Course Support. Past Paper Questions by Topic. For Free.
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
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
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 \(75 \times 7 + 16 = 541\)
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 |
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 |
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 \((-1, 3)\)) |
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 |
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 |
No questions match these filters.