Pre-course task for 2015 entry
Welcome to AS Maths!
I hope you’re enjoying a good rest this summer following your hard efforts in the GCSE exams. We want
you to arrive refreshed and ready for the challenging “step-up” from GCSE Maths to A Level Maths, but we
also want to make sure that you are absolutely prepared for the topics you will be studying and that you
understand what lies ahead.
The aim of this task is to give you a little more information about AS Maths, as well as allow you to
complete some work which will make your start with us much, much easier!
Studying A Levels is very different from GCSEs. You will be expected to actively seek out help if you are
stuck, be organised, hand in work on time and make good use of all the support that your new teachers will
offer you. You will need to get into some good habits to become a successful A Level Maths student and
completing this task is the first step.
You should have this task completed to the best of your ability by Monday 29th June which is our
Welcome Day and should hand it in to your subject workshop leader on this day. You will shortly be
receiving an invite to this event. Please ensure your name is written on the top of every page of
work you produce. This work comprises an element of the assessment used during induction.
Wishing you the best of luck for your GCSE results and looking forward to meeting you in September.
Mark Webber
Head of Department, Mathematics
What will happen when I arrive in September?
Your first couple of weeks will begin by going over the higher tier GCSE algebra topics that are essential
for being able to study AS Maths. You will be able to get a feel for whether AS Maths is right for you, or
whether you would be better suited to our courses AS Use of Maths or AS Statistics.
You will be given tests over this time and we will also look at your attendance, your reliability in handing in
homework and the quality of the work you submit. All of these will be taken into consideration to decide
whether you will continue on AS Maths, or move onto one of our other Mathematics courses.
What if I am asked to move onto a different Maths course?
Our aim is for you to be as successful as possible during your time at Huddersfield New College – this
means ensuring that you are on a course that is best suited to your skills and abilities.
AS Maths suits those students who enjoy and have ability in algebra. Read through the section ‘AS Maths
Content’ and you will see just how much algebra is involved, unlike at GCSE.
If your skills are in using and applying mathematics, for instance using a graphical calculator, then AS Use
of Maths may be a better option. If you are studying Psychology, Geography or Science then AS Statistics
could be an even better match.
What if I am stuck and really struggle with this task?
During your time at Huddersfield New College we want you to develop as independent learners – that
means researching topics you are struggling with in your own time so that you can keep up with the
lessons.
Every exercise in this task also lists some handy websites you can go to if you are stuck and need some
extra help.
What will I study in AS Maths?
We use the AQA exam board. You will study three modules called Pure Core 1, Pure Core 2 and Decision
1. Each of these is assessed by a 90 minute exams at the end of the year (Pure Core 1 is non-calculator).
Pure Core 1 and Pure Core 2 are both algebra modules – you will spend your time solving equations and
manipulating algebra, and understanding the link between graphs and their equations.
Decision 1 is a module about algorithms: sets of instructions used for solving real-world problems that may,
for instance, be used in a computer program.
You can find more information at http://www.aqa.org.uk/subjects/mathematics/a-level/mathematics-6360
Pure Core 1 Content
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Surds
Solving quadratic equations by factorising, completing the square, and using the quadratic formula.
The Factor Theorem and the Remainder Theorem for polynomials.
Solution of nonlinear simultaneous equations.
Graphs of quadratic and cubic functions.
Simple transformations of graphs.
Coordinate geometry for straight parallel, perpendicular and intersecting lines.
Equations of circles.
Differentiation.
Equations of tangents and normals to curves.
Maxima and minima of functions.
Integration and areas beneath curves.
Pure Core 2 Content
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Transformations of graphs.
Sequences and series: recursive rules; nth term rules; geometric and arithmetic progressions; the
sum of a series; the limit of a sequence.
The binomial expansion of (1 + 𝑥)𝑛 .
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Trigonometry: the Sine and Cosine Rules; area of a triangle = 2 𝑎𝑏 sin 𝐶; formulae for arc length and
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area of sector; solution of trigonometric equations; knowledge and use of trigonometric identities.
Exponentials and logarithms.
Approximation of area beneath curves using the Trapezium Rule.
1
Decision 1
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The Bubble, Shuttle, Shell and Quick Sort algorithms for sorting data.
Graphs of networks: Eulerian trails and Hamiltonian cycles.
Finding minimum spanning trees using Prim’s and Kruskal’s algorithms.
Use of an ‘alternating path’ to obtain a matching from a bi-partite graph.
Finding the shortest distance between two points using Dijkstra’s Algorithm.
Route Inspection.
The Travelling Salesperson Problem.
Linear programming.
Exercise 1: Surds (Non-Calculator)
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You must show your working to score any marks on this exercise!
This work is typical of the skills you need to access the module Core 1, which is non-calculator.
If you are stuck on any questions in this exercise the videos at the following link may help:
http://www.waldomaths.com/video/Surds01/Surds01.jsp
Q1 Write
in the form k
, where k is an integer.
