drawn horizontally from point of intersection to vertical axis or \(F\) coordinate of intersection given |
| Their line does not intersect given line or they have no line | B0 |
| If their graph intersects given line at more than one point and they give all the \(C\) coordinates of the intersections | B1 |
| If their line is correct the answer should be approximately \(-25\) | |
| If their line is correct the \(F\) coordinate should be approximately \(-12\) | |
| Both their \(-25\) and their \(-12\) given eg correct line seen and \((-25, -12)\) or \((-12, -25)\) | B1 |
10 The number of items, \(n\), made in 1 hour by a machine is given by \(\quad n = \dfrac{60}{t}\)
\(t\) is the time in minutes the machine takes to make one item.
The value of \(t\) changes for different types of item.
Use your graph to estimate the value of \(n\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Plots the points (1, 60), (2, 30), (3, 20) and (4, 15) | M1 | \(\pm \dfrac{1}{2}\) small square |
| Correct smooth curve through correct four points | A1 | \(\pm \dfrac{1}{2}\) small square |
Ignore any calculations and mark the graph only
Points cannot be implied by a bar chart or vertical line graph, but condone crosses at the top of a vertical line graph for M1 and the correct curve superimposed for M1A1
For M1, ignore the curve outside the domain \(1 \leqslant t \leqslant 4\)
For A1, whether or not the curve extends outside the domain \(1 \leqslant t \leqslant 4\) it must not have a positive gradient at any point
If there is no curve, for M1 there must be no other points with \(x\)-coordinate 1, 2, 3 or 4
The curve should be a single line with no feathering
Unless it affects the shape of the curve (in which case A1 cannot be awarded), ignore incorrect evaluations of \(60 \div\) a non-integer value
eg \(60 \div 1.5 = \ldots\)
| Answer | Mark | Comments |
|---|---|---|
| Vertical line from \(3\dfrac{1}{2}\) minutes to their graph | M1 | \(\pm \dfrac{1}{2}\) small square implied by mark at correct place on the graph or on the vertical axis (but not on the horizontal axis) or by correct reading from their graph |
| Correct reading from their graph for \(t = 3.5\) | A1ft | ft their graph \(\pm \dfrac{1}{2}\) small square |
| Correct reading for their graph, with or without evidence of using graph | M1A1 |
| No graph in (a) | M0A0 |
| To score any marks, their graph must be decreasing in the domain \(1 \leqslant t \leqslant 4\), but may be a straight line or series of connected straight lines | |
| Answer from \(60 \div 3.5\) with no graph, or which does not match graph | M0A0 |
| Reading from 3.3 | M0A0 |
8 On the axes, sketch the curve \(\quad y = x^3 - 2\)
You must show the coordinates of the \(y\)-intercept. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Fully correct curve and point \((0, -2)\) indicated | B2 | B1 fully correct curve or partially correct curve with point \((0, -2)\) indicated |
| A partially correct curve must start in the 3rd quadrant and finish in the 1st quadrant, passing through the 4th quadrant not include a section with negative gradient | |
| A fully correct curve must have all the properties of a partially correct curve have only a decreasing gradient to the left of the \(y\)-axis   Condone straight line segments at each end of the curve | |
| Fully correct curve with \(y\)-intercept labelled \(-2\) | B2 |
| Partially correct curve with \(y\)-intercept labelled \(-2\) | B1 |
| \(y\)-intercept labelled \((-2, 0)\) is incorrect and can score a maximum of B1 | |
| Ignore any numbers on the axes other than the \(y\)-intercept | |
| \(y\)-intercept \((0, -2)\) stated does indicate the point \((0, -2)\) | |
| Unlabelled notches do not indicate the point \((0, -2)\) | |
| A table of values does not indicate the point \((0, -2)\) | |
| Graph consisting only of straight lines | B0 |
A fully correct curve but point \((0, -2)\) is not indicated | B1 |
Partially correct curve with point \((0, -2)\) indicated | B1 |
Fully correct curve with point \((0, -2)\) indicated | B2 |
Partially correct curve with point \((0, -2)\) indicated | B1 |
Curve includes a negative gradient so not partially correct | B0 |
6 Five points are plotted on a centimetre grid.
The points are five of the vertices of a hexagon.
Each side of the hexagon has the same length.
Work out one possible pair of coordinates of the other vertex. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| (2, 5) or (8, 5) | B2 | B1 correct point indicated on grid or \((x, 5)\) or \((2, y)\) or \((8, y)\), where \(x\) can be \(x\) or blank or any number other than 13 and \(y\) can be \(y\) or blank or any number |
Mark answer line first, then if no marks scored, check grid for B1 plot
No tolerance on values of 2 or 8 for B2 but allow half a square tolerance on plotting for B1
1 Circle the point that lies on the curve \(\quad y = x^2 - 4x + 1\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \((-1, 6)\) | B1 |
28 Izzy runs an 80-metre race in 14 seconds.
During the first 6 seconds her speed increases at a constant rate.
During the last 8 seconds her speed increases at a different constant rate.
Her speed at 14 seconds is 2 m/s more than her speed at 6 seconds.
Here is a sketch of her speed-time graph.
Not drawn accurately
State the units of your answer. [2 marks]
Work out the value of \(v\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| 0.25 or \(\dfrac{1}{4}\) or \(\dfrac{2}{8}\) | B1 | |
| m/s2 or ms−2 or m/s/s or \(\dfrac{\text{m}}{\text{s}^2}\) | B1 | oe eg metres per second per second SC2 acceleration and unit not in m/s2 eg 25 cm/s2 or 3240 km/h2 |
| \(\dfrac{2}{14 - 6}\) with no further simplification | (1st) B0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{1}{2} \times 6 \times (v - 2)\) or \(\dfrac{1}{2} \times (14 - 6) \times (v + v - 2)\) or \((14 - 6) \times (v - 2)\) or \(\dfrac{1}{2} \times (14 - 6) \times 2\) or 8 | M1 | oe partial area any letter |
| \(\dfrac{1}{2} \times 6 \times (v - 2)\) \(+\; \dfrac{1}{2} \times (14 - 6) \times (v + v - 2)\) or \(3(v - 2) + 8(v - 2) + 8\) or \(11v - 14\) | M1dep | oe full area in one variable eg \(14 \times v - \dfrac{1}{2} \times 6 \times (v - 2)\) \(-\; \dfrac{1}{2} \times 2 \times (6 + 14)\) implies M2 |
| \(\dfrac{1}{2} \times 6 \times (v - 2)\) \(+\; \dfrac{1}{2} \times (14 - 6) \times (v + v - 2) = 80\) or \(94 \div 11\) | A1 | oe full area in one variable equated to 80 |
| 8.5(4…) or 8.55 or \(\dfrac{94}{11}\) or \(8\dfrac{6}{11}\) | A1 | |
| Alternative method 2 | ||
| \(\dfrac{1}{2} \times 6 \times x\) or \(\dfrac{1}{2} \times (14 - 6) \times (x + x + 2)\) or \((14 - 6) \times x\) or \(\dfrac{1}{2} \times (14 - 6) \times 2\) or 8 | M1 | oe partial area \(x\) is the speed at 6 seconds any letter |
| \(\dfrac{1}{2} \times 6 \times x\) \(+\; \dfrac{1}{2} \times (14 - 6) \times (x + x + 2)\) or \(3x + 8x + 8\) or \(11x + 8\) | M1dep | oe full area in one variable eg \(14 \times (x + 2) - \dfrac{1}{2} \times 6 \times x\) \(-\; \dfrac{1}{2} \times 2 \times (6 + 14)\) implies M2 |
| \(\dfrac{1}{2} \times 6 \times x\) \(+\; \dfrac{1}{2} \times (14 - 6) \times (x + x + 2) = 80\) or \(72 \div 11\) or 6.5(4…) or 6.55 or \(\dfrac{72}{11}\) or \(6\dfrac{6}{11}\) | A1 | oe full area in one variable equated to 80 |
| 8.5(4…) or 8.55 or \(\dfrac{94}{11}\) or \(8\dfrac{6}{11}\) | A1 | |
First M1 Do not allow 8 from 14 − 6
Ignore units throughout
28 Here is a sketch of a quadratic curve.
The turning point is \((3, {-2})\)
Not drawn accurately
Circle the correct statement about the gradient of the curve for \(\quad x < 3\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| gradient is negative | B1 |
26 Here is a quadratic graph.
Circle the \(x\)-coordinate of the turning point of the graph. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| 1 | B1 |
25 A quadratic curve intersects the axes at \((-3, 0)\), \((3, 0)\) and \((0, 18)\)
Not drawn accurately
Work out the equation of the curve. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(a(-3)^2 + b(-3) + c = 0\) or \(a(3)^2 + b(3) + c = 0\) | M1 | oe |
| any two of \((-)6b = 0,\ c = 18\) and \(9a + 18 = 0\) | M1dep | oe |
| \(y = 18 - 2x^2\) | A1 | oe equation |
| Alternative method 2 | ||
| \(y = 18 - 2x^2\) | B3 | oe equation B2 correct equation missing \(y =\) eg \(18 - 2x^2\) B1 equation of a quadratic curve that passes through \((-3, 0)\) or \((3, 0)\) or \((0, 18)\) condone missing \(y =\) eg \((y =)\ 18 - x^2\) or \((y =)\ (3 + x)(3 - x)\) or \((y =)\ x^2 - 2x - 3\) or \((y =)\ (x + 3)(x - 3)\) |
Correct equations include
\(y = 2(3 + x)(3 - x)\)
\(y = -2(x + 3)(x - 3)\)
\(y = (6 + 2x)(3 - x)\)
\(y = (3 + x)(6 - 2x)\)
For B3, B2 or B1 ignore incorrect expansion after correct equation or expression seen
22
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 |
| \(y\) |
| Answer | Mark | Comments | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B1 |
| Answer | Mark | Comments |
|---|---|---|
| Plots their points correctly or restarts with 4 or 5 correct points plotted | M1 | \(\pm \dfrac{1}{2}\) square tolerance allow one error |
| Correct graph | A1 | smooth quadratic curve through points |
Allow \(\pm \dfrac{1}{2}\) square tolerance for curve passing through points
If their points do not form a quadratic curve, it is maximum M1
The ‘base’ of the quadratic curve should be a smooth fairly flat curve, not a pointed shape
Ignore additional points beyond \(x = 2\) and \(x = -2\)
Ignore extended graph beyond \(x = 2\) and \(x = -2\)
| Answer | Mark | Comments |
|---|---|---|
| Draws a horizontal line from 2.6 on the \(y\)-axis to their graph | M1 | implied by correct vertical line down to the \(x\)-axis from correct point or at least one correct value seen for their graph |
| Correct readings from their graph | A1ft | must see both values |
| Positive value only or negative value only given | M1A0 |
| Tolerance on readings of \(\pm \dfrac{1}{2}\) square | |
| It is sufficient, for M1, for the horizontal line to meet the graph once | |
| No graph and answer of 1.6 | M0A0 |
14 Lee wants to draw the graph of \(\quad y = x \quad\) for values of \(x\) from \(-5\) to 5
Here is his graph.
Make two different criticisms of his graph. [2 marks]
Criticism 1
Criticism 2
| Answer | Mark | Comments |
|---|---|---|
| The graph only goes from \(x = -4\) to \(x = 4\) and the graph shown is \(y = -x\) up to 0 | B2 | oe B1 one correct criticism SC1 correct graph drawn from \(x = -5\) to \(x = 5\) |
| For one criticism, accept eg it doesn’t reach 5 / 5 not plotted / it doesn’t start at –5 only starts at –4 / only reaches 4 it should go to (5, 5) / (5, 5) not plotted / (–5, –5) not plotted it isn’t long enough | B1 |
| Do not accept eg it isn’t finished (–5, 5) not plotted | B0 |
| For the other criticism, accept eg it’s the wrong line up to 0 it’s the wrong equation for the first part \(y\) does not equal \(x\) at the beginning it should go through (–4, –4) / (–5, –5) not plotted / (–1, –1) should be plotted it should be / it’s not a straight line it shouldn’t be a V-shape worked out the negative numbers wrong / no negative \(y\)-coordinates he should have plotted … and correct table of values | B1 |
| Do not accept eg it isn’t correctly drawn / it isn’t \(y = x\) / the points are plotted wrong it should be symmetrical / it shouldn’t be symmetrical one line should go below the \(x\)-axis | B0 |
| NB (–5, –5) should be plotted is valid for either (but not both) criticisms | B1 |
| Both criticisms may be in one answer space | |
| Ignore irrelevant statements but any additional statements must be correct eg It goes from –4 to 5 not –5 to 5 | B0 |
12 Lewis wants to draw the graph \(\quad y = x^3 \quad\) for values of \(x\) from \(-2\) to 2
Here is his graph.
Make one criticism of his graph. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Graph should be a curve | B1 | oe eg Should not be straight lines Not a curve Not smooth Too straight Need more points plotted |
28 \(P\) is a point on the circle with equation \(\quad x^2 + y^2 = 80\)
\(P\) has \(x\)-coordinate 4 and is below the \(x\)-axis.
Not drawn accurately
Work out the equation of the tangent to the circle at \(P\). [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(4^2 + y^2 = 80\) or \(y = \sqrt{64}\) | M1 | oe May be implied from 8 on diagram |
| \(y = -8\) | A1 | Accept \((4, -8)\) |
| \(\dfrac{\text{their } -8}{4}\) or \(-2\) | M1 | oe gradient of radius \(OP\) |
| \(-1 \div\) their \(-2\) or \(\dfrac{1}{2}\) or \(-1 \div\) their gradient | M1 | gradient of tangent at \(P\) |
| \(y = \dfrac{1}{2}x - 10\) or \(\;y + 8 = \dfrac{1}{2}(x - 4)\) | A1 | oe Ignore further working |
| \(y + 8 = \dfrac{1}{2}(x - 4)\) followed by error expanding and/or collecting terms | M1A1M1M1A1 |
| \(y = \dfrac{1}{2}x - 10\) in working and \(\dfrac{1}{2}x - 10\) only on answer | M1A1M1M1A1 |
| \(\dfrac{1}{2}x - 10\) | M1A1M1M1A0 |
| (\(y = \sqrt{64}\)) \(y = 8\) Gradient \(OP = 2\) Perpendicular gradient \(= -\dfrac{1}{2}\) | M1 A0 M1 M1 A0 |
24 The speed-time graph shows 20 seconds of a car journey.
Harry wants to estimate the distance the car travels in this time.
He uses a triangle and a trapezium, as shown, to estimate the area under the graph.
Tick one box. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{1}{2} \times 10 \times 20\) or 100 | M1 | oe Area of triangle on left |
| \(\dfrac{1}{2} \times (20 + 30) \times 10\) or 250 or \(20 \times 10\) or 200 and \(\dfrac{1}{2} \times 10 \times 10\) or 50 | M1 | oe Area of trapezium on right |
| 350 | A1 | |
| Alternative method 2 | ||
| \(\dfrac{1}{2} \times 10 \times 10\) or 50 | M1 | oe Area of triangle on top right |
| \(\dfrac{1}{2} \times (20 + 10) \times 20\) or 300 or \(10 \times 20\) or 200 and \(\dfrac{1}{2} \times 10 \times 20\) or 100 | M1 | oe Area of trapezium across bottom |
| 350 | A1 | |
| \(\dfrac{1}{2} \times (0 + 2 \times 20 + 30) \times 10\) (using Trapezium rule) | M1M1 |
| Beware of 300 from incorrect working | |
| Beware \((30 - 20) \times (20 - 10) = 100\) is incorrect working |
| Answer | Mark | Comments |
|---|---|---|
| It works out an underestimate of the distance | B1 |
22 A circle, centre \(O\), passes through (5, 0).
What is the equation of the circle?
