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Edexcel Awards
Mathematics
Sample Assessment Materials
Edexcel Level 2 Award in Algebra (AAL20)
Edexcel Level 3 Award in Algebra (AAL30)
For first teaching from October 2012
Pearson Education Limited is a registered company number 872828 with its registered office at Edinburgh Gate, Harlow, Essex CM20 2JE
Contents
General Marking guidance .................................................................................. 3
Level 2 ............................................................................................................
Level 2 Paper .................................................................................................. 5
Level 2 Mark scheme ......................................................................................... 21
Level 3 ............................................................................................................
Level 3 Paper ................................................................................................. 27
Level 3 Mark scheme ......................................................................................... 43
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
1
2
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
NOTES ON MARKING PRINCIPLES
1
Types of mark
M marks: method marks
A marks: accuracy marks
B marks: unconditional accuracy marks (independent of M marks)
2
Abbreviations
cao – correct answer only
isw – ignore subsequent working
oe – or equivalent (and appropriate)
indep - independent
3
No working
If no working is shown then correct answers normally score full marks, unless indicated
in the mark scheme.
If no working is shown then incorrect (even though nearly correct) answers score no
marks.
4
With working
If there is a wrong answer indicated on the answer line always check the working in the
body of the script (and on any diagrams), and award any marks appropriate from the
mark scheme.
If working is crossed out and still legible, then it should be given any appropriate
marks, as long as it has not been replaced by alternative work.
If it is clear from the working that the “correct” answer has been obtained from
incorrect working, award 0 marks.
If there is no answer on the answer line then check the working for an obvious answer.
Any case of suspected misread loses A (and B) marks on that part, but can gain the M
marks.
If there is a choice of methods shown, then no marks should be awarded, unless the
answer on the answer line makes clear the method that has been used.
5
Follow through marks
Follow through marks which involve a single stage calculation can be awarded without
working since you can check the answer yourself, but if ambiguous do not award.
Follow through marks which involve more than one stage of calculation can only be
awarded on sight of the relevant working, even if it appears obvious that there is only
one way you could get the answer given.
ft – follow through
SC: special case
dep – dependent
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
3
4
6
Ignoring subsequent work
It is appropriate to ignore subsequent work when the additional work does not change
the answer in a way that is inappropriate for the question: e.g. incorrect cancelling of a
fraction that would otherwise be correct
It is not appropriate to ignore subsequent work when the additional work essentially
makes the answer incorrect e.g. algebra.
Transcription errors occur when candidates present a correct answer in working, and
write it incorrectly on the answer line; mark the correct answer.
7
Parts of questions
Unless allowed by the mark scheme, the marks allocated to one part of the question
CANNOT be awarded in another.
8
Use of ranges for answers
If an answer is within a range this is inclusive, unless otherwise stated.
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
Write your name here
Surname
Other names
Centre Number
Candidate Number
Edexcel Award
Algebrain Algebra
Award
Level 2
Calculator NOT allowed
Sample Assessment Material
Time: 1 hour 30 minutes
Paper Reference
AAL20/01
You must have:
Ruler graduated in centimetres and millimetres, pen, HB pencil,
eraser.
Total Marks
Instructions
black ink or ball-point pen.
• Use
Fill
in
boxes at the top of this page with your name,
• centrethe
number and candidate number.
all questions.
• Answer
the questions in the spaces provided
• Answer
– there may be more space than you need.
• Calculators may NOT be used.
Information
total mark for this section is 80
• The
marks for each question are shown in brackets
• The
– use this as a guide as to how much time to spend on each question.
Advice
each question carefully before you start to answer it.
• Read
an eye on the time.
• Keep
Try
to
every question.
• Checkanswer
• your answers if you have time at the end.
S42815A
©2012 Pearson Education Ltd.
3/3/
*S42815A0116*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
Turn over
5
Answer ALL questions.
Write your answers in the spaces provided.
You must write down all stages in your working.
1
Pencils are sold in packets of 4 and boxes of 12.
Box of
12 pencils
Packet of 4
Didi buys p packets of pencils and b boxes of pencils.
Write an expression, in terms of p and b, for the number of pencils Didi bought.
..............................................................
