Contents
3. A guide to Maths Courses at OSFC
5. Indices
6. Indices - Exercise A
7. Fractional indices – Exercise B
8. Fractional Indices – Exercise C
9. Negative Indices – Exercise D
10. Two special powers – Exercise E
11. Surds
12. Surds - Exercise F
13. Quadratics – Exercise G
14. The Difference of two squares - Exercise H
14. Factorising other quadratic expressions – Exercise I
15. Solving quadratic equations – Exercise J
16. Solving quadratic equations - Exercise K
17. Simultaneous Equations- Ex L
18. Common Misconceptions – Exercise M
19. Note for Further Mathematicians
19. Harder Equations – Exercise N
20. Graphs of Quadratic Equations
21. Graphs of Quadratic Equations -Exercise O
22. Changing the subject of the formula- Exercise P
23. Answers
2
A Guide to A-Level Mathematics Courses at
Oldham Sixth Form College
The Maths Department offers 4 different AA-Levels designed to match the abilities and
ambitions of students. Experience shows that enjoyment of the subject and eventual exam
success depends to a large extent on your being enrolled onto the correct one of these four
A-Levels.
All four of our A-Levels require you to purchase a graphical calculator (Casio fx-9860GII).
These can be purchased through the college at the start of the year. Students who qualify for
the bursary receive a 50% discount on the cost of the calculator.
A-Level Mathematics
•
Minimum GCSE grade B.
You will develop your existing knowledge of topics from GCSE in algebra, geometry,
trigonometry and vectors. Additionally you will either study a Statistics module or a Mechanics
module if you choose to do Physics.
AS Mathematics is a challenging and demanding subject. Algebraic competency is
extremely important, since the Core units rely heavily on this skill. A high grade at GCSE is
not always an automatic indicator of this, and even students achieving an A or A* at GCSE
will need to work especially hard to succeed.
A-Level Maths combines well with most other subjects, particularly academic ones.
A-Level Use of Maths
•
Minimum GCSE grade B
You will study an Algebra module, a Decision module and either a Statistics module or a
Dynamics module if you choose to do Physics. All the applied modules are FSMQs (Free
Standing Maths Qualifications), which, if passed, count as qualifications in their own right,
gaining UCAS points.
In each module there is a strong emphasis on relating the theory covered to real life
situations. This makes the course especially suitable for those who like to see the practical
relevance of what they learn or who might be studying the subject to support their other
choices.
A-Level Use of Maths combines particularly well with Sciences, Humanities, Social Sciences
and Arts.
3
A-Level Statistics
•
Minimum GCSE grade B.
This is a pure Statistics A-Level and you will study three modules of Statistics in your first
year. Statistics is about applying mathematical techniques to describe, predict and
understand numerical behaviour in a wide variety of real life situations. Successful
students are those able to work methodically and accurately and who can relate their
results to the context of the situation being studied.
This is a great option for students who want to see Maths made relevant to real-life
situations. Statistics may be taken alongside A-Level Mathematics and combines very well
with subjects that include some element of data analysis, such as Psychology, Biology,
Sociology, Economics, Geography and Business Studies.
A-Level Further Mathematics
•
Minimum GCSE grade A, plus a strong GCSE profile.
In addition to the topics covered in A-Level Mathematics you will study additional modules
in algebra, mechanics and statistics.
Further Mathematics must be studied alongside A-Level. It is suitable only for those students
with a passion and natural flair for Mathematics. Typically, students on the course will have
achieved very high grades in all their mathematical and scientific subjects at GCSE, simply
because they will not have found them at all challenging!
It is important to remember that choosing Mathematics and Further Mathematics counts as
two subjects on your timetable. Further Mathematics is an essential choice for those who
want to study Mathematics at university and a good choice for those who want to study
related courses such as Engineering or Physics
Any queries please contact Claudia Broad, Head of Maths, email: [email protected]
4
Indices
An index is another name for power.
power
The index is the number of times a (base) number is multiplied by itself.
base
aⁿ
index or power
E.g 53 = 5 × 5 × 5 = 125
The following rules only apply when multiplying or dividing powers of the
same number or variable (same base).