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + y^2 = 25\) | B1 |
17 A ball is thrown vertically upwards.
The graph shows the height of the ball above the ground after it is thrown.
| Answer | Mark | Comments |
|---|---|---|
| [2.9, 3] | B1 |
| Answer | Mark | Comments |
|---|---|---|
| [1.4, 1.6] | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{[4.55, 4.65] - 0}{3.5 - [1.5, 1.6]}\) or \(\dfrac{[4.55, 4.65]}{[1.9, 2]}\) or \(\dfrac{[4.55, 4.65] - 0}{[1.5, 1.6] - 3.5}\) or \(\dfrac{[4.55, 4.65]}{[-2, -1.9]}\) or \([-2.45, -2.275]\) | M1 | oe |
| [2.275, 2.45] | A1 |
15 The graph of \(\quad y = 4 - x \quad\) for values of \(x\) from \(-2\) to 5 \(\quad\) is shown on the grid.
| Answer | Mark | Comments |
|---|---|---|
| Any two of \((-2, -9)\), \((-1, -7)\), \((0, -5)\), \((1, -3)\), \((2, -1)\), (3, 1), (4, 3), (5, 5) | M1 | gives at least two correct pairs of coordinates, may be in a table implied by points plotted \(\pm\dfrac{1}{2}\) small square |
| At least two correct points plotted or at least two of their points plotted correctly | M1dep | implied by correct line which does not have to extend from \(x = -2\) to \(x = 5\) \(\pm\dfrac{1}{2}\) small square |
| Correct line from \((-2, -9)\) to (5, 5) | A1 | \(\pm\dfrac{1}{2}\) small square ignore ends of line outside \([-2, 5]\) |
Ignore extra points that are incorrect
| Answer | Mark | Comments |
|---|---|---|
| 3 | B1ft | correct or ft the intersection of their graph with the given graph \(\pm\dfrac{1}{2}\) small square |
| Answer 3 with or without correct graph | B1 |
| Answer (3, 1) | B0 |
| Answer \((x =)\) 3, \(y = 1\) | B1 |
| If their graph intersects the given graph at more than one point they need to give the correct \(x\)-coordinate of each point of intersection | B1ft |
14 Draw the graph of \(\quad y = 0.8^x \quad\) for values of \(x\) from 0 to 6 [3 marks]
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \(y\) |
| Answer | Mark | Comments |
|---|---|---|
| At least 4 of (\(x = 0\)) \(\;y = 1\) (\(x = 1\)) \(\;y = 0.8\) or \(\dfrac{4}{5}\) (\(x = 2\)) \(\;y = 0.64\) or \(\dfrac{16}{25}\) (\(x = 3\)) \(\;y = [0.51, 0.512]\) or \(\dfrac{64}{125}\) (\(x = 4\)) \(\;y = [0.40, 0.41]\) or \(\dfrac{256}{625}\) (\(x = 5\)) \(\;y = [0.32, 0.33]\) or \(\dfrac{1024}{3125}\) (\(x = 6\)) \(\;y = [0.26, 0.262144]\) or \(\dfrac{4096}{15\,625}\) | M1 | oe May be seen in the table or a list or implied from their graph |
| 6 or 7 correct points plotted | A1 | tolerance of \(\pm\dfrac{1}{2}\) small square |
| Fully correct smooth curve through all seven correct points | A1 | tolerance of \(\pm\dfrac{1}{2}\) small square |
| Ignore extra points plotted | |
| Ignore any curve drawn for \(x \lt 0\) or \(x \gt 6\) | |
| Curve passing through all correct points within tolerance | M1A1A1 |
| Ruled straight lines | A0 |
29 Here is the graph of \(\;y = \mathrm{f}(x)\;\) where \(\mathrm{f}(x)\) is a quadratic function.
Write down all the integer solutions of \(\;\mathrm{f}(x) \geqslant 0\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| −3 −2 −1 0 1 2 | B2 | B1 for 5 correct and 0 incorrect or 6 correct and 1 incorrect |
Do not accept coordinates
28 \(A\), \(B\) and \(C\) are points on the circle \(\quad x^2 + y^2 = 36 \quad\) as shown.
\(A\) is on the \(y\)-axis.
\(B\) is on the \(x\)-axis.
\(M\) is the midpoint of \(AB\).
\(COM\) is a straight line.
Give your answers in surd form. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| (\(0^2 +\)) \(6^2 = 36\) or (\(OA =\)) radius = 6 or \(\sqrt{36} = 6\) | B1 | oe |
| \(0 + 36 = 36\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| (6, 0) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{6 - \text{their } 0}{0 - \text{their } 6}\) or \(\dfrac{\text{their } 0 - 6}{\text{their } 6 - 0}\) or \(\dfrac{6}{-6}\) or \(\dfrac{-6}{6}\) or \(-1\) | M1 | gradient \(AB\) |
| gradient \(OM \times\) gradient \(AB = -1\) and gradient \(OM = 1\) (and \(y = x\)) | A1 | must see correct working for M1 |
| Alternative method 2 | ||
| \(\left(\dfrac{6 + 0}{2}, \dfrac{0 + 6}{2}\right)\) or (3, 3) | M1 | coordinates of \(M\) |
| gradient \(OM = 1\) (and \(y = x\)) or (0, 0) and (3, 3) (and \(y = x\)) | A1 | must see correct working for M1 |
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + x^2 = 36\) or \(2x^2 = 36\) or \(y^2 + y^2 = 36\) or \(2y^2 = 36\) or (–)\(6 \cos 45°\) or (–)\(6 \sin 45°\) | M1 | oe equation |
| (–)\(\sqrt{\dfrac{36}{2}}\) or (–)\(\sqrt{18}\) or (–)\(3\sqrt{2}\) or (–)\(\dfrac{6\sqrt{2}}{2}\) or (–)\(\dfrac{6}{\sqrt{2}}\) | M1 | |
| \(\left(-\sqrt{18}, -\sqrt{18}\right)\) or \(\left(-3\sqrt{2}, -3\sqrt{2}\right)\) or \(\left(-\dfrac{6\sqrt{2}}{2}, -\dfrac{6\sqrt{2}}{2}\right)\) or \(\left(-\dfrac{6}{\sqrt{2}}, -\dfrac{6}{\sqrt{2}}\right)\) | A1 | oe surd form |
25 Liquid is leaking out of a container.
The graph shows the depth of the liquid for 60 seconds.
Use the graph to work out an estimate of the rate of decrease of depth at 10 seconds.
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Attempt to draw a tangent | M1 | |
| Attempt at slope of a tangent drawn at (10, 15) | M1dep | tolerance \(\pm \frac{1}{2}\) square Must be an attempt at change in \(y\) divided by change in \(x\) Accept positive or negative |
| [0.6, 0.8] from tangent drawn at (10, 15) | A1ft | Condone −[0.6, 0.8] from tangent drawn at (10, 15) ft from their tangent drawn at (10, 15) |
| Tangent drawn at incorrect point | M1M0A0 |
| No tangent | M0 |
| Tangent drawn at (10, 15) \(10 \div 15 = 0.6\) | M1 M0 A0 |
| Misread of scale for tangent drawn at (10, 15) could score M1M1 |
24 Beth ran a 200 metre race.
Here is a graph of the first 8 seconds of her race.
She completed the race at a constant speed of 9 m/s
Amy completed the race in 27 seconds.
Did Beth finish before Amy?
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(0.5 \times 8 \times 9\) or 36 or \((27 - 8) \times 9\) or \(19 \times 9\) or 171 | M1 | May be seen on graph |
| \(0.5 \times 8 \times 9 + (27 - 8) \times 9\) or 207 | M1dep | M2 \(0.5 \times (27 + 19) \times 9\) |
| 207 and Yes | A1 | |
| Alternative method 2 | ||
| \(0.5 \times 8 \times 9\) or 36 | M1 | May be seen on graph |
| \(\dfrac{200 - \text{their } 36}{9}\) or \(\dfrac{164}{9}\) or 18.2… | M1dep | |
| 26.2… and Yes or 18.2… and 19 and Yes | A1 | |
| Alternative method 3 | ||
| \(0.5 \times 8 \times 9\) or 36 | M1 | May be seen on graph |
| \(\dfrac{200 - \text{their } 36}{27 - 8}\) or \(\dfrac{164}{19}\) or 8.6… | M1dep | |
| 8.6… and Yes | A1 | |
| Alternative method 4 | ||
| \(0.5 \times 8 \times 9\) or 36 | M1 | May be seen on graph |
| Attempt at total distance for Beth for \(26.\dot{2} \leqslant\) total time \(\lt 27\) | M1dep | eg (time 26.5s) \(0.5 \times 8 \times 9 + (26.5 - 8) \times 9\) |
| Correct total distance for Beth for \(26.\dot{2} \leqslant\) total time \(\lt 27\) and Yes | A1 | eg (time 26.5s) 202.5 and Yes |
23 Anil’s home is 1 km from a shop.
He walked from home to the shop at a constant speed in 10 minutes.
He stayed at the shop for 5 minutes.
He walked home at a constant speed in 8 minutes.
Anil drew this distance-time graph to represent his journey.
Make two criticisms of his graph. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Valid criticism referring to the line from (0, 0) to (10, 1) | B1 | eg there shouldn’t be a curve need to be specific about the line shape, it is not sufficient to simply say it is wrong |
| Valid criticism referring to the line from (15, 1) | B1 | oe eg he never goes 2 km from home |
| Criticisms can be in either order | |
| A correct diagram takes precedence over statements, otherwise ignore diagram | |
| For first B1: | |
| The first part is curved | B1 |
| The curve should be a straight line | B1 |
| He has drawn a curve for constant speed | B1 |
| The line is curved which shows his speed was not consistent/constant | B1 |
| He’s not going at a constant speed to the shop (correct referral to graph) | B1 |
| All lines should be straight | B1 |
| Constant speed should be a diagonal/straight line | B1 |
| The line shouldn’t curve | B1 |
| The constant speed should be [a sloping straight line sketched] | B1 |
| The curved line shows he decreased speed | B1 |
| It should be a straight line from 0 to 10 | B1 |
| It should be a straight line at the start | B1 |
| A distance-time graph shouldn’t have curves | B0 |
| It should be a straight line ( ‘It’ seems to be referring to the whole graph) | B0 |
| The curved line shows he increased and decreased speed | B0 |
| He was walking at a range of speeds, so not consistent (referral to whole graph) | B0 |
| The constant speed is drawn incorrectly (how?) | B0 |
| The lines should be curved or straight, not both | B0 |
| The curve should be a line of best fit | B0 |
| It should be a straight line from 0 to 15 (it should be to 10) | B0 |
| The curve is wrong (how?) | B0 |
| For 2nd B1: | |
| The line should go down at the end | B1 |
| He isn’t walking home, he’s walking further away | B1 |
| He has walked away from home when he hasn’t | B1 |
| The line should go back to the bottom of the graph | B1 |
| The graph should return to zero | B1 |
| The last part should be decreasing (instead of increasing) | B1 |
| The line for him walking home should have negative gradient | B1 |
| The graph shows he didn’t walk home | B1 |
| The line for him walking home should have negative correlation | B0 |
| The line for the journey home goes the wrong way | B0 |
| The graph does not show his journey home | B0 |
| His house is 2 km away from the shop | B0 |
| The line should be decreasing instead of increasing (which line?) | B0 |
| His home is 1 km from the shop not 2 km | B0 |
21
She draws the graph of \(\quad y = 2x^2 \quad\) and a straight line graph on the same grid.
Here is the graph of \(\quad y = 2x^2\)
Complete her method to solve \(\quad 2x^2 - 3x = 0\) [2 marks]
He uses this method.
\[\begin{aligned} 2x^2 + 5x &= 0 && \text{subtract } 5x \text{ from both sides} \\ 2x^2 &= -5x && \text{divide both sides by } x \\ 2x &= -5 && \text{divide both sides by } 2 \\ x &= -2.5 && \end{aligned}\]Evaluate his method and his answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Draws \(y = 3x\) and (\(x =\)) [−0.1, 0.1] and (\(x =\)) [1.4, 1.6] | B2 | B1 Draws \(y = 3x\) or states \(y = 3x\) \(\pm\dfrac{1}{2}\) square tolerance for drawing graph Graph must be seen for \(x\) values from 0 to 1.5 |
| Ignore any \(y\) values seen | |
| Solutions from a non-graphical method | B0 |
| Ignore other lines drawn on grid |
| Answer | Mark | Comments |
|---|---|---|
| Full evaluation of method and answer | B2 | eg1 Cannot divide by \(x\) as it could be zero eg2 Should have factorised and then he would have also found that \(x = 0\) eg3 Should have used the formula and then he would have also found that \(x = 0\) eg4 Should have used a graphical method then he would have also found that \(x = 0\) eg5 Should have completed the square then he would have also found that \(x = 0\) B1 Partial evaluation eg1 \(x = 0\) has been omitted eg2 Should have factorised eg3 Should have used the formula eg4 Should have drawn a graph eg5 Only found one solution eg6 Cannot divide by zero |
| For B2 there needs to be an evaluation of the method and an indication that \(x = 0\) has been omitted from the answer | |
| \(x(2x + 5) = 0\) \(x = 0\) and \(x = -2.5\) | B2 |
| Should be two solutions | B1 |
| What about \(x = 0\) | B1 |
| The answer is wrong | B0 |
| Ignore non-contradictory further work |
14 On the grid, draw the graph of \(\quad x + y = 2 \quad\) for values of \(x\) from –3 to 3 [2 marks]
| Answer | Mark | Comments | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Identifies or plots any two correct points | M1 | points with integer values are
may be in a list ignore incorrect plots | ||||||||||||||||
| Correct straight ruled line from (–3, 5) to (3, –1) | A1 | ignore incorrect plots if correct line drawn |
| Correct line, but not extending from (–3, 5) to (3, –1) | M1A0 |
| Two lines, one correct and one incorrect | M1A0 |
26
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(y\) | \(-2\) | \(-2\) | \(4\) |
| Answer | Mark | Comments | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B2 | B1 1 or 2 values correct |
| Answer | Mark | Comments |
|---|---|---|
| 5 or 6 points plotted correctly | M1 | Correct or ft their table in (a) Tolerance of \(\pm 1\) small square Points can be implied by graph passing through them |
| Correct smooth parabolic curve and \(y\)-coordinate of minimum point in the range \(-2.5 \leqslant y \leqslant -2.1\) | A1 | Tolerance of \(\pm 1\) small square for the six correct points from the table No further tolerance for the minimum |
Tolerance of \(\pm 1\) small square means it is on the edges of or within the shaded area | |
| Ignore extra points plotted | |
| If their table in (a) has points that are beyond the grid these points will not be able to be plotted correctly | |
| Ignore any curve drawn for \(x \lt -2\) or \(x \gt 3\) | |
| Curve passing through all correct points within tolerance | M1A1 |
| Ruled straight lines | A0 |
24 A ball is thrown from a point 6 metres above the ground.
The graph shows the height of the ball above the ground, in metres.
Estimate the speed of the ball, in m/s, after 1 second.
You must show your working. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Draws a tangent at 1 second | M1 | |
| Their gradient at 1 second | A1ft | Must see a tangent on the graph ft their tangent \(\pm 0.2\) tolerance on vertical reading \(\pm 0.1\) tolerance on horizontal reading |
23 Lily goes on a car journey.
For the first 30 minutes her average speed is 40 miles per hour.
She then stops for 15 minutes.
She then completes the journey at an average speed of 60 miles per hour.