(Total for Question 1 is 2 marks)
2
(a) Simplify
2a2 + 7b3 + 5 + 6a2 – 8 – b3
..............................................................
(2)
(b) Expand
5(2g3 + 6)
..............................................................
(2)
(c) Expand
6k(1 + 2k – k2)
..............................................................
(2)
(Total for Question 2 is 6 marks)
2
6
*S42815A0216*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
3
The nth term of a sequence is given by the expression 6n – 5
(a) Write down the first two terms of the sequence.
...........................
...........................
(2)
Here are the first five terms of another sequence
5
9
13
17
21
(b) Write down an expression, in terms of n, for the nth term of this sequence.
..............................................................
(2)
(Total for Question 3 is 4 marks)
4
Here is a formula
v = u + at
(a) Find the value of v when u = 0, a = 6 and t = 8
..............................................................
(2)
(b) Find the value of v when u = 10, a = –5 and t = 3
..............................................................
(2)
(c) Find the value of t when v = 20, u = 10 and a = 2
..............................................................
(3)
(Total for Question 4 is 7 marks)
*S42815A0316*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
3
7
Turn over
5
(a) Solve
x
+ 6 = 15
2
x = ..............................................................
(2)
(b) Solve
6p – 5 = 3p + 13
p = ..............................................................
(2)
(c) Solve
3(t + 5) = 10
x = ..............................................................
(3)
(Total for Question 5 is 7 marks)
4
8
*S42815A0416*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
6
The line L is shown on the grid.
(i) Work out the gradient of the line L.
...............................
(ii) Find an equation of the line L.
..............................................................
y
L
10
8
6
4
2
–2
–1
O
1
2
3
4
x
5
–2
–4
–6
(Total for Question 6 is 4 marks)
*S42815A0516*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
5
9
Turn over
7
(a) Simplify
x4 × x6
...............................
(1)
(b) Simplify
y8 ÷ y3
...............................
(1)
(c) Simplify
(p2)4
...............................
(1)
(d) Simplify
5r5s4 × 4r3s
..............................................................
(2)
(e) Expand and simplify
5(2a + 3) + 2(a – 4)
..............................................................
(3)
(Total for Question 7 is 8 marks)
6
10
*S42815A0616*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
8
–1 < y  2
(a) On the number line below, show the inequality
–4 –3 –2 –1
0
1
2
3
4
5
y
(2)
Here is an inequality, in x, shown on a number line.
–4 –3 –2 –1
0
1
2
3
4
5
x
(b) Write down the inequality.
..............................................................
(2)
(c) –3  n < 2
n is an integer.
Write down all the possible values of n.
..............................................................
(2)
(d) Solve the inequality
4t – 3 > 7
..............................................................
(3)
(Total for Question 8 is 9 marks)
*S42815A0716*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
7
11
Turn over
9
y = –2x + 4
(a) Complete the table of values for
x
–1
0
1
2
3
4
y
(2)
y
10
8
6
4
2
–2
–1
O
1
2
3
4
x
5
–2
–4
–6
(b) On the grid draw the line with equation y = –2x + 4 for values of x from –1 to 4
(2)
(Total for Question 9 is 4 marks)
8
12
*S42815A0816*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
10 Here is a travel graph for Susan’s journey from her house to the library and back to her
house.
20
18
16
14
Distance 12
from
Susan’s 10
house
8
(km)
6
4
2
0
09 00
09 30
10 00
10 30
11 00
11 30
Time
Susan stopped at some road works at 09 30
(a) How far is Susan from her house at 09 30?
...............................
(1)
km
The library is 20 km from Susan’s house.
(b) (i) At what time did Susan get to the library?
...............................
(ii) How long did Susan stay at the library?
...............................
(2)
minutes
(c) At what time did Susan arrive back at her house?
...............................
(1)
(Total for Question 10 is 4 marks)
*S42815A0916*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
9
13
Turn over
11 Sketch the graph y = x2 – 4
You must label relevant information on the sketch.
(Total for Question 11 is 2 marks)
12 (a) Factorise
6ab – 9a
..............................................................
(2)
(b) Factorise
12p2 – 18p3
..............................................................
(2)
(Total for Question 12 is 4 marks)
10
14
*S42815A01016*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
13 Here is part of a travel graph of Tom’s journey from his house to the shops and back.
It shows his journey to the shops and how much time he was at the shops.