Rule for multiplying numbers in index form (add the powers):
1. 34 × 35 = 3(4+5) = 39
2. 104 × 10-2 = 10(4+-2) = 102
3. am × an = a(m+n)
Rule for dividing numbers in index form (subtract the powers):
1. 45 ÷ 42 = 4(5-2) = 43
2. 10-2 ÷ 10-4 = 10(-2--4) = 102
3. am ÷ an = a(m-n)
Rule for raising a power term to a further power(multiply the powers):
1. (62)4 = 62x4 = 68
2. (7-2)4 = 7-2x4 = 7-8
3. (am)n = amn
5
Indices
Exercise A (Non
NonNon-calculator)
Write as a single power:
1. 56 × 5-3 =
2. 4 × 42 =
3. 64 ÷ 6-2 =
4. 6-3 ÷ 64 =
5. (42)3 =
6. (43)-2 =
7. (4-2)-3 =
8. (47)0 =
9. 42 × 25 =
10.
(22)3 × 25 =
Simplify the following expressions:
1. 6LM ÷ 2LN =
2. 2LN O M × 4LM O =
3.
PQR
4.
SQR TU
MQ
=
MQT
=
5. Can you simplify this?
NQW TX W × PQTX U
YQTW X
6
Fractional Indices
[
When m = 1
i.e Indices of the form
Z\
1
n
m is the power and n is the root
- this means ‘the nth root of’.
Examples
_
1. 64U = √64
U
= √4 × 4 × 4
=4
U
_
2. 9W = √9 = 3
_
3. La = √L
a
Exercise B (Non
(Nonon-calculator)
Evaluate:
_
1. 27U =
_
2. 81R =
_
3. 125U =
_
4. 100 000b =
_
5.
Nd W
cMPe
6.
Nf U
cgNde
=
_
=
7
h
a
Fractional Indices
Indices of the form L : the nth root of L raised to the power of m
Examples:
M
U
W
1. 16 = i√16j = 4M = 64
R
Y
2. 32b = i√32j = 2Y = 16
b
h
3. L a = ( √L)k lm √Lk
a
a
Exercise C (Non
(Nonon-calculator)
Evaluate
1. 1252/3 =
4. (27p3) 2/3
2. 163/2 =
5. (x4 y10)3/2
3. 2434/5 =
Rewrite in index form:
N
6. i √oj =
U
7. pqM =
R
P
O
W
E
R
n2
n3
n4
n5
n6
n7
N
U
M
B
E
R (n
n)
2
3
4
5
6
7
8
9
10
11
4
9
16
25
36
49 64 81 100 121
8
27
64 125 216 343 512 729 1000 1331
16 81 256 625 1296
32 243 1024
64 729
128
8
Negative Indices
Consider the index rule for dividing: 52 = 5 2 – 3 = 5 -1
53
Numerically, 52 ÷ 53 = 25 ÷ 125 = ⅕
Therefore, 5-1 = ⅕
i.e The negative sign means ‘one over’ (reciprocal)
1. 4sM =
2. 9sg =
3. t sQ =
g
YU
=
g
PY
g
S
g
uv
4. 4qsM = 4 × qsM = 4 ×
g
kU
=
Y
kU
Exercise D. (N
Nonon-calculator)
Write each of these in fraction form.
1) 10-5 =
2) 8-2 =
3) t -1 =
4) 4q-4 =
M
5) t sd
Y
9
Indices – Two Special Powers
A number raised to the power 1 stays the same number.
e.g
51 = 5
A number raised to the power 0 is always equal to 1.
e.g
40 = 1
Because:
43 ÷ 43 = 43-3 = 40 and
64÷64 = 1
Exercise E. (Non
(Nonon-calculator)
Simplify the following
_
1. 81R
2. 27w
xU
3. 4 W
4. y gw × y × y d
_
5.
gNd U
c z e
6.
gP W
cSe
xU
10
Surds
Surds are roots of numbers: i.e. √2 , √5, √10
Rules for multiplying numbers in surd form.
1. √4 × √9 = √4 × 9 = √36 = 6
2. 2√3 × 5√7 = 2 × 5√3 × 7 = 10√21
3. √L × √O = √LO
4. t √L × y√O = ty√LO
Rules for dividing numbers in surd form.