The total journey time is 1 hour.
| Answer | Mark | Comments |
|---|---|---|
| Joins (0, 0) to (30, 20) | B1 | Line does not need to be straight but must start and finish at correct points and not be decreasing Mark intention |
| Horizontal line for 15 minutes from their (30, 20) | B1ft | Mark intention |
| Line with gradient 1 or a curve from their (45, 20) and stops at 60 minutes or stops at top edge of grid or higher but not beyond 60 minutes | B1ft | A curve must not be decreasing and must start and finish at two points that could be joined by a line with gradient 1 Condone a horizontal or vertical line from 60 minutes Mark intention |
 | B3 |
| Allow any horizontal line between 30 minutes and 45 minutes if first part of journey is blank eg  | B0B1 |
| Do not allow second mark if their first line is followed by a drop back towards the horizontal axis before she stops eg  | B1B0 (top two graphs) B0B0 (lower graph) |
| If there are more than 3 lines or curves, assume the last part is the part where she completes her journey eg  | B1B0B1ft |
| If their (45, 20) is too high to fit a line of gradient 1 ending at 60 minutes, allow the final line to stop at the top of the grid or higher, but not beyond 60 minutes eg  | B0B1ftB1ft |
| Points but no lines | B0 |
| Ignore any lines that could be working for part (a) or part (b) |
| Answer | Mark | Comments |
|---|---|---|
| 35 | B1ft | Correct or ft total distance travelled for their graph at 60 minutes |
| 35 from any or no graph | B1 |
| If their graph extends beyond 60 minutes, read off at 60 minutes for ft | |
| Follow through total distance travelled eg (a)  | |
| (b) answer 25 | B0ft |
| (b) answer 55 | B1ft |
| Ignores the stationary parts | B0 |
| Do not follow through a graph above the grid at 60 eg (a)  | |
| (b) answer 55 | B0ft |
23 Here is a velocity-time graph of a motorbike for 25 seconds.
| Answer | Mark | Comments |
|---|---|---|
| [6, 6.5] | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{1}{2} \times (22 + 18) \times (25 - 10)\) or \(15 \times 18 + \dfrac{1}{2} \times 15 \times 4\) | M1 | oe |
| 300 | A1 | |
| Alternative method 2 | ||
| \(20 \times 15\) | M1 | |
| 300 | A1 | |
Alternative method 2 uses average velocity \(\times\) time
20 Work out the diameter of the circle \(\quad x^2 + y^2 = 64\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| 16 | B1 |
18 Nick sketches the graph of \(\quad y = 0.5^x \quad\) for \(\;x \geqslant 0\)
Make one criticism of his sketch. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Valid criticism | B1 | eg (\(y =\)) 0.5 should be (\(y =\)) 1 \(y = 0.5\) should be when \(x = 1\) When \(x = 0 \;\; y = 1\) 0.5 is incorrect Crosses \(y\) axis in wrong place Graph should start at 1 \(0.5^0 = 1\) |
| Do not accept statements which are contradictory | |
| He does not have a scale on the \(x\) axis | B0 |
| It does not pass through zero | B0 |
| The line should meet the \(x\) axis | B0 |
15 The graph shows the depth of water in a harbour for 12 hours.
\(d\) is the depth of water in a harbour in metres
\(t\) is the number of hours after 9 am
| Answer | Mark | Comments |
|---|---|---|
| 8 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| 3 | B1 | Accept –3 |
8 Lily goes on a car journey.
| Answer | Mark | Comments |
|---|---|---|
| Joins (0, 0) to (30, 20) | B1 | Line does not need to be straight but must start and finish at correct points and not be decreasing Mark intention |
| Horizontal line for 15 minutes from their (30, 20) | B1ft | Mark intention |
| Line with gradient 1 or a curve from their (45, 20) and stops at 60 minutes or stops at top edge of grid or higher but not beyond 60 minutes | B1ft | A curve must not be decreasing and must start and finish at two points that could be joined by a line with gradient 1 Condone a horizontal or vertical line from 60 minutes Mark intention |
 | B3 |
| Allow any horizontal line between 30 minutes and 45 minutes if first part of journey is blank eg  | B0B1 |
| Do not allow second mark if their first line is followed by a drop back towards the horizontal axis before she stops eg  | B1B0 (first two) B0B0 (third) |
| If there are more than 3 lines or curves, assume the last part is the part where she completes her journey eg  | B1B0B1ft |
| If their (45, 20) is too high to fit a line of gradient 1 ending at 60 minutes, allow the final line to stop at the top of the grid or higher, but not beyond 60 minutes eg  | B0B1ftB1ft |
| Points but no lines | B0 |
| Ignore any lines that could be working for part (a) or part (b) |
| Answer | Mark | Comments |
|---|---|---|
| 35 | B1ft | Correct or ft total distance travelled for their graph at 60 minutes |
| 35 from any or no graph | B1 |
| If their graph extends beyond 60 minutes, read off at 60 minutes for ft | |
| Follow through total distance travelled eg (a)  (b) answer 55 | B0ft B1ft |
| Ignores the stationary parts | B0 |
| Do not follow through a graph above the grid at 60 eg (a)  | B0ft |
6
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | \(-2\) | 4 |
| Answer | Mark | Comments | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B2 | B1 1 or 2 values correct |
| Answer | Mark | Comments |
|---|---|---|
| 5 or 6 points plotted correctly | M1 | Correct or ft their table in (a) Tolerance of \(\pm 1\) small square Points can be implied by graph passing through them |
| Correct smooth parabolic curve and \(y\)-coordinate of minimum point in the range \(-2.5 \leqslant y \leqslant -2.1\) | A1 | Tolerance of \(\pm 1\) small square for the six correct points from the table No further tolerance for the minimum |
Tolerance of \(\pm 1\) small square means it is on the edges of or within the shaded area | |
| Ignore extra points plotted | |
| If their table in (a) has points that are beyond the grid these points will not be able to be plotted correctly | |
| Ignore any curve drawn for \(x \lt -2\) or \(x \gt 3\) | |
| Curve passing through all correct points within tolerance | M1A1 |
| Ruled straight lines | A0 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{2}\) or 0.5 | B1 | Ignore any \(y\)-coordinate |
| \((-2.25, 0.5)\) | B0 |
| Ignore their graph drawn in (b) – there is no ft | |
| Condone \(0.5, -2.25\) | B1 |
23 A pot of tea is made using boiling water and left to cool.
The temperature, \(T\) °C, of the pot of tea after \(n\) minutes is given by the formula
\(T = ab^n\).
The graph shows some information about the temperature of the pot of tea.
Not to scale
Find the value of \(a\) and the value of \(b\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [\(a\) = ] 98 [\(b\) =] 0.949[9…] or 0.95[0] | 3 | B1 for [\(a\) = ] 98 AND B2 for [\(b\) =] 0.949[9…] or 0.95[0] or M1 for \(84 = 98b^3\) or better or \(84 = ab^3\) | SC2 if correct answers transposed unless correctly identified in body |
21 The graph of a function is drawn on the grid for \(0 \leqslant x \leqslant 4\).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 3.8 – 4.2 nfww | 2 | M1 for \(\dfrac{\textit{their}\,15 - \textit{their}\,7}{3 - 1}\) oe | condone \(6.8 \leqslant\) their\(7 \leqslant 7.2\) and \(14.8 \leqslant\) their\(15 \leqslant 15.2\) e.g \(\frac{14.8 - 7.2}{3 - 1} = 3.8\) This part cannot be answered with a tangent at \(x = 2\). |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Ruled tangent drawn at \(x = 2.5\) | B1 | Tangent should not cross the curve, should touch at \(x = 2.5\) (condone slight daylight between tangent and curve) and condone running along the curve slightly. Ignore any tangent at \(x = 2\) for this part. | |
| using two points on their tangent at \(x = 2.5\) | M1 | this might be in the form of a triangle or two points clearly identified or in a calculation | The remaining 3 marks depend on a reasonable ruled attempt at a tangent clearly at \(x = 2.5\) so B0 M1 M1 A1FT is possible. |
| use of \(\dfrac{\textit{difference in } y}{\textit{difference in } x}\) | M1 | M1 for attempting to divide vertical distance by horizontal distance on their tangent at \(x = 2.5\) even if the readings are inaccurate | |
| accurate calculation from their tangent | A1FT | The final mark (A1) is easily checked for accuracy by using the ‘ruler’ which is lined up on the tangent and extended as far as possible, calculate the gradient, or translate the ruler to put one end on the origin, and use a tolerance of ± 0.2 and if their figure is within range award A1 | |
20 A container takes 8 minutes to be filled with water.
The graph shows the rate of flow, in litres per minute, of the water into the container during the 8 minutes.
Not to scale
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 18 × (7.5 – 0.5) or 18 × 7 | 1 | 126/18 = 7 only scores if 7 is related to the context 126/7 = 18 only scores if 18 is related to the context | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 16.87[5] or 16.88 or 16.9 oe | 4 | M3 for \(\frac{126 + 2 \times \frac{1}{2} \times 0.5 \times 18}{8}\) oe or M2 for \(126 + 2 \times \frac{1}{2} \times 0.5 \times 18\) oe may be implied by 135 or M1 for \([2\ \times]\ \frac{1}{2} \times 0.5 \times 18\) oe may be implied by 4.5 for one triangle or 9 for two triangles If 0 scored, SC1 for \(\frac{126 + 2 \times 9}{8}\) or \(\frac{126 + 18}{8}\) | e.g. \(\frac{\frac{1}{2} \times (7 + 8) \times 18}{8}\) e.g. \(\frac{1}{2} \times (7 + 8) \times 18\) or \(18 \times 8 - 2 \times \frac{1}{2} \times 0.5 \times 18\) |
16
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct shape and orientation, reasonable symmetry in y-axis, intent to pass through origin | 2 | M1 attempt at correct curve shape and orientation not necessarily through the origin | See Appendix Should be a single smooth curve that appears symmetrical by eye For M1 condone translation in any direction, minor feathering and curvature errors Plots should be curved and not straight-line segments |
2 marks: Correct shape and orientation, reasonable symmetry in y-axis, intent to pass through origin
1 mark: Attempt at correct curve shape and orientation not necessarily through the origin; condone translation in any direction, minor feathering and curvature errors
Plots should be curved and not straight-line segments
| 2 marks sym in y-axis, origin  | 2 marks sym in y-axis, origin, min feathering  | 2 marks reasonable.sym in y-axis, origin, min curvature  | 2 marks sym y-axis, origin, borderline feathering and min curvature  |
| 1 mark shape, not origin  | 1 mark Shape, not origin, minor curvature error  | 1 mark (best fit), shape, poor symmetry, origin, curvature issues | 0 marks |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [\(y\) =] [\(k\)] \(\tan x\) where \(k \gt 0\) | 1 | Condone [y =][\(k\)] \(\tan x^\circ\), \(\tan\theta\) etc Do not accept [y =][\(k\)] tan, \(\tan y\), \(\tan\frac{o}{a}\), tan(a number) etc Ignore attempts at domain | |
15 For each graph below, select its possible equation from this list.
| A \(\quad x = 4\) | B \(\quad y = 4\) | C \(\quad y = x\) |
| D \(\quad y = x^2\) | E \(\quad y = \dfrac{1}{x}\) | F \(\quad y = x^3\) |
Write the letter of the equation beneath each graph.
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| A E | 2 | B1 for each | Accept the full correct equation |
2 Point A is shown on this grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((-4, 3)\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Point plotted at \((2, -2)\) | 1 | Clear intent | |
22
Not to scale
[1]
Not to scale
[1]
Not to scale
The equation of the graph is \(y = \sin(x + p)\), where \(0^\circ \leqslant x \leqslant 360^\circ\).
The \(x\)-intercepts are 130° and \(r\)°.
Write down the value of \(p\) and the value of \(r\). [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [\(y =\)] \(\left(\frac{1}{4}\right)^x\) | 1 | ||
| (ii) [\(y =\)] \(\sqrt{4^2 - x^2}\) | 1 | Accept [\(y =\)] \(\sqrt{16 - x^2}\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [\(p =\)] 50 | 1 | ||
| [\(r =\)] 310 | 1 | ||
21
Complete the diagram by adding the values of the three intercepts with the axes. [2]
Write down the coordinates of the point T. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + 3x - 5x - 15\) [\(= x^2 - 2x - 15\)] | 1 | All four terms must be seen Could be seen in a grid | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-15\) [\(y\)-intercept] | 1 | Allow \((0, -15)\) | |
| \(-3\) and 5 [roots] | 1 | Must be in correct place. | Allow \((-3, 0)\) and \((5, 0)\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((1, -16)\) | 2 | B1FT for 1 | B1FT is mid-point of their two roots from 21(b) provided one is positive and one negative. Their 1 must be > 0 |
| B1FT for \(-16\) | B1FT for their value of \(x\) their \(-16\) must be \(< 0\) | ||
19 This graph shows the depth of the water, in centimetres, at a particular point in a river over a period of 10 hours.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −0.5 or +0.5 | 2 | M1 for \(\frac{27 - 32}{10}\) or \(\frac{32 - 27}{10}\) oe | For M1 condone one error in the figures |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Tangent drawn at time 17.00 | 1 | Tangent should not cross the curve and should touch at 17.00. Condone slight daylight between tangent and curve. | |
| 1.6 to 2.0 nfww and dep on acceptable tangent drawn | 3 | M1 for using two points on their line M1 for use of \(\frac{\text{difference in } y}{\text{difference in } x}\) | |
13 Zayn drives from home to a shopping centre.
The graph shows information about the journey and the time spent at the shopping centre.
Complete the graph to show this information. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Zayn travels at a constant speed | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 50 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 25 | 2 | M1 for their (b) \(\div\) 2 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Line or curve from (11 00, 50) to (12 45, 0) with no horizontal or vertical sections | 2 | B1 for line or curve reaching time axis after 11 00 or for 12 45 soi | eg a line ending at time 12 45 |
10 Here is a table of values for \(y = x^2 - x - 8\).
| \(x\) | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| \(y\) | 12 | 4 | -2 | -6 | -8 | -8 | -6 | -2 | 4 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| correct curve within tolerance | 3 | B2 for 8 points accurately plotted or B1 for 5 points accurately plotted | tolerance \(\pm\frac{1}{2}\) small square radially for curve and points, some daylight between \(y = -8\) and curve, condone a wobbly curve and slight feathering or tram lines in no more than 3 sections but no ruled lines and no dashed lines |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x = 0.5\) oe | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| –2.3 or –2.4 3.3 or 3.4 | 2 | B1 for either answer SC1 for an answer to more than 1 dp in each of –2.4 to –2.3 and 3.3 to 3.4 If 1 scored FT their curve for 2 marks or if 0 scored FT their curve for 1 or 2 marks or for SC1 | tolerance \(\pm\frac{1}{2}\) small square radially |
22
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch with \(y\) – intercept 1 indicated | 2 | B1 for increasing exponential curve with no ruled sections or for any sketch with a single \(y\) – intercept at 1 | For 2 marks, condone curve touching but not crossing \(x\)- axis For 2 marks or B1 mark intention For 2 marks no ruled sections See AG |
| Example A |  | Correct shape, mark intention passes through \(y = 1\) 2 marks |
| Example B |  | Correct shape B1 |
| Example C |  | \(y\) – intercept at 1 B1 |
| Example D |  | Ruled wrong shape B0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch of \(y = \cos x\) with 90 and 270 indicated and starting at 1 on \(y\)-axis | 2 | B1 for cos shape graph with period 360 starting at (0, \(k\)), \(k \gt 0\) and amplitude \(k\) Or for cos shape graph starting at (0, 1) with amplitude 1 and with a period a factor of 360 Or for cos shape graph with period 360 with 90 and 270 indicated where graph crosses \(x\) - axis | For 2 marks mark intention e.g. \(y = 2\cos x\) For B1 allow poor curvature, mark intention e.g. \(y = \cos 2x\), May be errors with amplitude See AG |
| Example A |  | Mark intention 2 marks |
| Example B |  | BOD mark intention 2 marks |
| Example C |  | Period not a factor of 360 B0 |
| Example D |  | Amplitude not 2 B0 |
| Example E |  | Period not a factor of 360 B0 |
| Example F |  | Period not a factor of 360 B0 |
| Example G |  | Cos shape with period 360 90 and 270 indicated B1 |
| Example H |  | Period not a factor of 360 B0 |
18
Calculate the distance travelled by the object during the 20 seconds. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 162 | 3 | M2 for \(\frac{1}{2} \times (\textit{their}7 + 20) \times \textit{their}12\) oe or M1 for one relevant area e.g. \(\frac{1}{2} \times \textit{their}8 \times \textit{their}12\) or their7 × their12 or \(\frac{1}{2} \times \textit{their}5 \times \textit{their}12\) | e.g. M2 for \(\frac{1}{2} \times \textit{their}8 \times \textit{their}12 + \textit{their}7 \times \textit{their}12 + \frac{1}{2} \times \textit{their}5 \times \textit{their}12\) implied by 48 + 84 + 30 |
| alternative method : M2 for \(\textit{their}12 \times 20 - \frac{1}{2} \times \textit{their}8 \times \textit{their}12 - \frac{1}{2} \times \textit{their}5 \times \textit{their}12\) or M1 for one relevant area e.g. their12 × 20 or \(\frac{1}{2} \times \textit{their}8 \times \textit{their}12\) or \(\frac{1}{2} \times \textit{their}5 \times \textit{their}12\) | implied by 240 – 48 – 30 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 14 | 2 | M1 for \(\frac{\textit{their}32 - \textit{their}4}{4 - 2}\) oe | Allow any two points on the line joining (2,4) and (4,32) |
| (ii) Ruled tangent drawn at 3 seconds | B1 | Tangent – mark intention to touch the curve at 3 seconds(condone thin daylight at 3), ignore other lines, condone dotty/dashed and condone line on one side only. If tangent is ruled and their value is outside the range apply FT to their tangent use accuracy of 2 sf | |
| 14.0 – 16.0 | B2dep | B2 dep on B1 correct or FT or M1 for rise ÷ run with values substituted | |
| (iii) [the speed] increases oe [because] the gradient is steeper/greater oe | 1 | Accept any correct response, accept accelerates for increase, see appendix | |