20
18
16
14
Distance
12
from
Tom’s
10
house
8
(km)
6
4
2
O
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Time in minutes
(a) Work out Tom’s speed for the first 30 minutes of his journey.
Give your answer in km/h.
......................................................... . . . . .
(2)
Km/h
Tom travels back to his house at 60 km/h.
(b) Complete the travel graph.
(2)
(Total for Question 13 is 4 marks)
*S42815A01116*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
11
15
Turn over
14 The graph shows the cost of using a mobile phone for one month on three different
tariffs.
60
...................... .
50
...................... .
40
Cost
(£s)
...................... .
30
20
10
0
0
20
40
60
Time (minutes)
80
100
120
The three tariffs are
Tariff A
Tariff B
Tariff C
Rental £20
every minute costs 20p
Pay as you go every minute costs 50p
Rental £25
first 60 minutes free then each minute costs 10p
(a) Label each line on the graph with the letter of the tariff.
12
16
*S42815A01216*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
(2)
Fiona uses her mobile phone for about 60 minutes each month.
(b) Explain which tariff would be the cheapest for her to use.
You must give the reasons for your answer.
. . . . . . . . . ............................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . ............................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 14 is 4 marks)
15 Make t the subject of the formula
p = 2(u + 5t)
t = ........................................ . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 15 is 3 marks)
*S42815A01316*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
13
17
Turn over
16 (a) Complete the table of values for
x
–4
y
9
y = x2 + x – 3
–3
–2
–1
–1
–3
0
1
2
(2)
(b) On the grid below, draw the graph of y = x2 + x – 3 for values of x from –4 to 2
y
10
8
6
4
2
–4
–3
–2
O
–1
1
2
x
–2
–4
–5
(2)
(c) Use your graph to find estimates for the solutions of
x2 + x – 3 = 0
........................................ . . . . . . . . . . . . . . . . . . . . . .
(2)
14
18
*S42815A01416*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
(d) Use your graph to find estimates for the solutions of
x2 + x = 7
........................................ . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 16 is 8 marks)
ToTAL for PAPer is 80 MArKs
*S42815A01516*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
15
19
20
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
21
4
v=0+6×8
v = 10 + –5 × 3
= 10 – 15
20 = 10 + 2 × t
20 – 10 = 2t
t = 10 ÷ 2
(b)
(c)
5
–5
48
4n + 1
(b)
(a)
1, 7
3
2
2
2
2
2
6k + 12k2 –6k3
(c)
(a)
2
10g3 + 30
(b)
3
2
8a2 + 6b3 – 3
(a)
2
Mark
2
Answer
4p + 12b
Working
1
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
M1 for 20 = 10 + 2 × t
M1 for evidence of – 10 or ÷ 2 or sight of 20 – 10 = 2t or
t = 10 or 10 – 5 = t
A1 for 5
M1 for v = 10 + –5 × 3
A1 for –5
M1 for V = 0 + 6 × 8
A1 for 48
B2 for 4n + 1
(B1 for a linear expression including 4n)
B2 for 1 and 7
(B1 for 1 or 7)
B2 for 6k +12k2 –6k3
(B1 for any two correct from 6k, 12k2 or –6k3)
B2 for 10g3 + 30
(B1 for 10g3 or 30)
B2 for 8a2 + 6b3 – 3
(B1 for any two correct from 8a2, + 6b3, – 3)
B2 for 4p + 12b oe
(B1 for 4p or 12b oe seen)
Notes
22
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
6
5
y = 3x – 2
(ii)
2
2
3
5
3
−
3
3x + 15 = 10
3x = 10 – 15
x = –5 ÷ 3
(c)
2
2
Mark
6
18
Answer
(i)
3p = 18
p = 18 ÷ 3
x=9×2
x
= 15 – 6
2
Working
(b)
(a)
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
x
x
= 15 – 6 or
= 9 oe
2
2
10
3
5
accept –1.6�
3
M1 ft for eqn of the form y = mx + c with m = “3” or c = –2
A1 cao
M1 for attempt to find difference is y divided by difference in x
can be evidenced by drawing a right angled triangle and
marking lengths on it
A1 for 3
A1 for −
10
–5
3
M1 for subtracting “15” from both sides or sight of –5 or
sight of x + 5 =
