MV16 Mathematics Department
Induction Booklet
We want you to start AS Maths with the skills that you need in order to be
successful. These are higher level GCSE skills, which this booklet provides
examples for you to practice. In order for us to assess your suitability for the
AS course, we will look carefully at the work you have done throughout, but
we will assess your ability by marking the yellow practice booklet test.
Please use separate file paper to answer the questions
CONTENTS
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Removing brackets
Linear equations
Simultaneous equations
Factorising
Change the subject of the formula
Solving quadratic equations
Indices
Practice booklet test
Page 2
2
3
3
4
5
6
7
To help you with these exercises you could consider using the following book.
AS-Level Maths Head Start
Published by CGP Workbooks
ISBN: 978 1 84146 993 5
Cost: £3.99 (Amazon)
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Chapter 1: REMOVING BRACKETS
EXERCISE A
Multiply out the following brackets and simplify.
1.
7(4x + 5)
7.
(x + 2)(x + 3)
2.
-3(5x - 7)
8.
(t - 5)(t - 2)
3.
5a – 4(3a - 1)
9.
(2x + 3y)(3x – 4y)
4.
4y + y(2 + 3y)
10.
4(x - 2)(x + 3)
5.
-3x – (x + 4)
11.
(2y - 1)(2y + 1)
6.
5(2x - 1) – (3x - 4)
12.
(3 + 5x)(4 – x)
EXERCISE B
Multiply out
1.
(x - 1)2
4.
(x + 2)(x - 2)
2.
(3x + 5)2
5.
(3x + 1)(3x - 1)
3.
(7x - 2)2
6.
(5y - 3)(5y + 3)
Chapter 2: LINEAR EQUATIONS
Exercise A: Solve the following equations, showing each step in your working:
1)
2x + 5 = 19
2) 5x – 2 = 13
4)
5 – 7x = -9
5) 11 + 3x = 8 – 2x
3) 11 – 4x = 5
6) 7x + 2 = 4x – 5
Exercise B: Solve the following equations.
1)
5(2x – 4) = 4
3)
8 – (x + 3) = 4
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2)
4(2 – x) = 3(x – 9)
4)
14 – 3(2x + 3) = 2
2
EQUATIONS CONTAINING FRACTIONS
Exercise C: Solve these equations
1)
3)
5)
7)
1
( x  3)  5
2
y
y
3 5
4
3
2)
4)
7x 1
 13  x
2
x  1 5x  3
2x 