| Response | Mark |
|---|---|
| Increases and gradient is steeper/greater | 1 |
| Accelerates as curve is getting steeper | 1 |
| Increases as the distance per second increases | 1 |
| Increases as the distance travelled in the same time increases | 1 |
| Increases and curve goes upwards | 0 |
16 Here is a table of values for \(y = 2x^2 - 3x\).
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 14 | 5 | 0 | \(-1\) | 2 | 9 |
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve through given points | 3 | B2 for 6 points correctly plotted or B1 for 4 points correctly plotted | Half square accuracy. Use overlay as guide For curve: No line segments used Condone minor feathering or doubling Max half square vertically or horizontally from any point |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| their \(-1.1\) and their 2.6 | 2 | Strict FT from their graph. B1 for each or ruled \(y = 6\) cutting their curve twice or points indicated on their curve where \(y = 6\) Must have graph to score | Do not accept coordinates Half square accuracy or better For thick lines mark centre of line Ruled line within half square of \(y = 6\) throughout Their curve may be a polygon or straight line If intersection at mid-point of small square e.g. \(-1.15\) accept \(-1.2\) or \(-1.1\) |
8
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(y\) | \(-20\) | 0 | \(-2\) | \(-4\) | 16 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| -4 and 0 | 2 | B1 for one correct value | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct curve | 3 | B2FT for 6 or 7 points accurately plotted or B1FT for 4 or 5 accurately plotted | Mark curve first. Curve must pass within ½ small square of the correct seven points FT their values from the table in (a) but accept only the correct curve Accuracy ± ½ small square radially If no points plotted mark the curve at the \(x\)-values Condone curve not having max at (0, 0) and min at (2, -4) as long as it passes through correct points Condone wobbly curve and slight feathering Do not condone straight line segments |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 3.4 | 1FT | Strict FT from graph to 1 d.p. | Do not accept answers to more than 1 d.p. or answers without a graph If curve is between two grid lines accept either value eg if crossing between 3.4 and 3.5 accept 3.4 or 3.5 If curve has more than one \(x\) value where \(y = 5\) then they must give all |
7 This grid shows a vertical line going through the point P.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \((-2, -3)\) | 1 | ||
| (ii) Plot at \((4, -1)\). | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x = -2\) | 1 | ||
1 Below are six graphs, numbered 1 to 6, that show different relationships between \(x\) and \(y\).
For each description below, write down the number of the graph that best represents the relationship between \(x\) and \(y\).
| \(x\): temperature in °C \(y\): temperature in °F, where \(y = 1.8x + 32\). | Graph ............ |
| \(x\): average speed when running 200 m \(y\): time taken to run 200 m. | Graph ............ |
| \(x\): mass of a solid object made from clay \(y\): volume of the same solid object. | Graph ............ |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2 5 1 | 3 | B1 for each | |
24 The \(x\)-coordinates of the intersections of the graphs of \(y = x^2 + ax + 7\) and \(y = 3x + b\) are the solutions to the equation \(x^2 + 8x + 13 = 0\).
Find the value of \(a\) and the value of \(b\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(a = 11\) \(b = -6\) | 3 | B2 for \(a - 3 = 8\) or \(7 - b = 13\) or better or M1 for \(x^2 + ax + 7 = 3x + b\) oe | B2 implied by one correct value e.g. \(0 = 3x + b - x^2 - ax - 7\) |
20 Here is a table of values for \(y = \dfrac{6}{x} + 2\).
| \(x\) | \(-6\) | \(-3\) | \(-2\) | \(-1\) | 1 | 2 | 3 | 6 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 1 | 0 | \(-1\) | \(-4\) | 8 | 5 | 4 | 3 |
Draw the graph of \(y = \dfrac{6}{x} + 2\) for \(-6 \leqslant x \leqslant 6\), \(x \ne 0\).
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct graph | 3 | Curves must not be joined, nor touch either axis except at \(x = -3\), B2 for 7 or 8 correct plots or B1 for 5 or 6 correct plots | Use overlay as a guide accuracy \(\pm \frac{1}{2}\) box radially Mark curve first Condone slight feathering No ruled segments If no plot, curve implies plot Condone continuation of graph after \((-1, -4)\) or before \((1, 8)\) if graph continues to decrease/increase and does not touch \(y\)-axis |
20 The diagram shows a circle, centre the origin, with the tangent to the circle at the point (4, 2).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + y^2 = 20\) | 2 | B1 for \(x^2 + y^2 = k\) | \(k\) could be \(r^2\) or \(\sqrt{20}^{\,2}\) but not 20 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [gradient =] \(\frac{2}{4}\) or \(\frac{1}{2}\) | M1 | ||
| \(m \times \textit{their}\ \frac{2}{4} = -1\) used or implied | M1 | If 0 scored SC1 for \(-\frac{4}{2}\) or \(-\frac{2}{1}\) without seeing \(\frac{2}{4}\) or \(\frac{1}{2}\) | for M1 allow \(-2 \times \frac{2}{4} = -1\) or e.g. \(-\frac{4}{2}\) after \(\frac{2}{4}\) seen or \(-\frac{2}{1}\) after \(\frac{1}{2}\) seen |
| (ii) \(y = -2x + 10\) oe | 2 | B1 for \(y = -2x + c\) seen oe or \(-2x + 10\) or \(c = 10\) | ‘c’ can be 0 but not 10 |
16 The graph of \(y = x^5 - 70x - 150\) is sketched below.
The root of the equation \(x^5 - 70x - 150 = 0\) is \(p\).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(3^5 - 70 \times 3 - 150 = -117\) and \(4^5 - 70 \times 4 - 150 = 594\) Sign change so solution between \(x = 3\) and \(x = 4\) | 3 | M2 for \(3^5 - 70 \times 3 - 150 = -117\) and \(4^5 - 70 \times 4 - 150 = 594\) or M1 for \(3^5 - 70 \times 3 - 150\) soi by −117 or \(4^5 - 70 \times 4 - 150\) soi by 594 | Accept other values of \(x\) used between 3 and 4 (see table in part (b)). For full marks, the two values need to produce a sign change or values either side of 150 if using alternative method. Examples just sufficient for third mark include: Change of sign −117 < 0 < 594 \(x = 3\) gives an answer < 0 and \(x = 4\) gives an answer > 0 Examples insufficient for third mark: so \(x\) lies between 3 and 4 |
| Alternative method After \(x^5 - 70x = 150\) seen M2 for \(3^5 - 70 \times 3 = 33\) and \(4^5 - 70 \times 4 = 744\) A1 for 33 < 150 and 744 > 150 so solution between \(x = 3\) and \(x = 4\) OR M1 for \(3^5 - 70 \times 3\) soi by 33 or \(4^5 - 70 \times 4\) soi by 744 | |||
| Alternative method SC3 for using an iterative equation that converges to a value in the range 3.25 and 3.35 and concluding statement that 3 < 3.25 to 3.35 < 4 oe or SC2 for using an iterative equation that converges to a value in the range 3.25 to 3.35 | If within part (a) candidates refer to their working in part (b), award marks for this final alternative method. | ||
| Answer | Marks | Part marks and guidance | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Examples: when \(x = 3.1\) \(y = -80[.7\ldots]\), so \(3.1 < p < 4\) when \(x = 3.5\) \(y = 130[.2\ldots]\), so \(3 < p < 3.5\) when \(x = 3.1\) \(y = -80[.7\ldots]\) and when \(x = 3.5\) \(y = 130[.2\ldots]\), so \(3.1 < p < 3.5\) | 3 | Dependent on achieving at least M2 M2 for one further value of \(y\) evaluated correctly, possibly rot or truncated to 2 or more sf, for a value of \(x\) such that \(3 < x < 4\) OR M1 for working shown to calculate one further value of \(y\) for a value of \(x\) such that \(3 < x < 4\) Alternative method After \(x^5 - 70x = 150\) seen Award marks as for the main method, but with one evaluation being < 150 and the other being > 150 Note after SC considered in part (a): if SC2 was awarded then they must use a value of \(x\) that produces a smaller interval than \(3 < x <\) their \(x\)-value in (a) or their \(x\)-value in (a) \(< x < 4\) If 0 scored, instead award SC1 or SC2 if evidence for M1 or M2 has not been credited in part (a) | Likely values: accept rot to 2+sf
Calculations in support of \(x = 3\) or \(x = 4\) need not be repeated from part (a) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
14 For each graph below, select its possible equation from this list.
| A \(y = -3\) | B \(y = x\) | C \(y = x + 1\) |
| D \(y = x^2\) | E \(x = -3\) | F \(y = x^3\) |
Write the letter of the equation beneath each graph.
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Left graph: B Right graph: F | 2 | B1 for each | Accept correct equations |
13
| \(x\) | −5 | −4.5 | −4 | −3 | −2 | −1 | 0 | 1 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | −25 | −10.1 | 0 | 9 | 8 | 0 | 5 |
[1]
[3]
Find the greatest possible value of \(k\). [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 3 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 3 | B2FT for 7 or 8 correct plots or B1FT for 5 or 6 correct plots | Use overlay as a guide Mark curve first accuracy ± 1mm, ½ small sq Condone slight feathering Condone ruled segment between \(x = -5\) and \(x = -4.5\) only If curve incorrect then mark plots accuracy ± 1mm If no plot, curve implies plot |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| –1 cao | 1 | ||
13
On the same axes, sketch the graph of \(y = 2^x\). [3]
Charlie says
| The equation of my graph is \(y = \sin x\). |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
 | 3 | B1 for correct shape and same \(y\)-intercept B1 for sketch drawn below \(y = 3^x\) for positive \(x\)-values or has shallower gradient (by eye) B1 for sketch drawn above \(y = 3^x\) for negative \(x\)-values | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) eg. \(\sin x\) should initially increase/be positive [after \(x = 0\)] | 1 | Other acceptable answers include: [The sketch is a] reflection of \(y = \sin x\) [in the \(x\)-axis]; First turning point should be a maximum; \(\sin x\) has a maximum at 90 [and then a minimum at 270]. | |
| (ii) eg. \(y = -\sin x\) | 1 | Accept any other possibly correct equations such as: \(y = -k \sin x\) or \(y = \sin(-x)\) or \(y = k\cos(x + 90)\) | |
26 Here is a table of values for \(y = \dfrac{6}{x} - 2x\).
| \(x\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 6.5 | 4 | 1 | \(-4\) | 4 | \(-1\) | \(-4\) | \(-6.5\) |
[3]
Give your answer to 1 decimal place. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Two accurate curves | 3 | B2 for 7 or 8 points plotted accurately or B1 for 5 or 6 points plotted accurately | Ignore the curve beyond points but the curve must not cross or touch the y-axis tolerance \(\pm\dfrac{1}{2}\) small square from correct points radially no excessive feathering, no ruled lines, no excessive ‘tram lines’ overlay gives guidance only |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| A correct and accurate reading from their graph | 1FT DEP. | Dep. on a graph in (a) with at least one positive solution and strict FT their curve. | If curve crosses x-axis between two grid lines accept either grid line value as correct answer If their curve has more than one positive solution accept any of their correct solutions Do not accept answers to more than 1d.p., \(\sqrt{3}\) or answers clearly rounded from this. Condone whole numbers where appropriate e.g. 2 for 2.0 Do not accept 0 as positive. |
24 The diagram shows the graph of \(y = kx - x^2 + 2\), where \(k\) is an integer.
Give your answers to 1 decimal place. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct substitution of \((x, y)\) from integer point on curve into equation leading to \(k = 3\) e.g. \((2, 4)\) \(4 = 2k - 2^2 + 2\) or \(4 = 2k - 4 + 2\) leading to \(k = 3\) with at least one correct intermediate step | 2 | M1 for correct substitution of \((x, y)\) from integer point on curve into \(y = kx - x^2 + 2\) or \(y = 3x - x^2 + 2\) OR M1 for e.g. \(x = 2\) correctly substituted in \(y = 3x - x^2 + 2\) and finding \(y = 4\) Max M1 if \(k = 3\) substituted | \((-1, -2)\) : \(-2 = -[1]k - (-1)^2 + 2\) \((1, 4)\) : \(4 = [1]k - 1^2 + 2\) \((2, 4)\) : \(4 = 2k - 2^2 + 2\) \((3, 2)\) : \(2 = 3k - 3^2 + 2\) \((4, -2)\) : \(-2 = 4k - 4^2 + 2\) Use of \((0, 2)\) scores 0 but may be replaced with another point (ie do not treat as a choice) Examples of intermediate steps \(4 = 2k - 2^2 + 2\) then \(4 = 2k - 4 + 2\) is a sufficient int step or \(4 = 2k - 2\) is a sufficient int step or \(6 = 2k\) is a sufficient int step \(3 = k\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0.4 and 2.6 | 2 | B1 for 0.4 or 2.6 or M1 for line \(y = 3\) drawn or for \((0.4, 3)\) and \((2.6, 3)\) indicated | Line to cut curve twice Treat \(x = 3\) drawn or multiple horizontal lines as choice unless \(y = 3\) clearly chosen Condone good freehand line eg circled or lines drawn down to \(x\)-axis |
19 The graph shows the velocity of a particle over the first 20 minutes of its motion.
Not to scale
Between 7 minutes and 15 minutes the velocity of the particle is \(v\) metres per minute.
The average velocity of the particle over the 20 minutes is 11.55 metres per minute.
Find the value of \(v\).
You must show your working. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 16.5 with correct working | 5 | M1 for 11.55 × 20 implied by 231 AND M2 for \(\frac{1}{2}\)(20+(15 – 7)) × \(v\) implied by \(14v\) or \(\frac{28v}{2}\) or M1 for an attempt at area e.g. one of the three areas correct e.g. \(\frac{1}{2}\) × 7 × \(v\) M1 for their 231 ÷ their 14 oe If 0 or M1 scored, instead award SC2 for answer 16.5 with no or insufficient working | “Correct working” requires evidence of at least M1 AND M2 Condone letters other than \(v\) and working could be on the diagram. in parts \(\frac{1}{2}\) × 7 × \(v\) + (15−7) × \(v\) + \(\frac{1}{2}\) ×(20−15) × \(v\) For M2 allow a proportionality argument e.g. \(\frac{1}{2}\) × 7 mins + (15 − 7) mins + \(\frac{1}{2}\) × (20 − 15) mins [= 14 mins at \(v\) m/min] their 231 is any number derived from 11.55 and implied by 0.825 and their 14 has got to be an attempt at the whole ‘area’ implied by e.g 20 condone the use of trials, we must see one at 16.5, award M2 for a trial at 16.5 and M1 for any other positive trial (see appendix) |
Trials are for values of \(v\) and evaluate the area ÷20 so here are some :