..........................................................
(2)
Q2
Expand (1 + √2 )(3 − √2 )
Give your answer in the form a + b √2 where a and b are integers.
........................................................... (2)
Q3
Write (5 − √5)2 in the form a + b√5, where a and b are integers.
........................................................... (2)
Q4
Rationalise the denominator of
14
.
√7
Give your answer in its simplest form.
...........................................................
(2)
Q5.
(a) Rationalise the denominator of
5
.
√2
..............................................................................................................................................
(2)
2
2
(b) Expand and simplify (2 + √3) − (2 − √3) .
..............................................................................................................................................
(2)
Q6
(a) Express 5√27 in the form √3 , where n is a positive integer.
..............................................................................................................................................
(2)
(b) Rationalise the denominator of
21
.
√3
..............................................................................................................................................
(2)
Q7
Rationalise the denominator of
(6−√5)(6+√5)
.
√31
Give your answer in its simplest form.
..............................................................................................................................................
(2)
Exercise 2: Indices (Non-Calculator)
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Q1.
You must show your working to score any marks on this exercise!
This work is typical of the skills you need to access the module Core 1, which is non-calculator.
If you are stuck on any questions in this exercise the videos at the following link may help:
http://www.waldomaths.com/video/Indices04/Indices04.jsp
Write these numbers in order of size. Start with the smallest number.
..............................................................................................................................................
(2)
Q2.
(a) Write down the value of 70
...........................................................
(1)
(b) Write down the value of 2−4
...........................................................
(1)
Q3.
(a) Write down the reciprocal of 5
...........................................................
(1)
(b) Evaluate 3−2
...........................................................
(1)
Q4.
(a) Write down the value of 10–1
..............................................................................................................................................
(1)
(b) Find the value of
..............................................................................................................................................
(2)
Q5.
(a) Find the value of
5°
..............................................................................................................................................
(1)
(b) Find the value of
27 1⁄3
..............................................................................................................................................
(1)
(c) Find the value of
2-3
..............................................................................................................................................
(1)
Exercise 3: Linear Equations
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You must show your working to score any marks on this exercise!
Every single topic of both of the modules Core 1 and Core 2, as well as many of the topics in
Decision 1, require you to manipulate algebra. You will need to be able to solve any linear equation
you could be shown.
Q1.
(a) Solve 2x + 3 = x – 4
x=..................
(2)
(b) Solve 4(x – 5) = 14
x=..................
(2)
Q2.
Solve
5𝑤−8
3
= 4𝑤 + 2
w= ......................
(3)
Q3.
Solve 3(x – 2) = x + 7
x=......................
(3)
(b) Solve
2−𝑦
5
=1
y=......................
(2)
Q4.
Solve
4𝑥−1
𝑥+1
+
5
2
=3
x=......................
(3)
Q5.
Solve
4(8𝑥−2)
3𝑥
= 10
...........................................................
(3)
Exercise 4: Quadratic Equations
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You must show your working to score any marks on this exercise!
Quadratic equations come up all the time in Core 1 and Core 2 topics. You need to be able to
solve them using all three methods: factorising, completing the square, using the quadratic formula.
If you are stuck follow the link to http://www.mathsrevision.net/gcse-mathsrevision/algebra/quadratic-equations
Q1.
Solve
3x2 – 4x – 2 = 0
Give your solutions correct to 3 significant figures.
..............................................................................................................................................
(3)
Q2.
Solve
2x2 + 5x – 3 = 0
..............................................................................................................................................
(3)
Q3.
Solve, by factorising, the equation
8x2 − 30x − 27 = 0
...........................................................
(3)
Exercise 5: Simultaneous Equations
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You must show your working to score any marks on this exercise!
Every single topic of both of the modules Core 1 and Core 2, as well as many of the topics in
Decision 1, require you to manipulate algebra. You will need to be able to solve any linear equation
you could be shown.
If you are unsure what to do in questions 4 and 5, watch the video at
http://www.waldomaths.com/video/SimEq03/SimEq03.jsp .
Q1.
Solve the simultaneous equations
4x + 7y = 1
3x + 10y = 15
x=......................
y=......................
(4)
Q2.
Solve the simultaneous equations
5x + 2y = 11
4x – 3y = 18
x=......................
y=......................
(4)
Q3.
Solve the simultaneous equations
4x + y = 25
x − 3y = 16
x =...........................................................
y =...........................................................
(3)
Q4.
Solve the simultaneous equations x2 + y2 = 9
x+y=2
Give your answers correct to 2 decimal places.
x=...............y=...............
or x = . . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . .
(6)
Q5.
Solve the simultaneous equations
x2 + y2 = 25
y = 2x + 5
x = . . . . . . . . . . . . . . and y = . . . . . . . . . . . . . .
or
x = . . . . . . . . . . . . . . and y = . . . . . . . . . . . . . .
(6)