2
3
6)
8)
2x
x
1   4
3
3
x2
3 x
 2
7
14
y 1 y 1 2y  5


2
3
6
5 10
1
2 
x x
Chapter 3: SIMULTANEOUS EQUATIONS
Exercise E:
Solve the pairs of simultaneous equations in the following questions:
1)
x + 2y = 7
3x + 2y = 9
2)
x + 3y = 0
3x + 2y = -7
3)
3x – 2y = 4
2x + 3y = -6
4)
9x – 2y = 25
4x – 5y = 7
5)
4a + 3b = 22
5a – 4b = 43
6)
3p + 3q = 15
2p + 5q = 14
Chapter 4: FACTORISING
Common factors
We can factorise some expressions by taking out a common factor.
Exercise A
Factorise each of the following
1)
2)
3x + xy
2x3 – 6x2
6)
8a5b2 – 12a3b4
7)
5y(y – 1) + 3(y – 1)
4x – 2xy
2
2
3)
pq – p q
4)
3pq - 9q2
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5)
2
3
Factorising quadratics
Exercise B
Factorise
1)
x2  x  6
8)
10 x 2  5 x  30
2)
x 2  6 x  16
9)
4 x 2  25
3)
2 x2  5x  2
10)
x 2  3x  xy  3 y 2
4)
2 x 2  3x
11)
4 x 2  12 x  8
5)
3x 2  5 x  2
12)
16m 2  81n 2
6)
2 y 2  17 y  21
13)
4 y 3  9a 2 y
7)
7 y 2  10 y  3
14)
8( x  1) 2  2( x  1)  10
Chapter 5: CHANGING THE SUBJECT OF A FORMULA
Exercise A
Make x the subject of each of these formulae:
1)
y = 7x – 1
2)
y
x5
4
3)
4y 
x
2
3
4)
y
4(3 x  5)
9
Rearranging equations involving squares and square roots
Exercise B:
Make t the subject of each of the following
wt
32r
1)
P
3)
1
V   t2h
3
5)
Pa 
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w(v  t )
g
2)
P
wt 2
32r
4)
P
2t
g
6)
r  a  bt 2
4
More difficult examples
Sometimes the variable that we wish to make the subject occurs in more than one place in
the formula. In these questions, we collect the terms involving this variable on one side of
the equation, and we put the other terms on the opposite side.
Exercise C
Make x the subject of these formulae:
1)
3)
ax  3  bx  c
2x  3
y
5x  2
2)
4)
3( x  a)  k ( x  2)
x
x
 1
a
b
Chapter 6: SOLVING QUADRATIC EQUATIONS
Method 1: Factorising
Method 2: Using the formula
Recall that the roots of the quadratic equation ax 2  bx  c  0 are given by the formula:
 b  b 2  4ac
x
2a
EXERCISE A
1) Use factorisation to solve the following equations:
a)
x2 + 3x + 2 = 0
b)
x2 – 3x – 4 = 0
c)
x2 = 15 – 2x
2) Find the roots of the following equations:
a)
x2 + 3x = 0
b)
x2 – 4x = 0
c)
4 – x2 = 0
3) Solve the following equations either by factorising or by using the formula:
a)
6x2 - 5x – 4 = 0
b)
8x2 – 24x + 10 = 0
4) Use the formula to solve the following equations to 3 significant figures. Some of the
equations can’t be solved.
a)
x2 +7x +9 = 0
b)
6 + 3x = 8x2
c)
4x2 – x – 7 = 0
d)
x2 – 3x + 18 = 0
e)
3x2 + 4x + 4 = 0
f)
3x2 = 13x – 16
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Chapter 7: INDICES
Exercise A
Simplify the following:
1)
b  5b
5
2)
3c 2  2c 5
3)
b c  bc
4)
2n 6  (6n 2 )
2
3
5)
8n8  2n3
6)
d 11  d 9
7)
a 
8)
 d 
8)
2
 
3
9)
82 / 3
10)
 0.04 
11)
 8 
 
 27 
12)
1
 
 16 
15)
x y 
3 2
4 3
More complex powers
Exercise B:
Find the value of:
1)
41/ 2
1/ 3
2)
27
3)
 19 
4)
1/ 2
2
1/ 2
52
5)
180
6)
7 1
7)
27 2 / 3
2/3
3 / 2
Simplify each of the following:
13)
14)
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2a1/ 2  3a 5 / 2
x3  x 2
2
4 1/ 2
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Practice Booklet Test
You may NOT use a calculator
If ax2 + bx + c = 0 then x =
1.
2.
3.
4.
 b  b 2  4ac
2a
Expand and simplify
(a) (2x + 3)(2x – 1)
(b) (a + 3)2
(c) 4x(3x – 2) – x(2x + 5)
Factorise
(a) x2 – 7x
(c) 2x2 + 5x – 3
(d) 6t2 – 13t + 5
Simplify
4x3 y
(a)
8x 2 y 3
(b) y2 – 64
(b)
3x  2 4 x  1
+
3
6
Solve the following equations
h  1 3h
(a)
+
=4
(b) x2 – 8x = 0
4
5
(c) p2 + 4p = 12
5.
Write each of the following as single powers of x and / y
1
x5
2 3
(a) 4
(b) (x y)
(c)  2
x
x
6.
Work out the values of the following, giving your answers as fractions
1
(a) 4-2
(b) 100
 8 3
(c)  
 27 
7.
Solve the simultaneous equations
3x – 5y = -11
5x – 2y = 7
8.
Rearrange the following equations to make x the subject
x2
1
(a) v2 = u2 + 2ax
(b) V = πx2h
(c) y =
3
x 1
9.
Solve 5x2 – x – 1 = 0 giving your solutions in surd form
With grateful thanks to Dr A Duncombe, MGSC for much of the content of this booklet
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