| \(v\) | average speed |
|---|---|
| 1 | 0.7 |
| 2 | 1.4 |
| 3 | 2.1 |
| 4 | 2.8 |
| 5 | 3.5 |
| 6 | 4.2 |
| 7 | 4.9 |
| 8 | 5.6 |
| 9 | 6.3 |
| 10 | 7 |
| 11 | 7.7 |
| 12 | 8.4 |
| 13 | 9.1 |
| 14 | 9.8 |
| 15 | 10.5 |
| 16 | 11.2 |
| 17 | 11.9 |
| 18 | 12.6 |
| 19 | 13.3 |
| 20 | 14 |
| 16.5 | 11.55 |
7 The diagram shows the graph of \(y = kx - x^2 + 2\), where \(k\) is an integer.
Give your answers to 1 decimal place. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct substitution of \((x, y)\) from integer point on curve into equation leading to \(k = 3\) e.g. (2, 4) \(4 = 2k - 2^2 + 2\) or \(4 = 2k - 4 + 2\) leading to \(k = 3\) with at least one correct intermediate step | 2 | M1 for correct substitution of \((x, y)\) from integer point on curve into \(y = kx - x^2 + 2\) or \(y = 3x - x^2 + 2\) OR M1 for e.g. \(x = 2\) correctly substituted in \(y = 3x - x^2 + 2\) and finding \(y = 4\) Max M1 if \(k = 3\) substituted | (-1, -2) : \(-2 = -[1]k - (-1)^2 + 2\) (1, 4) : \(4 = [1]k - 1^2 + 2\) (2, 4) : \(4 = 2k - 2^2 + 2\) (3, 2): \(2 = 3k - 3^2 + 2\) (4, -2): \(-2 = 4k - 4^2 + 2\) Use of (0, 2) scores 0 but may be replaced with another point (ie do not treat as a choice) Examples of intermediate steps \(4 = 2k - 2^2 + 2\) then \(4 = 2k - 4 + 2\) is a sufficient int step or \(4 = 2k - 2\) is a sufficient int step or \(6 = 2k\) is a sufficient int step \(3 = k\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0.4 and 2.6 | 2 | B1 for 0.4 or 2.6 or M1 for line \(y = 3\) drawn or for (0.4, 3) and (2.6, 3) indicated | Line to cut curve twice Treat \(x = 3\) drawn or multiple horizontal lines as choice unless \(y = 3\) clearly chosen Condone good freehand line eg circled or lines drawn down to \(x\)-axis |
7 Here is a table of values for \(y = \frac{6}{x} - 2x\).
| \(x\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 6.5 | 4 | 1 | \(-4\) | 4 | \(-1\) | \(-4\) | \(-6.5\) |
Give your answer to 1 decimal place. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Two accurate curves | 3 | B2 for 7 or 8 points plotted accurately or B1 for 5 or 6 points plotted accurately | Ignore the curve beyond points but the curve must not cross or touch the \(y\)-axis tolerance ± \(\frac{1}{2}\) small square from correct points radially no excessive feathering, no ruled lines, no excessive ‘tram lines’ overlay gives guidance only |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| A correct and accurate reading from their graph | 1FT DEP. | Dep. on a graph in (a) with at least one positive solution and strict FT their curve. | If curve crosses \(x\)-axis between two grid lines accept either grid line value as correct answer If their curve has more than one positive solution accept any of their correct solutions Do not accept answers to more than 1d.p., \(\sqrt{3}\) or answers clearly rounded from this. CONDONE whole numbers where appropriate e.g. 2 for 2.0 Do not accept 0 as positive. |
5 This is a one-centimetre square grid.
Find the length of the line AB. [1]
Find the coordinates of point D. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((-3, 3)\) plotted correctly | 1 | Use overlay if in doubt Take centre of mark as position If A used as plot, scores 1 if centre of A in correct position | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 5 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Ruled vertical line \(x = 2\) | 1 | Accept good freehand and dashed line If more than one vertical line and no clear choice mark the worst, ignore line at \(x = -3\) If \(y = 2\) also drawn and not rejected, treat as choice Use overlay to check if in doubt At least 2 cm long. If line more than 2 cm then must be in overlay throughout length. | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((-3, -2)\) | 2 | B1 for answer \((2, -2)\) or \((-3, -2)\) seen or plotted on grid If 0 scored, SC1 for answer \((-3, \ldots)\) or \((\ldots, -2)\) | Ignore other points |
21 The graph of \(y = \dfrac{1}{x - 2}\) is drawn on the grid for \(-2 \leqslant x \leqslant 6\).
Find the value of \(k\). [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [k = ] 0 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (b)(i) | |||
| \(y = 3x - 1\) ruled | M2 | M1 for correct freehand or short line or for \(y = 3x - k\) ruled or \(y = ax - 1\) ruled but not \(y = -1\) | For M2 must cross curve twice Accuracy ± 1mm at (0, -1) and (1, 2) |
| 0.1 to 0.3 and 2.1 to 2.3 | A2 | A1 for each After A0, SC1 for both values correct | Only award if M2 scored previously 0.15287… , 2.1804…. |
| (b)(ii) | |||
| \(1 = (3x - 1)(x - 2)\) | M1 | Allow recovery from missing brackets for M1 or ‘ = 1’ | |
| \(3x^2 - x - 6x + 2\) | B2 | For correctly expansion of brackets B1 for 3 terms correct in expansion | For B2 accept \(3x^2 - 7x + 2\) For B1 \(-7x\) counts as two terms |
| \(3x^2 - 7x + 1 = 0\) | A1 | Dep on M1B2 with no errors or omissions | |
19
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch with \(y\) – intercept indicated at 1 | 2 | B1 for correct increasing shape or any sketch with \(y\) – intercept at 1 | For 2 marks, condone curve touching but not crossing \(x\)- axis |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch through (0, 0), (180, 0) and (360, 0) indicated | 2 | B1 for three correct sections but joined and/or 180 not indicated | |
18 A circle has equation \(x^2 + y^2 = 100\).
The sketch shows the circle and two points, A and B, which lie on the circumference of the circle.
Find the exact value of the \(y\)-coordinate of point B. [3]
Find the two possible pairs of coordinates for point C.
Give your answers in exact form.
You must show your working. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (10, 0) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-\sqrt{91}\) | 3 | B2 for \(\sqrt{91}\) or −9.539[…] or −9.54 or B1 for 9.539[…] or 9.54 or M1 for \(9 + y^2 = 100\) oe | ± answers treat as choice |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((\sqrt{2},\ 7\sqrt{2})\) and \((-\sqrt{2},\ -7\sqrt{2})\) with correct working | 5 | B4 for \((\sqrt{2},\ 7\sqrt{2})\) with correct working or \((-\sqrt{2},\ -7\sqrt{2})\) with correct working or M2 for \(x^2 + (7x)^2 = 100\) oe A1 for [\(x = \pm\)] \(\sqrt{2}\) or M1 for 7\(x\) seen If 0 or 1 or 2 scored, instead award SC3 for answer \((\sqrt{2},\ 7\sqrt{2})\) and \((-\sqrt{2},\ -7\sqrt{2})\) with no working or insufficient working If 0 or M1 scored, instead award SC2 for answer \((\sqrt{2},\ 7\sqrt{2})\) or \((-\sqrt{2},\ -7\sqrt{2})\) or for \(\left(\frac{5\sqrt{2}}{2}, \frac{35\sqrt{2}}{2}\right)\) and \(\left(-\frac{5\sqrt{2}}{2}, -\frac{35\sqrt{2}}{2}\right)\) with no working or insufficient working If 0 scored, instead award SC1 for [\(x = \pm\)] \(\sqrt{2}\) or for \(\left(\frac{5\sqrt{2}}{2}, \frac{35\sqrt{2}}{2}\right)\) or \(\left(-\frac{5\sqrt{2}}{2}, -\frac{35\sqrt{2}}{2}\right)\) with no working | “Correct working” requires evidence of at least M2 Allow \(\sqrt{98}\) for \(7\sqrt{2}\) Condone for M1 if seen as \(x^2 + 7x^2 = 100\) |
16 The graph shows the distance travelled by a particle over the first 20 seconds of its motion.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 36 ÷ 20 | 1 | Accept \(\frac{36}{20}\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Tangent drawn to graph at \(t\) = 10 | B1 | Tangent - mark intention but no daylight | |
| 1[.0] to 1.5 | B2 | Dep on tangent or close attempt M1dep for rise/run with values substituted | |
16 A ball of dough is left to rise before it is baked.
The graph shows the height of the ball of dough over the first 20 minutes.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0.2 or 0.19 to 0.20 | 2 | M1 for \(\dfrac{3.8 \text{ to } 4}{20}\) oe If 0 scored, SC1 for triangle with incorrect values marked and their \(\dfrac{\text{rise}}{\text{run}}\) or \(\dfrac{\text{rise}}{\text{run}}\) with answer in range \(0.18 \leqslant g \lt 0.19\) | e.g. \(\dfrac{1.9 \text{ to } 2}{10}\) e.g. \(\dfrac{13}{20}\) with no triangle scores 0 e.g. \(\dfrac{13}{20}\) with 13 and 20 marked on triangle scores SC1 e.g \(\dfrac{2.1}{11.6}\) and answer 0.181... SC1 |
| cm per minute or cm/min | 1 | Mark for units still available even if wrong gradient | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(5 \times\) their gradient | M1 | FT their gradient which is a number | Alternative method M2 for \(25 \times (0.19\) to \(0.2) + 9\) with answer that rounds to 14 or M1 for 25 × their gradient + 9 with answer that does not round to 14 |
| 13 + their 1 giving answer that rounds to 14 | M1 | ||
| (ii) Continues to rise at same rate oe or Rises 0.2 cm every minute oe | 1 | FT their gradient if figures quoted | Must imply constant increase [in height] or continued pattern If figures used must be correct or from their gradient Accept The pattern continues oe The dough rises consistently oe The dough will rise I cm in 5 minutes |
12 The graph shows the speed of a car during the first 40 seconds of a journey.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 12.37[5] or 12.38 or 12.4 all oe and with correct working | 5 | M3 for \(\frac{(15 + 40) \times 18}{2}\) or M2 for attempt at total area with one reading or one calculation error or M1 for one area calculation AND M1dep for their 495 ÷ 40 If 0 or 1 scored, instead award SC2 for answer 12.37[5] or 12.38 or 12.4 all oe with no working or insufficient working If 0 scored SC1 for 495 with no working or insufficient working | “Correct working” requires evidence of at least M3 Areas may be found in stages e.g. M3 for \(\left(\frac{1}{2} \times 10 \times 18\right) + (15 \times 18) + \left(\frac{1}{2} \times 15 \times 18\right)\) implied by 90 + 270 + 135 e.g. use of 16 rather than 18 throughout is one error but both triangles incorrect is two errors e.g. one of the above soi by 90, 270 or 135 Dep on at least M1 |
11
Show clearly the value of any intercepts.
[2]
Show clearly the value of any intercepts.
[3]
[1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch with -2 indicated as \(y\)-intercept | 2 | B1 for a horizontal line below the \(x\) axis | Condone good freehand Line must be at least 10mm |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch with -3 indicated as \(y\)-intercept and 3 indicated as \(x\)-intercept | 3 | B1 for line with positive gradient and B1 DEP for a line with -3 indicated as \(y\)-intercept or for a line with 3 indicated as \(x\)-intercept | Condone good freehand Line must be at least 10mm |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct sketch | 1 | curve, correct shape | Condone intention to curve at point (0,0) and length of curve. |
4 Point A is shown on this grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (4, 0) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Point plotted at (3, -3) | 1 | Accept unlabelled if no others | |
23 This is a sketch of the graph of \(y = x^2 - 10x + 16\).
Not to scale
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 16 | 1 | Do not accept coordinates Condone \(y = 16\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 5 | 1 | Do not accept coordinates Condone \(x = 5\) | |
22
| \(x\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|---|
| \(y\) | \(4\) | \(-1\) | \(-4\) | \(-4\) | \(-1\) | \(4\) |
[2]
[3]
Give your answers to 1 decimal place. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 11 \(-5\) | 2 | B1 for each | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 3 | B2ft for 7 or 8 points accurately plotted or B1ft for 5 or 6 points accurately plotted | Tolerance \(\pm\tfrac{1}{2}\) small square radially |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-2.3\) or \(-2.2\) 2.2 or 2.3 | 2 FT | Strict FT B1 for either FT their graph | If curve is between 2 grid lines accept either value as correct answer Do not accept answers to more than 1d.p. Do not allow \(\pm\sqrt{5}\) or answers with no graph |
19 The graph below shows a circle with centre (0, 0) and equation \(x^2 + y^2 = 169\).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(([-]12)^2 + 5^2\) | 1 | Accept equivalent reasoning e.g. For first mark \(13^2 - ([-]12)^2\) | If \(-12^2 + 5^2\) do not allow first mark |
| 144 + 25 = 169 or \(\sqrt{144 + 25} = 13\) and \(\sqrt{169} = 13\) | 1 | e.g. For second mark 169 – 144 = 25 \(\sqrt{25} = 5\) | If \(\sqrt{169}\) evaluated then it must be 13 For 2 marks there must be no errors leading to the answer |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = \frac{12}{5}x + \frac{169}{5}\) final answer | 5 | B4 for answer \(\frac{12}{5}x + \frac{169}{5}\) oe (no \(y =\) ) or correct 3 – term answer in different form e.g. \(5y - 12x = 169\) OR M1 for \(-\frac{5}{12}\) oe and M1 [tangent gradient = ] \(-1 \div \textit{their} -\frac{5}{12}\) oe | Accept e.g. \(y = 2.4x + 33.8\) \(\frac{12}{5}\) oe implies M1M1 unless contradicted |
| AND M1dep for \(y - 5 = \textit{their } \frac{12}{5}(x - (-12))\) oe | Dep on at least M1 oe e.g. \(5 = \textit{their } \frac{12}{5} \times -12 + c\) Do not allow M1 for e.g. grad \(-\frac{5}{12}\) used if the gradient is then changed subsequently to e.g. \(\frac{12}{5}\) | ||
| or M1dep for \(y = \textit{their } \frac{12}{5}x + c\) | Dep on at least M1 Allow ‘\(c\)’ or any value including 0 Answer \(y = \frac{12}{5}x\ [+ c]\) oe implies M1M1M1 | ||
18 The diagram shows a circle, centre the origin.
Write down the equation of the circle. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + y^2 = 3^2\) oe final answer | 2 | B1 for \(x^2 + y^2 = k\) (not 9) | \(k\) could be \(r^2\) |
16 Frankie sketches this quadratic graph.
Not to scale
Frankie says
The \(y\)-intercept is 15.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| E.g. [\(y =\)] \(-(x + 3)(x - 5)\) AND either [\(y = -x^2 - 3x + 5x\)] + 15 or [Constant/\(y\)-intercept =] \(-3 \times -5 = 15\) | 3 | M2 for [\(y =\)] \(-(x + 3)(x - 5)\) or M1 for \(k(x + 3)(x - 5)\) | See Appendix For full marks, all shown parts of their expansion must be correct Accept \(k\) implied as 1 |
| Alternative method using simultaneous equations \(y = -x^2 + bx + c\) \(0 = -(-3)^2 - 3b + c\) and \(0 = -5^2 + 5b + c\) [\(b = 2\)] \(c = 15\) | M2 for \(0 = -(-3)^2 - 3b + c\) and \(0 = -5^2 + 5b + c\) or M1 for \(y = -x^2 + bx + c\) or for \(0 = (-3)^2 - 3b + c\) and \(0 = 5^2 + 5b + c\) | ||
| Response | Judgement | Mark |
|---|---|---|
| \((x + 3)(x - 5)\) \(= x^2 - 2x - 15\) But the quadratic is upside down so it will be \(-x^2 + 2x + 15\) [there then was an arrow from the +15 to the intercept] | The response doesn’t quite fit the scheme but is thought worthy of full marks This line on its own scores M1 But this line makes it equivalent to the M2 and it also a correct expansion with + 15 The linking of +15 to the intercept is an added bonus |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| The equation could be a multiple E.g. [\(y =\)] \(-k(x + 3)(x - 5)\) So the intercept could be a multiple of 15 | 2 | B1 for giving an example in the form \(-k(x + 3)(x - 5)\) (where \(k > 0\), \(k \ne 1\), \(k\) need not be an integer) or stating that the intercept could be a multiple of 15 | Allow full or part marks for a fully correct algebraic example E.g. [\(y =\)] \(-2(x + 3)(x - 5)\) would have a \(y\)-intercept of 30 |
12 An athlete goes for a training run.
The graph shows their speed as they run.