1. √4 ÷ √9 =
√Y
√S
Y
={ =
S
2. 2√18 ÷ 5√2 =
N√gz
d √N
N
M
N
= {
d
gz
N
N
N
P
g
= √9 = × 3 = = 1
d
d
d
d
Q
3. √L ÷ √O = {
T
u
Q
4. t √L ÷ y√O = {
| T
In Maths, we would prefer to write the square root of 300 in surd form rather
than as a decimal because it is EXACT.
√300 = √(100 × 3) = √100√3 = 10√3
Similarly
and
√50 = √(25 × 2) = √25√2 = 5√2
√80 = √(16 × 5) = √16√5 = 4√5
11
Exercise F (Non
(Nonon-calculator)
Surds
Simplify each expression. Leave your answer in surd form where necessary
(do not use decimals).
1)
√2 × √3
2)
√6 × √6
3)
√54 ÷ √6
4)
√12 ÷ √3
5)
√5 × √8 × √8
6)
4√2 × 5√3
7)
4 × 2√2 × 3√18
8)
√75
9)
√700
10)
√128
Can you simplify these?
11)
2√5 × 3√6 ÷ √30
12)
L√O ÷ ~√O
NB it is acceptable to write the square root of 10t as √10t or √(10t) but not
√10t; the line above acts as a bracket!
12
Quadratic Expansion
Quadratic Expressions
Expressions such as 5p(2p-3) and (3y + 2)(4y – 5) can be expanded
(multiplied together) to give quadratic expressions.
and
(3y + 2)(4y -5) = 12y2 -7y -10
5p(2p – 3) = 10p2 – 15p
There are many methods used for expanding such expressions as
(t + 5)(3t – 4) but the rule is to multiply everything in one bracket by
everything in the other bracket.
Example: Expand (x + 3)(x + 4)
Now simplify:
= x ² + 4x + 3x + 12
= x² + 7x + 12
Example: Expand (x - 3) x + 5)
Now simplify:
= x² + 5x - 3x - 15
= x² + 2x - 15
Example: Expand (x - 7)(x - 4)
Now simplify:
= x² - 4x - 7x + 28
= x² - 11x + 28
You must be careful with negative signs.
Remember positive × positive = positive
positive × negative = negative
negative × positive = negative
negative × negative = positive
Exercise G. (Non
(Nonon-calculator)
Expand the following expressions:
1.
2 x (x - 1)
2.
-3 x (5 – 2 x)
3.
(x + 3)( x + 2)
4.
(x – 3)( x + 4)
5.
(3 x – 2)(2 x + 5)
6.
(2 - x)(1 - 3 x)
7.
(7 x – 1)(7 x + 1)
13
Quadratic Expressions - Factorising
A quadratic expression with only two terms, both of which are perfect squares
separated by a minus sign, is called the difference of two squares.
Eg. x² - 9, x² - 25, x² - 100, etc.
Recognise the pattern as x² minus a square number n²
Its factors are (x + n)(x – n)
Factorise x² - 36. Recognise 36 as 6², this then becomes (x + 6)( x – 6)
Similarly, recognise 9x ² - 100 as 3x squared minus ten squared
which will factorise to (3x + 10)(3x – 10)
Exercise H (Non
(Nonon-calculator)
Factorise the following expressions.
1. x ² - 9
2 k² – 100
3. 16 x ² - 9
4. 16y² - 25 x ²
Now consider this example :
Factorise 3x ² + 8x + 4
Both signs are positive, so both bracket signs must be positive
3 has only 3 × 1 as factors, the brackets must start (3x + )(x + )
Factors of 4 are 4 × 1 and 2 × 2
We could try (3x + 4)(x + 1) or (3x + 1)( x + 4) or (3x + 2)(x + 2)
There are many ways to test which of these works but the safest is to expand
them and see! The correct factorisation is (3x + 2)(x + 2)
Note : You should always check your solution by expanding the brackets.
**Be careful with negative signs!
Exercise I
Factorise the following expressions.
(1) x 2 + 6 x + 8
(5) 3 x 2 + 3 x - 36
(2) z2 + 10z + 16
(6) 5 x 2 - 9 x - 2
(3) y2 + 8y + 7
(7) 3y2 - 14y + 8
(4) 2 x ² + 5 x + 2
(8) 7 x ² + 8 x + 1
14
(9) 4 x 2 - 25
(10) 5 x 2 + 10 x
(11) ¼ x 2 – 9
(12) 3t²- 16t – 12
Solving Quadratic Equations
There are three main methods used to solve quadratic equations
algebraically:
1. By factorising.
2. Using the quadratic formula (at AS this needs to be memorised)
3. By completing the square (to taught in AS)
1.By Factorising
Factorising.
ing.