M1 for expanding bracket or sight of 3x + 15 or divide by 3 or
M1 for attempt to collect variables on one side or collect
constant terms on one side
A1 for 6
A1 for 18
side by 2 or sight of
M1 for attempt to subtract 6 from each side or multiply each
Notes
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
23
8
7
1
2
p8
20r 8s 5
12a + 7
(c)
(d)
(e)
–3, –2, –1, 0, 1
t > 2.5
(c)
(d)
4t > 7 + 3 = 10
t > 10 ÷ 4
2
–2 ≤ x < 3
(b)
3
2
2
Correct inequality
(a)
3
1
y5
(b)
10a + 15 + 2a – 8
1
Mark
x
Answer
(a)
Working
10
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
M1 for adding 3 to each side
M1 for dividing by 4
A1 for t > 2.5
B2 cao
(B1 for at least 4 correct and not more than one incorrect
integer)
B2 for –2 ≤ x < 3
(B1 for –2 ≤ x or x < 3)
M1 for a straight line joining –1 to 2 with either an open circle
at –1 or a closed circle at 2
A1 for a straight line joining –1 to 2 with an open circle at –1
and a closed circle at 2
Accept appropriate alternative notation where meaning is clear
M1 for 10a + 15 or 2a – 8
A1 for 12a
A1 for + 7
B2 for 20r8s6
(B1 for two elements correct from 20 or r8 or s5)
B1 cao
B1 cao
B1 cao
Notes
24
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
11
10
9
2
11 20
Sketch of graph
(c)
Graph sketch: includes labelled
axes, correct orientation,
intersection points with the yaxis labelled
(x2 – 4)
1
30
(b)(ii)
1
1
10 00
(b)(i)
1
10
4
Mark
(a)
Correct straight line
(b)
Answer
6, 4, 2, 0, –2, –4
Working
(a)
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
B1 for U – shaped curve and axes labelled x and y
B1 for intersection with y –axis labelled at (0, -4)
B1 cao
B1 cao
B1 cao
B1 cao
M1 for plotting all points correctly from their table values or
straight line with gradient –2 or line that passes through (0, 4)
A1 for a correct line lying between x = –1 and x = 4
M1 for table of values with 2 values correct
A1 for fully correct table of values
Notes
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
25
14
13
12
20 ÷ 0.5 =
20 × 2 =
2
Correct explanation
(b)
Example explanation:
For 60 minutes each month, on
Tariff A, Fiona would pay
20 + 60 × 0.2 = £32, on B, she
would pay 60 × 0.5 = £30, and
on C she would pay £25 only.
2
2
2
Correct labels
Line from (45, 20)
to (65, 0)
40
(a)
(b)
(a)
2
6p 2(2 – 3p)
(b)
Mark
2
Answer
3a(2b – 3)
Working
(a)
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
1
2
B2 for selecting Tariff C supported by explanation using
calculations for 60 minutes or for reading correct values from
the graph
B2 for lines labelled B, A, C
(B1 for labelling one graph correctly)
M1 for drawing a straight line from (45, 20) to time axis
A1 if line meets time axis at (65, 0)
A1 for 40
M1 for 20 × 2 or 20 ÷
M1 for partial correct factorisation of the expression using 2, 3,
6, p, p2 or any combination used
A1 for 6p 2(2 – 3p)
M1 for 3 outside a single bracket with linear term in p inside
the bracket
A1 for 3a(2b – 3)
Notes
26
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
16
15
Correct curve
1.3, –2.3
2.2, –3.3
(b)
(c)
(d)
3
p − 2u
10
2
2
2
2
Mark
Answer
3, –3, –1, 3
p = 2u + 10t
10t = p – 2u
Working
(a)
Question
Award in Algebra AAL20
Level 2 (AAL20) mark scheme
p − 2u
10
M1 for x2 + x – 3 = 4
A1 for 2 to 2.4 and –3 to –3.4
B1 for 1.3 ± 0.2 ft from their line
B1 for –2.3 ± 0.2 ft from their line
B2 for plotting all points correctly joined by a curve
(B1 for plotting their points correctly)
B2 for all 4 missing values correct
(B1 for 1 missing value correct)
A1 for
M1 for multiplying out bracket or sight of 2u+ 10t
M1 for attempt to take “2u” to other side of formula
Notes
Write your name here
Surname
Other names
Centre Number
Candidate Number
Edexcel Award
Algebrain Algebra
Award
Level 3
Calculator NOT allowed
Sample Assessment Material
Time: 2 hours
Paper Reference
AAL30/01
You must have:
Ruler graduated in centimetres and millimetres, pen, HB pencil,
eraser.