Not to scale
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 3 with correct working | 5 | “correct working” requires at least M3 | |
| M3 for method for three areas correct or for 3600 or 800 and 2400 and 400 nfww or M2 for method for two areas correct (method towards two from 800, 2400, 400) or M1 for method for one area correct (method towards 800 or 2400 or 400) or for 2400 seen AND M1dep for their 3600 ÷ 1200, dep on at least M2 | Treat as three separate areas, so trapezium from 0 to 1200 is three areas, trapezium from 0 to 1000 is 2 areas, trapezium from 400 to 1200 is 2 areas. For M marks, mark method not accuracy, so M1 for \(\frac{400 \times 4}{2}\) even if not evaluated as 800. | ||
| If 0 or 1 scored instead award SC2 for 3 with no or insufficient working but not from wrong working | |||
8 Here is a table of values for \(y = x^2 + 2x - 2\).
| \(x\) | −4 | −3 | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|---|
| \(y\) | 6 | 1 | −2 | −3 | −2 | 1 | 6 |
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| accurate curve | 3 | B2 for 6 or 7 points accurately plotted or B1 for 4 or 5 points accurately plotted | tolerance \(\pm \frac{1}{2}\) small square radially for curve and points, condone a wobbly curve and slight feathering or tram lines in no more than 3 sections but no ruled lines |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x = -1\) oe | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −2.8 or −2.7 0.7 or 0.8 | 2 | B1 for either If 0 or 1 scored FT their curve for 1 or 2 marks or SC1 for an answer in each of −2.8 to −2.7 and 0.7 to 0.8 | tolerance \(\pm \frac{1}{2}\) small square radially |
21
| \(x\) | \(-3\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \(y\) | 1 | \(-2\) | \(-2\) | 1 | 6 |
[2]
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 6 and \(-3\) | 2 | B1 for each | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 3 | B2FT for 6 or 7 correct plots B1FT for 4 or 5 correct plots | Tolerance \(\pm 1\) small square for plotting and curve through correct points. Condone slight feathering – must not be ruled If large blob for plot, check centre of blob |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-2.3\) to \(-2.2\) and 2.2 to 2.3 | 2FT | Strict FT B1 for either FT their graph | Tolerance \(\pm\tfrac{1}{2}\) a small square Do not allow exact answers or answers with no graph Do not FT from a straight line graph If more than 2 intersections, B1 for each correct intersection on the answer line. If more than 2 answers, mark the worst. |
19 The point \((5, 7\sqrt{2})\) lies on the circumference of a circle, centre (0, 0).
Find the equation of the circle. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + y^2 = 123\) final answer | 4 | B2 for 123 or B1 for 98 or M1 for \(5^2 + (7\sqrt{2})^2\) oe | |
| B1 for \(x^2 + y^2 = k\) as final answer | Accept ‘= \(k\)’ or numeric value where \(k > 0\) | ||
18
| \(y = x\) | \(y = x^2\) | \(y = \dfrac{1}{x}\) |
| \(y = \sin x\) | \(y = \cos x\) | \(y = \tan x\) |
| \(y = 3^x\) | \(y = \left(\dfrac{1}{3}\right)^x\) |
On the axes below, sketch the graphs of
[1]
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(\tan x\) | 1 | ||
| (ii) \(3^x\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) Graph of \(y = -x^3\) | 1 | Mark intent | |
| (ii) \(y = x^3\) translated vertically 8 down | 1 | ||
| \(y\)-intercept −8 | 1 | Intercepts must be marked on graph and accept given as coordinates | |
| \(x\)-intercept 2 | 1 | ||
15 For each graph below, select its possible equation from this list.
| A \(y = x^3\) | B \(y = -2\) | C \(y = -x\) |
| D \(x = -2\) | E \(y = x^2\) | F \(y = 2x + 1\) |
Write the letter of the equation beneath each graph.
[4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| E C B F | 4 | B1 for each | Accept correct equations |
8 Point G is shown on this grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((3, -4)\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| H plotted at \((-2, 4)\) | 1 | Must be labelled if more than 1 point plotted | |
8
| \(x\) | −6 | −3 | −2 | −1 | 1 | 2 | 3 | 6 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | −1 | −2 | −6 | 6 | 3 | 2 | 1 |
[1]
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −3 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct graph | 3 | Curves must not be joined or touch either axis | Mark in 50% zoom, use overlay, mark curve first For 3 marks, curve must pass through or touch circles on overlay Condone slight feathering, no ruled segments |
| B2FT for 7 or 8 correct plots Or B1FT for 5 or 6 correct plots | If curve incorrect, mark the plots use the overlay, plots must lie inside or touch circles. If large blob for plot, check centre of blob | ||
22
| \(x\) | \({}^{-}2\) | \({}^{-}1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 6 | 0 | 2 | 6 |
[2]
On the same grid, use part (a) to draw the graph of \(y = x^2 - x\) for values of x from \({}^{-}2\) to 3.
[3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2 0 | 2 | B1 for each | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
Correct curve | 3 | B2FT for all points correctly plotted or B1FT for 4 or 5 points correctly plotted | FT their values from the table in (a) for points but accept only the correct curve. Accuracy ± half small square Correct curve must have at least one square of daylight below \(x\)-axis at minimum point and not intended straight |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \({}^{-}[0].4\) and 2.4 | 2 | Correct answer or FT their graph for both B1 for each | -0.45 to -0.35 and 2.35 to 2.45 FT from their line with half square accuracy (may be straight) |
20 A pattern is made from four congruent right-angled triangles.
Not to scale
The line AC is parallel to the x-axis.
The point A has coordinates (4, 9) and the point B has coordinates (17, 12).
Work out the coordinates of point C and point D. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| C (24, 9) D (10, 2) | 5 | B4 for three correct ordinates or B3 for two correct ordinates or B2 for one correct ordinate from 24, 10, 2 or for longer length of triangle = 7 soi or B1 for 9 as \(y\)-coordinate for C or for shorter length of triangle = 3 soi OR M1 for long = 17 – 4 – 2 × their short oe A1FT for C (4 + 2 × their short + 2 × their long, 9)) A1FT for D (4 + 2 × their short, 9 – their long) | For part marks, check ordinates first (may be on diagram if answer line blank) .If B2 or fewer check alt method and mark to candidates’ advantage B4, B3, B2, B1 May be on diagram For M1 and A1FT, their short and their long needs to be clear in working or on diagram |
19
Calculate the distance travelled by the vehicle during the 40 seconds. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 975 | 3 | M2 for \((\frac{40 + 25}{2}) \times 30\) oe or \(\frac{1}{2} \times 15 \times 30 + 25 \times 30\) or M1 for one relevant area e.g. \(\frac{1}{2} \times 15 \times 30\) or 25 × 30 or 40 × 30 | Alternative: \(40 \times 30 - \frac{1}{2} \times 15 \times 30\) for M2 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 9 | 2 | M1 for \(\frac{24 - 6}{4 - 2}\) oe | For M1 condone one error in the figures |
| (ii) tangent drawn at x = 4 | 1 | Tangent should not cross the curve and should touch at 4, condone slight daylight between tangent and curve | |
| 12 nfww and dep on tangent drawn | 3 | M1 for using two points on their line M1 for use of \(\frac{\textit{difference in } y}{\textit{difference in } x}\) | accept range of 11 to 13 for answer which should be checked from their tangent |
19
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x - 5)^2 - 3\) final answer | 3 | B1 for \((x - 5)^2\) B2 FT for – 3 or M1 for \(22 - (-5)^2\) oe | M1 FT \(22 - (\textit{their}\ {-5})^2\) oe |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct sketch with TP at (5, −3) in 4th quadrant and \(y\) – intercept at (0, 22) | 4 | FT their (a) for TP M1 for U shaped curve B2FTdep their (a)for TP at (5, –3) in correct quadrant or B1FTdep for turning point at (\(k\), –3) or (5, \(k\)) soi FT for B2 or B1 dep on answer of form \((x - a)^2 - b\) in part (a), \(a, b \ne 0\) B1 for \(y\) – intercept at 22 indicated | Be generous for the U shape condone broken line Values for \(y\) - intercept and TP must be shown but could be marked on axes. Mark intention Accept turning point = (5, –3)FT written in working provided no contradiction on sketch If point (5, –3)FT only plotted on graph in 4th quadrant and no sketch then B2 only |
19 The graph of \(y = 2x^2 + 3x - 9\) is drawn below.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [x =] -3, 1.5 | 2 | B1 for 1 correct | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [a =] 2 [b =] -5 | 2 | B1 for each or for \(2x - 5\) seen | |
| (ii) \(y = 2x - 5\) or FT \(y = \textit{their}\ ax + \textit{their}\ b\) ruled on grid | M2 | M2 and M1 apply to \(y = 2x - 5\) or FT \(y = \textit{their}\ ax + \textit{their}\ b\) M1 for ‘correct’ \(y\)-intercept or for ‘correct’ gradient or for freehand or broken ‘correct’ line or for at least 3 ‘correct’ plots and no ‘incorrect’ plots | For M2 line must cross curve For M2 and M1, accuracy 1 small square at y-intercept (extended if necessary provided it fits on the grid) and gradient ±1 small square vertically for a run of 1 unit horizontally Do not FT if a = 0 or b = 0 |
| 1.1 to 1.3 and -1.8 to -1.6 | A1 | Only award if M2 scored | |
14
[2]
[2]
Make two comments about the accuracy of his sketch. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(x = 3\) sketched correctly with 3 indicated on \(x\)-axis as \(x\) – intercept | 2 | M1 for a vertical line or a dotted vertical line passing through 3 | Condone good freehand |
| (ii) \(y = x^2 + 1\) sketched correctly with 1 indicated as \(y\)-intercept | 2 | M1 for correct shape or \(y\)-intercept at 1 but not \(y = 1\) | Condone good freehand |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| It should not touch the axes oe | 1 | Accept responses on the graph | |
| It should also have a curve in the 3rd quadrant oe | 1 | ||
3 The graph shows the cost of electricity with Company A.
If Company B’s cost of electricity was plotted on the same axes as Company A’s cost of electricity, which line would be steeper?
Explain how you know. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 55.5[0] to 57.[0] nfww | 3 | M1 for 1 kwH costs £0.125 oe soi M1 for their 0.125 × 450 or cost × \(\frac{450}{\textit{their equivalent electricity use}}\) OR M1 for finding costs of electricity that sum to 450 kwH M1 for addition of these costs | eg 225 kwH costs £28.125 ± £0.5[0] 200 kwH costs £25 ± £0.5[0] 150 kwH costs £18.75 ± £0.5[0] 100 kwH costs £12.5 ± £0.5[0] 50 kwH costs £6.25 ± £0.5[0] eg \(25 \times \frac{450}{200}\) eg award M2 for 4 × (£12.5 ± £0.5[0]) + (£6.25 ± £0.5[0]) For the first M1 mark, costs must be from graph within ±½ small square ie ±£0.5[0] |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Method 1: By performing a calculation for A and comparing with B: | Method 2: By performing a calculation for B and comparing with A: | ||
| 12.5 or 0.125 or “cost ÷ usage” using a reading from the graph or their (a) ÷ 450 | 1 | 14.3 × their chosen usage soi or [0].143 × their chosen usage soi May be implied by line on graph | Mark method not accuracy for first two marks e.g. 6.75 ÷ 50 = 0.135 but not 0.135 with no working |
| 14.3 or 0.143 stated or comparison against it clearly implied in a statement | 1dep | Cost in p or £ from graph for their chosen usage stated | Dep on first mark scored. Costs for A and B may be in different units. e.g. 1kwH costs 12.5p which is less than Company B’s charge |
| “B” with 12.5 and 14.3 or “B” with 0.125 and 0.143 or “B” with 12.5 or 0.125 and a comparison against 14.3 or 0.143 clearly implied | 1 | “B” with both costs correct and in same units | No follow through e.g. B because 1kwH costs 12.5p which is less than Company B’s charge |
21 In this question all units are in cm.
A circle has equation \(x^2 + y^2 = 36\).
Work out the length AB. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 6 | 1 | ||
| (0, 0) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 10 | 4 | B3 for 5 and –5 Or M2 for \(x^2 = 25\) Or M1 for \(x^2 + (\sqrt{11})^2 = 36\) | Accept \(a\) or \(b\) for \(x\) |
20
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x - 3)^2 + 2\) final answer | 3 | B1 for \((x - 3)^2\) B2 FT for 2 or M1 for \(11 - (\textit{their} -3)^2\) If 0 scored, SC2 for final answer \((x - 3) + 2\) | FT can be implied, check \(11 - (\textit{their} -3)^2\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| U shaped parabola with minimum value indicated in 1st quadrant at (3, 2) | 3 | FT U-shaped parabola with turning point at their (– \(a\), \(b\)) from part (a) dep on answer of form \((x + a)^2 + b\) where \(a \ne -3\) and/or \(b \ne 2\) B1 for U shape curve B1 for turning point at (3, \(k\)) or FT for turning point at (–\(a\), \(k\)) dep on answer of form \((x + a)^2 + b\) in part (a) B1 for turning point at (\(k\), 2) or FT for turning point at (\(k\), \(b\)) dep on answer of form \((x + a)^2 + b\) in part (a) | Be generous for the U shape condone broken line Sketch takes priority when marking Do not allow all 3 marks if (3, 2) indicated on U shaped parabola but TP on sketch is in wrong quadrant Values must be shown but could be marked on axes. Mark intention Accept turning point = (3, 2) written in working provided no contradiction on sketch If point (3, 2) only plotted on graph and no sketch then B0B1B1 |
(corrected from the printed mark scheme: the form of the part (a) answer is printed as \((x + a)^2 - b\) in three places; it is \((x + a)^2 + b\), matching the turning point (–\(a\), \(b\)) and the answer \((x - 3)^2 + 2\))
19 The graph shows the distance travelled by a particle over 8 seconds.
Estimate the speed of the particle at 5 seconds. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 1.0 to 1.4 dep. on a tangent drawn | 4 | B3 for an answer within the given range with a slightly inaccurate tangent, i.e. touches curve in range \(t\) = 4.5 to 5.5 but not at \(t\) = 5 or an answer of 1.3 using a symmetric chord about \(t\) = 5 e.g. from (4, 2) to (6, 4.6). OR B2 for an answer from using a symmetric chord about \(t\) = 5, e.g. from (4, 2) to (6, 4.6), with at most one error OR M1 for an attempt to draw a tangent touching curve at \(t\) = 5 and M2 for their distance ÷ their time e.g. (7.4 – 1.2) ÷ (8 – 3) or M1 for an attempt at their distance ÷ their time with one error or a correct attempt to find the gradient of their line which is not a tangent e.g. line from (0, 0) to (5, 3) which gives the average speed of \(\frac{3}{5}\) or 0.6 | Note: 5 ÷ 3 = 1.66… scores 0 |
17 Sketch the graph of \(y = 3^x\).
Give the value of the \(y\)-intercept.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
 | 2 | B1 for either an acceptable curve and no/incorrect \(y\) intercept marked or any curve with 1 marked at the \(y\) intercept | |
13
[2]
[2]
Explain the error in her answer. [1]
Make two comments about the accuracy of his sketch. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(y = 2\) sketched correctly with 2 indicated on \(y\)-axis as y-intercept | 2 | M1 for a horizontal line | Condone good freehand |
| (ii) \(y = x + 1\) sketched correctly with 1 indicated as \(y\)-intercept | 2 | M1 for any straight line with positive gradient or for \(y\)- intercept at 1 | Condone good freehand |
| (iii) \(y\)-value where they cross has to be 2 oe | 1 | Isw extra statements. Accept eg (2, 3) is not on \(y = 2\) as the y coordinate is 3 they cross at (1, 2) they cross when \(x = 1\) See AG | |
Exemplar responses for Q13aiii
| Response | Mark | |
|---|---|---|
| 1 | y = 2 won’t go higher than 2 to cross at 3 (implies y values) | 1 |
| 2 | Her answer wouldn’t be accurate because y = 2 so wouldn’t cross the point of 3 | 1 |
| 3 | the first graph only equals to two ( it shows y = 2 ) the x = 1 | 1 |
| 4 | y = 2 only goes through y = 2 not y = 3 so not (2,3) | 1 |
| 5 | Because it needs to go through x = 1 (can BOD imply the x coordinate is 1) | BOD 1 |
| 6 | y = 2 will not show a cross at (2,3) due to not being 1 and being y = 2 | 0 |
| 7 | Because they are not on the same graph – they are both on different graphs | 0 |
| 8 | Y is in y axis meanwhile y = x+1 does not = to (2,3) coordinates | 0 |
| 9 | she added 2+1 = 3 and used 2 for x (if they had also stated ‘instead of y’ it would score 1) | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Should go through (0, 0) oe Should be a curve oe No numbers on axis/axes oe It is symmetrical oe | 2 | B1 for each to a max of 2 | If more than two comments, mark the best two See AG |
Exemplar responses for Q13b
| Response | Mark | |
|---|---|---|
| 1 | Its meant to be a smooth curve | 1 |
| 2 | He hasn’t placed the numbers on the graph | 1 |
| 3 | The curve is not a perfect curve. | 1 |
| 4 | The curve isn’t accurate as it has edges. | 1 |
| 5 | it doesn’t flow, has used a ruler to connect his points. | 1 |
| 6 | His lines are not curvy they are crooked | 1 |
| 7 | It’s not an actual curve | 1 |
| 8 | “central point isn’t on O” | 1 |
| 9 | The bottom should start at zero as there is no numbers for it to pass through | 1 |
| 10 | It doesn’t touch the centre of the graph (‘origin’ implied) | 1 |
| 11 | it is sketched with a ruler. | 1 |
| 12 | He has used a ruler so it will affect the reliability (this is ok to imply ‘it is not a curve’) | 1 |
| 13 | The line should be drawn freehand not with a ruler (BOD implies a curve) | BOD 1 |
| 14 | There are pointy sides with each coordinate (‘pointy sides’ implies ‘not a curve’) | BOD 1 |
| 15 | The line he makes was meant to be round and his was a straight line (BOD ‘straight line’ rather than ‘ straight lines’) | BOD 1 |
| 16 | does not fully meet the x axis and lines are not fully straight (they are) | 1, 0 |
| 17 | The curved line (U) is not touching the x-axis | 0, 1 |
| 18 | The curve is symmetrical (‘it is not a curve but it is symmetrical’ scores 1,1) | 0, 1 |
| 19 | the line is too blacky and show a stiff movement of the graph (‘stiff movement’ unclear) | 0, 0 |
| 20 | The bottom of the arch needs to be on the line (‘arch’ is wrong way up, ‘on the line’ could be x axis or y axis therefore choice) | 0, 0 |
| 21 | Each side is the same (the sides are ‘mirror images’ would score) | 0 |
| 22 | It is a curve (it’s not) | 0 |
| 23 | No marking on the graph (‘marking’ not enough to imply scale/numbers) | 0 |
| 24 | This sketch is not one line they have not done one single line what flows (‘single line’ will not score, ‘single curve’ will) | 0 |
| 25 | Should be a smooth line (‘smooth line’ does not imply curve) | 0 |
| 26 | The sketch does not start at the point O | 0 |
| 27 | He was accurate with the shape of the graph as it should look like a ‘U’ (‘wasn’t accurate’ in this statement would be fine) | 0 |
| 28 | He has joined each point together (‘with straight lines’ missing) | 0 |
| 29 | connected the points instead of a fluient line (‘with straight lines’ missing) | 0 |
| 30 | It doesn’t match y = x squared (needs to state why) | 0 |
| 31 | It is not going fully around, it stops | 0 |
7
| \(x\) | –1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| \(y\) | 1 | –2 | 6 |
Give your answers to 1 decimal place. [2]
| Answer | Marks | Part marks and guidance | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Completes table with
| 2 | B1 for at least 2 correct values | |||||||||||||||||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 3 | B2 for 6 or 7 points correctly plotted FT their table or B1 for 4 or 5 points correctly plotted FT their table | Tolerance ±2 mm for plotting and curve through the correct points. Strict marking of ‘smooth curve’ – must not be ruled or ‘feathered’ |
| Answer | Marks | Part marks and guidance | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Straight line passing through (0, –6) and (3, 0) | 3 | M2 for a correct unruled line or a straight line of gradient 2 or a straight line passing through (0, –6) or two correct points correctly stated or plotted or M1 for one correct point stated or plotted |
| ||||||||||||||||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 1.6 and 4.4 | 2FT | B1 for each or both answers as decimals to a greater accuracy Correct answer or FT their straight line | Tolerance ±1 mm. Do not allow exact answers \(3 + \sqrt{2}\) and \(3 - \sqrt{2}\) |
19 The graph of \(y = x^3 - x^2 - 2\) is drawn on the grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 1.7 cao | 1 | Mark at most accurate i.e. do not allow 1.65 = 1.7 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [\(a\) =] –5 [\(b\) =] 4 | 2 | B1 for each or for \(-5x + 4\) seen | |
| (ii) Correct line ruled on grid | M2 | Strict FT \(y =\) their \(ax\) + their \(b\) M1 for correct \(y\)–intercept (FT their \(b\)) or correct gradient (FT their \(a\)) or for freehand or broken ‘correct’ line FT \(y =\) their \(ax\) + their \(b\) or for at least 3 correct plots and no incorrect plots FT \(y =\) their \(ax\) + their \(b\) | For M2 line must cross curve For M2 or M1, accuracy ± 1 small square at \(y\) – intercept (extend line if necessary provided it fits on the grid) and gradient ± 1 small square vertically for a run of 1 unit horizontally Do not follow through if \(a = 0\) and/or \(b = 0\) |
| 1.1 or 1.2 | A1 | Only award if M2 scored previously | |
19 The point (–5, 2) lies on the circumference of a circle, centre (0, 0).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + y^2 = 29\) oe | 4 | B2 for 29 or \(\sqrt{29}\) or 5.38(5…) to 5.39 or M1 for \(2^2 + 5^2\) or \(\sqrt{2^2 + 5^2}\) or \(2^2 + (-5)^2\) or \(\sqrt{2^2 + (-5)^2}\) AND B1 for \(x^2 + y^2 = k\) where \(k\) is a number \(\gt 0\) or \(x^2 + y^2 = r^2\) | Condone poor use of or missing brackets for M1 e.g. \(-5^2 + 2^2\) or \(2^2 + -5^2\) earns M1, but \(2^2 - 5^2\) does NOT earn M1 Condone other letters instead of \(r\), except \(x\) and \(y\). |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2.5 or \(\frac{5}{2}\) oe | 2 | M1 for \(-\dfrac{2}{5}\) oe or –0.4 seen or use of \(m_1 m_2 = -1\) with their radius gradient | M1 for [\(y =\)] \(\frac{5}{2}x\) [\(+\ c\)] oe Condone \(-\frac{2}{5}x\) seen for M1 |
17
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(y\) | –2 | 2 | 6 | 14 |
[1]
[2]
Find the equation of this straight line. [3]
| Answer | Marks | Part marks and guidance |
|---|---|---|
| 10 | 1 |
| Answer | Marks | Part marks and guidance |
|---|---|---|
| Correct continuous ruled line from \(x = 0\) to \(x = 4\) | 2 | Tolerance 2 mm by eye B1 for 2 points correctly plotted FT their table |
| Answer | Marks | Part marks and guidance |
|---|---|---|
| \(y = -3x + 11\) oe | 3 | B2 for \(-3x + 11\) or \(y = mx + 11\) (\(m \ne -3\) or 0, \(m\) does not need to be numeric) or \(y = -3x + c\), (\(c\) can be 0, \(c\) does not need to be numeric) OR M1 for \(\dfrac{11 - -1}{0 - 4}\) oe OR B1 for \(mx + 11\), (where \(m \ne -3\) or 0, \(m\) does not need to be numeric) or \(-3x + c\) (where \(c \ne 11\), \(c\) can be 0, \(c\) does not need to be numeric). |
17
[4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x + 4)^2 - 13\) final answer | 3 | B1 for \((x + 4)^2\) B2FT for [+] 3 – their (a)\(^2\) after \((x +\) their \(a)^2\) correctly evaluated or B1 for [+] 3 – their \(a^2\) shown If 0 scored, SC2 for final answer \((x + 4) - 13\) | FT can be implied eg \((x + 2)^2 - 1\) gets B2FT |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| U shaped parabola with minimum value indicated in 3rd quadrant at (–4, –13) and intercepts positive \(y\) – axis at 3 | 4 | FT U-shaped parabola with turning point at their \((-a, -b)\) from part (a) dep on answer of form \((x + a)^2 - b\) where \(a \ne 4\) and/or \(b \ne 13\) | Be generous for the U shape condone broken line TP values must be shown but could be marked on axes. Mark intention Sketch takes priority when marking Accept turning point = (– 4, –13) written in working or in table provided no contradiction on sketch |
| B1 for U shape curve B1 for their curve or line intercepts positive \(y\) – axis at 3 | Must be stated on graph, 3 or (0, 3) Do not accept just in a table | ||
| B1 for turning point at \((-4, k)\) or FT for turning point at \((-a, k)\) dep on answer of form \((x + a)^2 - b\) in part (a) B1 for turning point at \((k, -13)\) or FT for turning point at \((k, -b)\) dep on answer of form \((x + a)^2 - b\) in part (a) | If point (– 4, –13) only plotted on graph and no sketch then can score these final 2 marks If more than one graph drawn treat as choice | ||
14 The graph shows the speed of a train during the first 60 seconds of motion.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 12 | 1 | condone “greater than 12” or “\(\gt 12\)” | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Acceleration is constant oe | 1 | see appendix for acceptable answers | |
| Response | Mark |
|---|---|
| Acceleration is constant | 1 |
| [Speed] increases by the same amount each time | 1 |
| It accelerates most over the first 15 seconds | 1 BOD |
| Increasing [its speed] | 1 BOD |
| steadily increasing pace | 1 |
| It is steady | 0 |
| [Speed] is constant | 0 |
| Increasing acceleration | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 15.8[3…] or 16 if no evidence of incorrect working | 5 | M1 for attempt to find an area under the graph (eg \(\frac{1}{2} \times 15 \times 20\)) soi by e.g. 150, 700, 100 or 950 M2 for \(\frac{1}{2}[(50 - 15) + 60] \times 20\) oe or 950 or M1 for a total area attempt with one error M1dep for their 950 ÷ 60, dep. on attempt at total area under graph | |
950 scores 3 marks.
13 Shirley is asked to sketch a graph of \(y = 5^x\) for \(x \geqslant 0\).
She produces the following.
The graph has two errors.
How should they be corrected? [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| It should be a curve with increasing gradient oe It should go through (0, 1) | 1 1 | Accept alternate forms e.g. correct sketch See AG Incorrect statements treat as choice. Incomplete statements ignore | |
| Response | Mark | |
|---|---|---|
| 1 | It should be a curve It should start at 1 not 0 | 0 0 |
| 2 | It should start at 1 on the \(y\) – axis It should be an exponential curve | 1 0 |
| 3 | Correct sketch shown of curve passing through 1 on \(y\) – axis | 2 |
| 4 | It should not start at 0 It should curve up | 0 1 |
| 5 | 5^0 = 1 not 0 so y -values should increase by 1 Line should curve sharply upward | 1 1 |
| 6 | Put numbers on axes Make gradient steeper | 0 0 |
| 7 | Graph should be steeper not directly proportional | 0 |
| 8 | As increases, y increases even more The graph should start above 0 and \(5^0 = 1\) BOD above implies starts at \(y = 1\) on y axis | 0 1 |
9 The graph shows Sarah’s journey from her home to a shopping centre.
Complete the graph to show this information. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| She travels at constant speeds oe | 1 | Any incorrect statement invalidates the reasoning eg implying a constant speed for the entire journey SEE APPENDIX | |
Exemplar responses for Q9a
| Response | Mark |
|---|---|
| Her speed was the same (this does not imply the whole journey) | 1 |
| She stayed at a constant speed (BOD as doesn’t refer to whole journey) | 1 BOD |
| Sarah drove at the same speed 2km from home all the way to the shopping centre. | 1 |
| The straight lines on the graph show steady speed. | 1 |
| She walked at the same pace (BOD, question doesn’t state form of transport) | 1 |
| She stayed at the same speed on the way to the shopping centre (implies whole journey) | 0 |
| That she was constantly moving and through no traffic. Therefore going at a constant speed. (The first sentence invalidates the reason) | 0 |
| She was going the same speed throughout. (implies whole journey so 0) | 0 |
| That she rest for 30 minutes after doing 30 minutes of her journey | 0 |
| She walked slowly at the beginning and fast at the end (doesn’t imply constant speed) | 0 |
| Her journey to the shopping centre would take less than 4 hours (reference to the size of the graph not relevant) | 0 |
| Her route did not alter. She went straight to the shopping centre (Incorrect) | 0 |
| When she left home she started off and then she goes into constant speed and then she accelerated. Or Sarah was at a constant acceleration. (reference to acceleration is wrong) | 0 |
| There was no traffic. (no indication of speed) | 0 |
| There are no major issues which cause delay. (Reference to conditions on the journey are irrelevant we need a comment about why we are drawing a straight line) | 0 |
| That Sarah did not stop off | 0 |
| Her journey is faster from 11:00 (true but nothing about constant speed) | 0 |
| She will leave exactly on time (answer MUST refer to constant speed) | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 36 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 10:30[am] and 11:00[am] | 1 | Accept. eg 11, 1100, 10 30, 10.30 Do not accept eg 11h, 10h30 | |
| distance from home stays the same or zero gradient oe | 1 | SEE APPENDIX | |
Exemplar responses for Q9c
| Response | Mark |
|---|---|
| It goes horizontally | 1 |
| Because it’s a straight line not moving up or down which means no distance is being made | 1 |
| A horizontal line with no gradient (BOD ‘no’ rather than ‘zero’) | 1BOD |
| It has a flat part | 1 |
| The line which goes straight across which means she stopped (word ‘across’ is needed) | 1 |
| The distance stayed the same | 1 |
| The line during this time does not move in distance but the line still moves on the time showing she was stationary | 1 |
| The graph stays on the same level | 1BOD |
| The line doesn’t carry on increasing it stays constant for half an hour. (BOD time referred to therefore ‘it’ must imply distance) | 1 |
| The line has no incline | 1 |
| Straight line which indicates when Sarah stopped | 0 |
| A straight line that does not increase (just ‘straight line’ is not enough – needs to imply distance hasn’t changed with time) | 0 |
| The straight parallel line (parallel to ‘time axis’ would be fine to score) | 0 |
| The consistent line between the two times meaning that she stopped for 30 minutes (‘consistent’ is not correct) | 0 |
| The slope on the graph shows that Sarah stopped (reference to, ‘no slope’ would score) | 0 |
| Not going up in a diagonal line and staying straight (‘straight across’ in this statement would be enough to score) | 0 |
| A steady line that does not go up (so could go down, horizontal not implied) | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) Horizontal line from (1140, 36) to (1300, 36) | 1 | Condone freehand line Ignore construction lines Mark endpoint as the vertex with their second line | |
| Line from (their 1300, 36) to reach time axis after their 1300 | 1 | Could be a curve provided no horizontal sections | If no/wrong horizontal section drawn assume (their 1300, 36) to be the start of their line with negative gradient |
| at (1340, 0) or FT (their 1300 + 40 mins, 0) | 1 | eg 2 marks for one line such as (1140, 36) to (1220, 0) | |
| (ii) 54 | 3 | M2 for \(\dfrac{36 \text{ or their (b)}}{40} \times 60\) oe or M1 for \(\dfrac{36 \text{ or their (b)}}{40}\) soi by 0.9 or for an equivalent distance to time ratio | Condone 36000 used for M2 and M1 eg 18 associated with 20 but not 36 to 40 |
21 The number of gannets on an island is assumed to follow this exponential growth model.
\[N = 0.45 \times 1.07^x\]\(N\) is the number of gannets, in thousands.