Total Marks
Instructions
black ink or ball-point pen.
• Use
Fill
in
boxes at the top of this page with your name,
• centrethe
number and candidate number.
all questions.
• Answer
the questions in the spaces provided
• Answer
– there may be more space than you need.
• Calculators are not allowed.
Information
total mark for this section is 90
• The
marks for each question are shown in brackets
• The
– use this as a guide as to how much time to spend on each question.
Advice
each question carefully before you start to answer it.
• Read
an eye on the time.
• Keep
Try
to
every question.
• Checkanswer
• your answers if you have time at the end.
S42816A
©2012 Pearson Education Ltd.
3/3/
*S42816A0116*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
Turn over
27
Answer ALL questions.
Write your answers in the spaces provided.
You must write down all stages in your working.
1
(a) Simplify
x 4 × x6
x3
.......................... . . . . . . . . . . . . . . . . . . . . . .
(2)
1
(b) Simplify
y2
3
y ×
5
y2
......................................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(c) Expand and simplify
(2x + 5)(x – 3)
......................................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(d) Factorise
12p2q3 – 18p3q2
......................................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 1 is 8 marks)
2
28
*S42816A0216*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
2
The straight line L1 passes through the points A and B with coordinates (1, –2) and (4, 4)
respectively.
(a) Find an equation of L1 in the form y = mx + c.
y = ............................................ . . . . . . . . . . . . . . . . . . . . . .
(3)
The line L2 is perpendicular to the line L1 and passes through the origin.
(b) Find an equation of L2
......................................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 2 is 5 marks)
*S42816A0316*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
3
29
Turn over
3
On the grid, shade the region that satisfies all these inequalities
y >–1
x +y <4
y < 3x + 2
y
10
8
6
4
2
–2
–1
O
1
2
3
4
5
x
–2
–4
–6
(Total for Question 3 is 5 marks)
4
30
*S42816A0416*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
4
(a) Factorise
x2 – 7x + 12
....................................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(b) Factorise
x2 + 5x – xy – 5y
....................................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(3)
(Total for Question 4 is 5 marks)
5
The equation x2 – 18x + p = 0 has two equal roots.
(i) Find the value of p
................................... . . . . . . . . . . . . . . . . . . . . . .
(ii) For this value of p, sketch the graph of y = x2 –18x + p showing the coordinates of
any points at which the graph meets the coordinate axes.
......................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 5 is 5 marks)
*S42816A0516*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
5
31
Turn over
6
(a) On the grid below, draw the graph of y = x3 – 2x2 – 4 for values of x from –2 to +4
(4)
(b) Use your graph to find estimates for the solutions of x3 – 2x2 = 4
.......................................... . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 6 is 6 marks)
6
32
*S42816A0616*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
7
Solve the simultaneous equations
x – 2y = –2
x2 – y2 = 7
........................................................................................................ . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 7 is 6 marks)
*S42816A0716*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
7
33
Turn over
8
Make k the subject of the formula
t =
2k + 1
k−2
k = .................................................................. . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 8 is 4 marks)
9
For a quadratic equation
the sum of its roots is 3.5
the product of its roots is 1.5
Find the quadratic equation in the form ax2 + bx + c = 0
where a, b and c are integers.
.......................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 9 is 3 marks)
8
34
*S42816A0816*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
10 The sum of the first two terms of an arithmetic series is 47.
The thirtieth term of this series is –62
(a) Find the first term of the series and the common difference.
First term ............. . . . . . . . . . . . . . . . . . . . . . .
Common difference ............. . . . . . . . . . . . . . . . . . . . . . .
(3)
(b) The sum of the first 60 terms of the series.
.............................................. . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 10 is 5 marks)
11 Write the quadratic expression x2 – 5x + 3 in the form (x + a)2 + b where a and b are
fractions.
.............................................................................................................. . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 11 is 2 marks)
*S42816A0916*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
9
35
Turn over
12 Here are some sketch graphs.
y
y
x
O
y
x
O
A
x
O
B
y
C
y
x
O
D
x
O
E
The table shows the equations of some graphs.
Equation
Graph
y = 4x
y = –x(x – 4)
y = x3 – x2 – 2x
xy = 8
y = x2 – 4x
Match the letter of the graph with its equation.
(Total for Question 12 is 3 marks)
10
36
*S42816A01016*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
13 Explain why any straight line that passes through the point (1, 2) must intersect the curve
with equation
x2 + y2 = 16
in exactly two points.
y
5
(1, 2)
–5
O
5
x
–5
(Total for Question 13 is 3 marks)
*S42816A01116*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
11
37
Turn over
14 Solve
x2 – 4x < –5
..................................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 14 is 3 marks)
15 Expandandsimplify (2+√2)(3+√8)
Give your answer in the form a + b√2wherea and b are integers.
...................................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 15 is 4 marks)
16 Solve the equation
2x2 + 5x – 7 = 0
....................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 16 is 3 marks)
12
38
*S42816A01216*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
17 Here is a speed time graph that shows Tim’s speed between two sets of traffic lights.
He travels between the two sets of traffic lights in 9 seconds.
35
30
25
Speed
(m / second) 20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
Time (seconds)
(a) Work out Tim’s acceleration in the first 3 seconds.
................................................................................
metres per second
(2)
(b) Work out the distance between the two sets of traffic lights.
.......................................... . . . . . . . . . . . . . . .
(2)
m
(Total for Question 17 is 4 marks)
*S42816A01316*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
13
39
Turn over
18 Solve
2
3
5
+
= 2
x +1 x −1 x −1
x = .................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 18 is 4 marks)
19 The force, F, between two magnets is inversely proportional to the square of the distance, x, between
them.
When x = 3, F = 4
(a) Find a formula for F in terms of x.
F = .................... . . . . . . . . . . . . . . . . . . . . . .
(3)
(b) Calculate the value of F when x = 2
............ . . . . . . . . . . . . . . . . . . . . . .
(1)
(c) Calculate the value of x when F = 64
............ . . . . . . . . . . . . . . . . . . . . . .
(1)
(Total for Question 19 is 5 marks)
14
40
*S42816A01416*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
20 The graph of y = f(x) is shown on the two grids.
(a) On this grid, sketch the graph of y = f(x + 2)
y
5
2
–5
O
–2
2
5
x
–2
–5
(2)
(b) On this grid, sketch the graph of y = – f(x)
y
5
2
–5
–2
O
2
5
x
–2
–5
(2)
(Total for Question 20 is 4 marks)
*S42816A01516*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
15
41
Turn over
21
y
10
8
6
4
2
–2
–1
O
1
2
3
4
5
x
–2
–4
–6
Use the trapezium rule to find the area of the region under the curve and between
x = 0, y = 0 and x = 4
Use 4 strips of equal width.