\(x\) is the number of years after 1st January 2010.
| \(x\) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \(N\) | 0.45 | 0.63 | 1.24 |
[2]
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [0].88 or [0].89 1.7[4] | 2 | B1 for each | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 2 | B1 for 3 or 4 correct points plotted FT their table | tolerance ± \(\frac{1}{2}\) square |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2021 or 2022 | 2 | B1 for \(x\) = 11 to 12 | |
11 Point P is shown on this grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (2, 4) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Q plotted at \(({}^{-}1, 2)\) | 1 | ||
10
[1]
[1]
[1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(x = 3\) | 1 | ||
| (ii) \(y = x\) | 1 | Condone \(y = x \pm 0\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct sketch of the graph of \(y = x^2\). | 1 | U shaped graph, approximately symmetrical going through the origin | |
10 Here is the graph of \(y = x^2 + x - 6\).
Use the graph to solve the equation \(x^2 + x - 6 = 0\). [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| –3 2 | 2 | B1 for each | |
8 The graph shows the speed of a cyclist during 20 seconds of a journey.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 2 | 2 | M1 for ‘rise’ ÷ ‘run’ e.g. 8 ÷ 4 | |
| (ii) 0 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 150 | 4 | M3 for complete area \(\left[\dfrac{4 \times 8}{2} + (10 \times 8) + \dfrac{(8 + 10) \times 6}{2}\right]\) or M2 for two areas \(\dfrac{4 \times 8}{2}\) oe, \((10 \times 8)\) oe, or \(\dfrac{(8 + 10) \times 6}{2}\) oe or M1 for one area \(\dfrac{4 \times 8}{2}\) oe, \((10 \times 8)\) oe, or \(\dfrac{(8 + 10) \times 6}{2}\) oe | For M2 combining a triangle and a rectangle into a trapezium \(\dfrac{(14 + 10) \times 8}{2}\) counts as “two areas” Look for answers of 16, 80 and 54. Allow M marks for calculations from other suitable splitting of the areas |
5 This graph shows the cost of buying potatoes from a farm.
How much will it cost to buy 180 kg of potatoes from the farm? [3]
Describe one misleading feature of the graph. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 28 | 1 | Allow ±0.50 | |
| (ii) 95 | 1 | Allow ±1 | |
| (iii) 72 | 3 | Allow ±2 B1 for 1kg costs £0.40 soi M1 their 0.4 \(\times\) 180 or their cost \(\times \dfrac{180}{\textit{their}\ \text{equivalent weight}}\) or M1 for finding costs of weights that sum to 180 kg M1 for addition of these costs | e.g. 10kg costs £4 e.g. 4 x \(\dfrac{180}{10}\) Note: award M2 for e.g. cost of 60 kg \(\times\) 3 attempted |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| one valid reason | 1 | Such as ‘the vertical scale is not linear’ | e.g. vertical scale is wrong vertical scale does not start from 0 |
17 For each graph below, select its possible equation from this list.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = \dfrac{1}{x}\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = \sin x\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = 2^x\) | 1 | ||
9
| \(x\) | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \(y\) | −18 | −2 | 0 | −2 | 2 | 18 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Fully correct graph | 3 | B2FT for 7 correctly plotted points or B1FT for 5 or 6 correctly plotted points | Mark in 70% zoom, use overlay, mark curve first For 3 marks, curve must pass through or touch circles on overlay Condone ruled sections for \(-3 \leqslant x \leqslant -2\), \(-1 \leqslant x \leqslant 1\) and \(2 \leqslant x \leqslant 3\) . No vertical section on curve of more than 5 small squares must have min and max Condone slight feathering If curve incorrect, mark the plots use the overlay, plots must lie inside or touch circles. If large blob for plot, check centre of blob |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2.5 to 2.7 | 1 | ||
8 The graph shows the speed of a tram as it travels from the library to the town hall.
Not to scale
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 0.3 oe | 2 | M1 for \(\dfrac{[\pm]6}{85 - 65}\) oe or answer −0.3 If 0 scored, allow SC1 for 0.092[3…] or \(\dfrac{6}{65}\) as final answer | Allow unsimplified equivalents for full marks eg. \(\dfrac{6}{20}\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 255 | 3 | M2 for valid method to find complete area under the graph using one or more parts OR M1 for attempt to find partial area below the graph | M2 examples: eg \(\dfrac{85 \times 6}{2}\) oe or two triangles soi by 195 and 60 or [rectangle] 6 × 85 – two triangles oe M1 examples a triangle between \(t\) = 0 and 65 or a triangle between \(t\) = 65 and 85 or [rectangle] 6 × 85 – one triangle M0 for [rectangle] 6 × 85 Allow full marks for equivalent with units stated eg. 0.255 km |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 21.6 or \(\dfrac{108}{5}\) or \(21\frac{3}{5}\) nfww | 4 | B1 for 6 soi AND M2 for \(\dfrac{\textit{their } 6 \times 60 \times 60}{1000}\) oe or M1 for their 6 × 60 × 60 oe soi 21 600 or their 6 ÷ 1000 oe soi 0.006 or \(\dfrac{60 \times 60}{1000}\) oe soi 3.6 | Condone missing or incorrect units in working eg 6 m for 6 m/s their 6 could be the average speed 255/85 21600 or 0.006 imply B1M1 |
4 This grid shows a horizontal line going through the point A.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) (4, 3) | 1 | ||
| (ii) \((-2, 3)\) plotted | 1 | Centre of mark within overlay around point | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = 3\) | 1 | Accept any alternative form: EG \(y - 3 = 0\), \(3 - y = 0\), \(-y = -3\) even \(2y = 6\) | |
17 The diagram shows a circle, centre the origin.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x^2 + y^2 = 100\) oe | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(8^2 + (-6)^2 = 100\), so it’s on the circle oe | 2 | M1 for \(8^2 + ([-]6)^2\) seen or for substituting \(x = 8\) and \(y = -6\) into their part (a) | Alternative using Pythagoras \(\sqrt{8^2 + 6^2} = 10\) their part (a) must be an equation in both \(x\) and \(y\). |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(3y - 4x + 50 = 0\) oe | 5 | B2 for [tangent gradient = ] \(\dfrac{4}{3}\) oe or M1 for \(\pm\dfrac{6}{8}\) or \(\pm\dfrac{8}{6}\) oe | Equivalents include: \(y = \dfrac{4}{3}x - \dfrac{50}{3}\) Condone decimals with at least 2 decimal places rot: Eg. \(y = 1.33x - 16.67\) |
| AND M2 for \(y + 6 = \textit{their } \dfrac{4}{3}(x - 8)\) oe or M1 for \(y = \textit{their } \dfrac{4}{3}x + \text{‘}c\text{’}\) | Equivalent for M2 includes \(y = \textit{their } \dfrac{4}{3}x + c\) and then attempt to find \(c\) by substituting in \(y = -6\) and \(x = 8\) | ||
16
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x - 3)^2 + 11\) final answer | 3 | B1 for \((x - 3)^2\) B2 for +11 or FT their \((x - 3)^2\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (3, 11) | 2 | B1FT for each part | FT their \((x - a)^2 + b\) e.g. \((a, b)\) |
15
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 | 3 | 4 |
| \(y\) | \(-4\) | \(-5\) | \(-4\) | 11 |
[2]
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-1\) [\(-4\)] [\(-5\)] [\(-4\)] \(-1\) 4 [11] | 2 | B1 for 1 correct | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 2 | B1 for 4 or more points correctly plotted FT their table | Tolerance half small square |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Ruled line \(y = -2\) drawn | 1 | Line from \(x = -2\) to \(x = 2\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-1.8\) to \(-1.6\) and 1.6 to 1.8 | 2 | B1 for 1 correct | FT from their graph \(\pm\) 0.1 for 2 or 1 mark Must have a curve and a straight line for FT |
13 This graph shows part of Lucy’s car journey from London to Sheffield.
The car made one stop at a service station.
Use the graph to answer these questions.
Complete the graph. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 20 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 60 | 2 | M1 for 50 miles in 50 min oe \(\dfrac{50}{50}\) [\(\times\) 60] | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Line from (13 10, 120) to (14 20, 180) | 2 | B1 for line from (13 10, 120) B1 for line to (14 20, 180) | Extra stop allowed A non-decreasing curve is OK |
13 The graph shows information about the speed of a vehicle during the final 50 seconds of a journey.
At the start of the 50 seconds the speed is \(k\) metres per second.
The distance travelled during the 50 seconds is 1.35 kilometres.
........................................ m/s [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 27 | 2 | M1 for 1350 ÷ 50 If 0 scored SC1 for answer figs 27 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 30 | 5 | B1 for 1350 M3 for 1350 = 40\(k\) + ½ ×10×\(k\) oe or M2 for 40\(k\) + ½ ×10×\(k\) oe or M1 for any attempt to find any relevant area under the graph | Condone figs 135 for M3 and variable other than \(k\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) –3 | 1FT | FT (–their (b) ÷ 10) | |
| (ii) [Constant] deceleration oe | 1 | Condone acceleration The rate at which the speed changes | |
| m/s\(^2\) | 1 | ||
11 A company tests a new battery for an electric car.
The distance the car travels, \(d\) km, and the charge left in the battery, \(C\) %, are measured.
Some measurements are shown in the table.
| Distance travelled, \(d\) km. | 0 | 50 | 100 | 150 |
| Charge left in the battery, \(C\) %. | 100 | 75 | 50 | 25 |
[2]
Give your equation in the form \(C = ad + b\). [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4 points plotted and a ruled line joining | 2 | B1 for 3 points correctly plotted | Line at least between (0, 100) and (150, 25) Use overlay as guide. ½ square accuracy |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 198 to 202 | 1 | Do not FT their line | |
| (ii) Battery usage remains the same or Battery can be used right to 0% or Trend or pattern continues | 1 | Accept For every 50 km it uses 25% | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(-\dfrac{1}{2}\) oe or \(-[0].5\) | 1 | Ignore units | |
| (ii) 100 | 1 | Accept 0, 100 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-\dfrac{1}{2}d + 100\) | 1 | FT their (c)(i)\(d\) + their (c)(ii) | Accept any letter for \(d\) (except \(c\)) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(-5\) | 2 | FT their (d) if linear in \(d\). B1 for correct substitution of 210 | Expect \(-\dfrac{1}{2} \times 210 + 100\) Accept any letter for \(d\) (except \(c\)) |
| (ii) Impossible [as battery cannot have negative charge] oe | 1 | FT their (i) only if their equation gives negative outcome | |
10 This graph shows the world population, in billions, between 1951 and 2015.
Use the graph to estimate the average rate of growth of the world population between 1951 and 2015.
Give suitable units for your answer. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 71 000 000 to 89 000 000 in figs or words | 2 | M1 for attempt to find ‘gradient’ using figures from the graph e.g. (7.4 – 2.6) ÷ (2015 – 1951) | Could be in billions Eg. (7 400 000 000 – 2 600 000 000) ÷ (2015 – 1951) For M1, condone incorrect conversion used consistently for both population figures. |
| and | |||
| people/year | 1 | ||
5
| \(x\) | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 2 | −4 | −4 | 2 |
You may use the table if you wish.
| \(x\) | −4 | ||
|---|---|---|---|
| \(y\) | 7 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 8 −2 −2 8 | 2 | B1 for any 2 correct | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| correct curve which dips below the line \(y = -4\) | 3 | B2 for 6 or 7 points correctly plotted FT their table or B1 for 4 or 5 points correctly plotted FT their table | tolerance ± 2 mm for plotting and the curve through the correct points |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −2.7 to −2.5 1.5 to 1.7 | 2 | B1 for each Correct answer or FT their graph | tolerance ± 2 mm |
| Answer | Marks | Part marks and guidance | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| correct ruled line | 3 | M2 for a correct unruled line or a line of gradient −2 or a line going through (0, −1) or two further correct points in the table or plotted or M1 for one point correctly plotted or one further correct point in the table | points are
tolerance ± 2 mm | ||||||||||||||||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −3.9 to −3.7 [0].7 to [0].9 | 2 | B1 for each Correct answer or FT their straight line | tolerance ± 1 mm |
21
| \(x\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|
| \(y\) | \(3\) | \(0\) | \(-1\) | \(0\) | \(3\) |
[1]
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 8 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 2 | B1FT for 4, 5 or 6 points plotted correctly | \(\frac{1}{2}\) square tolerance B1 max if line ruled (between any points) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-0.9\) to \(-0.6\) 2.6 to 2.9 | 2 | B1 for each If 0 scored SC1 for (\(-0.9\) to \(-0.6\), 2) and (2.6 to 2.9, 2) If 0 scored SC1 for answer as an inequality Eg \(-0.8 \leqslant x \leqslant 2.7\) | If more than two answers mark the worst two Condone for 2 marks when both answers in body but only one given on answer line |
20 The graph shows the speed, in metres per second, of a particle over the first four seconds of motion.
Use the graph to estimate the distance travelled by the particle in the four seconds. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| evidence of finding the area under the graph | M1 | M1 for evidence of any correct method to find the area under the graph e.g. counting squares (numbers or dots in squares) or use of triangles, trapeziums or rectangles | |
| 15 to 22 | B1 | accept answer as a fraction | |
19
[3]
Not to scale
Find the values of \(a\), \(b\) and \(c\). [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| U shaped parabola with minimum value indicated at (2, –3) | 3 | B1 for U shape curve B1 for turning point at \((2, k)\) B1 for turning point at \((k, -3)\) | Be generous for the U shape condone broken line Values must be shown but could be marked on axes. Mark intention Accept turning point = (2, –3) written in working provided no contradiction on sketch If point (2, –3) only plotted on graph and no sketch then B0B1B1 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4, 16, 12 | 5 | B4 for \(a = 4\) and \(b = 16\) OR B3 for \(c = 12\) and either \(a = 4\) or \(b = 16\) OR M1 for \((x + 3)(x + 1)\) seen isw A1 for \(x^2 + x + 3x + 3\) or better seen isw B1 for \(c = 12\) | |
| OR B1 for \(c = 12\) soi M1 for \((-1)^2a - 1b + 12 = 0\) oe and \((-3)^2a - 3b + 12 = 0\) oe | Alt method uses simultaneous equations with \(c = 12\) Allow recovery for omission of brackets if negatives correctly dealt with | ||
17 Describe fully the graph which has the equation \(x^2 + y^2 = 9\). [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| circle | 1 | condone circular | |
| centre (0, 0) oe and radius 3 | 1 | accept origin or O for (0,0) | |
15 The graph shows the speed, \(v\) metres per second (m/s), of a car at time \(t\) seconds.
Use 1 mile = 1.6 kilometres. [5]
Find a formula linking \(v\) and \(t\). [3]
Make one comment to show that this statement is incorrect. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 14 – 15 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 30 from graph | B1 | accept any correct method e.g. | |
| Starting with their 30 (using mph) These method marks could be awarded in any order | Starting with 60 (using m/s) These method marks could be awarded in any order M1 × 1.6 soi by 96 M1 × 1000 soi M1 \(\div 60^2\) soi B1 for 30 from graph A1 for 26.6 to 26.7 and 30 so yes | (using km/hr) M1 for 60 × 1.6 soi by 96 B1 for 30 from graph Starting with their 30 These method marks could be awarded in any order M1 \(\times 60^2\) soi M1 ÷ 1000 soi A1 for 96 and 108 so yes | |
| \(\times 60^2\) soi | M1 | ||
| ÷ 1000 soi | M1 | ||
| ÷ 1.6 soi | M1 | ||
| 67.5 so yes | A1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Attempt at a tangent drawn at \(t = 7\) | B1 | Accept answer as a fraction and tolerance on reading from graph ± ½ small square Gradient for M1 could be from a chord. Ignore any negative sign | |
| 4.0 to 4.5 oe | B2 | M1 for an attempt at speed ÷ time, could be on the graph e.g. 30 ÷ 10 or their (a) ÷ 7 soi 2.07… or 2.14.. | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(v = kt^2\) where \(0.25 \leqslant k \leqslant 0.33\) | 3 | SC2 for \(v \propto kt^2\) where \(0.25 \leqslant k \leqslant 0.33\) or B1 for \(v = kt^2\) AND M1 for \(30 = k(10)^2\) or FT their reading from the graph for values of \(v\) and \(t\) or B1 for \(0.25 \leqslant k \leqslant 0.33\) | Condone use of other letters especially \(s\) for speed Can be implied by eg \(30 = k(10)^2\) \(k\) could be a fraction e.g. \(\frac{15}{49}\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| any correct comment e.g. graph only valid/information only available up to 10 secs or car will eventually reach max. speed | 1 | See appendix | |
| Response | Mark |
|---|---|
| graph only valid/information only available up to 10 secs | 1 |
| car will eventually reach max. speed | 1 |
| It could stay at a constant speed | 1BOD |
| It gains enough acceleration its speed becomes constant | 0 |
| He drove the whole route | 0 |
7
| \(x\) | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| \(y\) | 3 | 0 | −1 | 0 | 3 |
[1]
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 8 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Correct curve | 2 | B1FT for 4, 5 or 6 points plotted correctly | ½ square tolerance B1 max if line ruled (between any points) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −0.9 to −0.6 2.6 to 2.9 | 2 | B1 for each If 0 scored SC1 for (−0.9 to −0.6, 2) and (2.6 to 2.9, 2) | If more than two answers mark the worst two Condone for 2 marks when both answers in body but only one given on answer line |
5 Points P and Q are shown on this grid.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \((4, 3)\) | 1 | ||
| (ii) \((-2, -1)\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Point plotted at \((3, -2)\) | 1 | Condone use of a letter (R) if clearly in correct position | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| line \(y = 3\) drawn | 1 | minimum length 1 square | |
No questions match these filters.