....................................................................................... . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 21 is 3 marks)
TOTAL FOR PAPER IS 90 MARKS
16
42
*S42816A01616*
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
43
2
1
m=-
1
2
2 × m = –1
(b)
2
1
x
2
y=–
3
2
6p 2 q 2(2q – 3p)
(d)
y = 2x – 4
2
2x2 – x –15
(c)
4 − −2 6
= =2
4 −1
3
y – 4 = 2(x – 4)
y – 4 = 2x – 8
2
y –5
(b)
Gradient is
2
x7
(a)
(a)
Mark
Working
Answer
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
or y
1
5
6
3
M1 for attempt to use mm = –1
A1
M1 for attempt to use y – y1 = m(x – x 1)
A1
or using formula or sight of
M1 for attempt to find gradient by using a right angled triangle
M1 for partial correct factorisation of the expression using any
two of 2, 3, 6, p, p2, q ,2q used correctly
A1 for 6p 2 q 2(2q – 3p) oe
M1 for expanding bracket to obtain 4 terms with all 4 correct
without considering signs or for 3 terms out of 4 correct with
correct signs
A1
(B1 for sight of y11/2)
B2 for y
–5
M1 for either x 4 + 6 or x 4 – 3 or x 6 –3 or x 10 – 3
A1 cao
Notes
44
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
3
(x + 5)(x – y)
(b)
2
(x – 3)(x – 4)
(a)
5
Mark
4
Answer
Correct region
shaded
Working
3
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
1 1
, 3 ); –1, –1)
2 2
M1 for x(x + 5) OR – y(x + 5) seen
M1 for x(x + 5) and – y(x + 5) seen
A1 oe
M1 for (x ± 3)(x ± 4)
A1 oe
(A1 for correct shading for one inequality)
((5, –1); (
M3 for drawing all 3 lines correctly
(M2 for drawing 2 lines correctly)
(M1 for drawing one line correctly)
A2 for Correct shading of triangle with vertices
Notes
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
45
6
5
Solution
(b)
Sketch graph
p = 81
Answer
Correct curve
OR
b2 = 4ac
182 = 4p
p = 324 ÷ 4
1
be – b
2
1
p = (– b)2
2
For equal roots they must both
Working
(a)
(ii)
(i)
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
2
4
5
Mark
1 2
b)
2
1
b
2
M1 for realisation that solutions lie on y = 0
A1 for x ft from their line
B1 for drawing suitable axes on grid
M1 for calculating points for values of x from x = –2 to +4
A1 for all points correct
A1 for drawing smooth curve through their correct points
M1 for drawing x and y axes with U shaped quadratic graph
and for showing the curve touching the positive x axis at one
point only
A1 for establishing turning point at (9, 0)
A1 if curve cuts y axis at (0, 81)
A1 for 81
OR
M1 for writing b2 = 4ac
M1 for substituting 182 = 4p
A1 for 81
M1 for p = (–
M1 for writing that for equal roots they must both be –
Notes
46
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
8
7
Question
Working
t(k – 2) = 2k + 1
tk – 2t = 2k + 1
tk – 2k = 2t + 1
k(t – 2) = 2t + 1
y= –
1
or y = 3
3
2
x = – 2 or x = 4
3
x = 2y – 2
(2y –2)2 – y2 = 7
4y2 – 8y + 4 – y2 = 7
3y2 – 8y – 3 = 0
(3y + 1)(y – 3) = 0
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
2t + 1
t −2
2
1
,y= –
3
3
k=
x=–2
x = 4, y = 3
Answer
4
6
Mark
A1 for
2t + 1
−2t −1
or
oe
t −2
2−t
M1 for attempt to multiply LHS by (k– 2) or sight of t(k –2) or
tk – 2t (ignore RHS)
M1 for attempt to subtract 2k from LHS or sight of tk – 2k
(ignore RHS) or attempt to subtract tk to give
–2t + 1 = 2k – tk (ignore LHS)
M1 for attempt to factorise for k e.g. k(t – 2) or k(2 – t)
A1 for y = –
1
or y = 3
3
2
A1 for x = – 2 or x = 4 linked to y values
3
M1 for rearranging the linear equation in terms of x or y
M1 for substituting rearranged linear equation into the
quadratic equation
M1 for simplifying to get a quadratic in one variable
M1 for factorising to obtain (3y + 1)(y – 3) = 0 oe
Notes
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
47
12
11
10
9
2a + d = 47
a + 29d = –62
2a + d = 47
2a + 58d = –124
57d = –171
d = –171 ÷ 57
2a – 3 = = 47 2a = 50
S = 30 (2 × “25” + 59 × ”–3”)
(b)
–b = 7, c = 3
b
= 3.5
a
c
= 1.5
a
–
Working
(a)
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
5 2 13
) –
2
4
E, B, A, D, C
(x –
–3810
a = 25
d = –3
2x –7x +3
2
Answer
3
2
2
3
3
Mark
b
c
= 3.5 and
= 1.5
a
a
B3 for all 5 correct
B2 for 3 or 4 correct
B1 for 1 or 2 correct
5
2
13
B1 for –
4
B1 for –
A1
M1 for substituting into S =
1
n(2a + (n– 1)d)
2
M1 for establishing 2a + d = 47 and a + 29d = –62
M1 for solving to find d (or a)
A1 for substituting to find a (or d)
A2 for 2x2 – 7x + 3
A1 for x2 ± 3.5x + 1.5 or 2x2 ± 7x + 3
M1 for establishing –
Notes
48
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
(2 + √2)(3 + √8) =
6 + 2√8 + 3√2 + √2×√8
=10 + 3√2 + 2√8
10 + 3√2 + 2√8
= 10 + 3√2 + (2 × 2 × √2)
15
OR
(2 + 2√2)(3 + √8) = (2 + √2)(3 +
2√2)
=6 + 4√2 + 3√2 + √ 2× 2√2
6 + 7√2 + √2 × 2√2
= 6 + 7√2 + 2 × 2=
x2 – 4x –5 ≤ 0
(x – 5)(x + 1) ≤ 0
Working
14
13
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
10 + 7√2
–1 ≤ x ≤ 5
Explanation given
Answer
4
3
3
Mark
4× 2
2 8
2 8 - terms may be simplified
OR
B1 √8 = 4 ×√2
M1 for 3 or 4 terms out of 4 correct in the expansion of
(2 + √2)(3 + 2√2)
A1 6 + 7√2 + √2 × 2√2
A1 cao
A1 10 + 7 2 cao
B1 =
8
A1 for 10 from 6 +
M1 3 or 4 of 6, 2 8 , 3 2 ,
and could be in a list
M1 for factorising (x – 5)(x + 1) ≤ 0
A1 ft for establishing the critical values
A1
M1 for recognising that the equation is a circle
A1 for drawing a circle centre (0, 0) and radius 4
A1 for attempt to draw a straight line through (1, 2) so that it
passes through the circle in two places with a statement of the
line passing through the circle twice or for explanation that the
circle is a closed curve and therefore the straight line has
infinite length and passes through the circle in two places oe
Notes
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
49
18
17
16
0.8
180
1
(3 + 9) × 30
2
(b)
2(x – 1) + 3(x + 1) = 5
5x + 1 = 5
10
1 or –3.5
Answer
30 ÷ 3 =
Alternative
(2x + 7)(x – 1)
2x + 7 = 0 or x + 1 = 0
−5 ± 25 + 56
4
−5 ± 9
4
−5 ± 52 − 4 × 2 × −7
2× 2
Working
(a)
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
4
2
2
3
Mark
−5 ± 9
4
M1 for attempt to multiply one term by a common
denominator or sight of 2(x – 1) or 3(x + 1) or 5
M1 for multiplying all terms by a common denominator or
sight of 2(x – 1) + 3(x + 1) = 5
M1 for attempt to clear brackets or sight of 5x + 1 = 5
A1 oe
M1 for attempt to find the area under the graph
A1 for 180
M1 for 30 ÷ 3
A1 for 10
A1 for 1 and –3.5
Alternative
M1 for (2x + 7)(x – 1)
M1 for 2x + 7 = 0 or x + 1 = 0
A1 for 1 and – 3.5
M1 for method to establish x =
−5 ± 52 − 4 × 2 × −7
2× 2
M1 for correct substitution into quadratic formula or sight of
Notes
16
50
EA033597 – Sample Assessment Materials – Edexcel Level 2 and Level 3 Awards in Algebra
© Pearson Education Limited 2012
21
20
19
k
9
26 sq units
Sketch
(b)
(½ (8 + 0)+(9 + 8 + 5)) × 1
= 4 + 22
Sketch
(a)
3
2
2
1
3
3
4
36
x2
(c)
F=
Mark
1
1
k
=F= 2
2
x
x
Answer
9
k = 36
4=
F ∝
Working
(b)
(a)
Question
Award in Algebra AAL30
Level 3 (AAL30) mark scheme
values multiplied by 1
A1 for 26 ± 2 units
M1 for
1
(first + last ordinate) or sum of other ordinate values
2
1
M1 for
(first + last ordinate) and sum of other ordinate
2
M1 for a reflection in a horizontal line
A1 for a reflection in the x axis
M1 for a horizontal translation by 2
A1 for translation by 2 to the left
B1 oe
B1 cao
1
k
or F = 2
2
x
x
M1 for method to establish k = 36 or substituting to get
k
4=
9
36
A1 for F = 2
x
M1 for F ∝
Notes