Example:
Solve 2 x 2 + 5 x = 3 firstly rearrange the equation to make it equal zero
2 x 2 + 5 x - 3 = 0 now factorise the left hand side
→ (2 x - 1) (x + 3) = 0
∴ 2 x - 1 = 0 or x + 3 = 0
∴ either x = ½ or x = -3
Exercise J
Solve the following quadratic equations:
(1) x 2 - 6 x + 5 = 0
(6) 13 x 2 = 11 – 2 x
(2) x 2 - 4 x = 5
(7) 4x2 - 4x = 35
(3) 4 x 2 + 4 x - 35 = 0
(8) 7 x 2 + 5 x + 4 = 2 x 2 - x + 3
(4) 4 x 2 - 25 = 0
(9) 4 x 2 + 2 x - 25 = 2 x
(5) 3 x 2 - 8 x - 3 = 0
(10) 2 x 2 - x = 6
2. Using the quadratic formula.
Many quadratic equations cannot be solved by factorisation. One way of
solving this type of equation is to use this formula:
The solution of the equation ax ² + bx + c = 0 is given by
− b ± b 2 − 4ac
x=
2a
15
Example: Solve the following equation using the quadratic formula. (Give your
answer in surd form.)
2x ² + x – 8 = 0
(note: a = 2, b = 1, c = -8)
Substituting into the formula gives
x =
−1±
1 − (4 × 2 × − 8)
−1±
=
4
4
65
(this is in surd form)
Exercise K
Solve the following equations, giving your answers in surd form.
1. x ² - x – 10 = 0
2. 4x ² + 9x + 3 = 0
3. 6x ²+ 12x + 5 = 0
4. 4x ² - 9x + 4 = 0
16
Simultaneous Equations
There are several methods for solving simultaneous equations. Where two
linear equations are involved, elimination is usually the best method.
Example: Solve the following
3x + 4y = 26
7x – y = 9
(1)
(2)
First make either the coefficient of the x’s or the y’s the same. In this case it is
easiest to make the y coefficients the same. So multiply equation (2) by 4. So
we get
28x – 4y = 36
(3)
Because the signs in front of the y’s are different, we add the equations (if they
were the same we would subtract).
(3) + (1)
31 x = 62
Now solve for x by dividing by 31.
x=2
Substitute into one of the original equations to find y.
7 × 2 – y = 9, so y = 5
Check by substituting into the other original equation.
3 × 2 + 4 × 5 = 26 (correct)
Exercise L :Solve the following
1)
5x + 3y = 1
3x – y = 9
2)
3x + y = 19
5x + 2y = 32
3)
2x + 3y = −1
7x + 4y = 16
4)
2x + 3y = −8
3x – 4y = 5
17
Common Misconceptions
A misconception is a false view of how things are. In mathematics, some things
that look sensible are, in fact, completely wrong. It is important that you are
aware of common mistakes some students make and don’t make them
yourself!
A good way to check is to substitute values.
For example, to show that: . 1 + . 1 is NOT equal to .
x
y
substitute x = 1 and y = 1 .
Then
1 1
+ =2
1 1
and
1 .
x+y
NOT the same.
1
1
=
1+1 2
Exercise M
Are the following true or false? If false write down the correct answer.
1. 3(t − y) = 3t − 3y
2. (t + y) N = t N + y N
3. (L + O) + 5 = L + 5 + O + 5
4.
u…g
5.
u
u
u…g
u
g
= + =1+
g
u
u
u
u
u
u
= + =1+ =1+t
u
g
g
6. p† + ‡ = p† + p‡
7. p†‡ = p†p‡
8. (6o)N = 6o N
9. 2t sM =
g
Nu U
18
Note :
The following pages cover work that your AS Maths teacher will be
introducing in the first two weeks of the course. Do as much as you can from
these exercises –there may be some questions you are unable to answer
simply because you have not covered the work at school.
If you have applied to study Further Maths at OSFC then we strongly recommend
you work through these exercises. As a Further Maths student you must be quicker
to take on board new concepts, particularly in manipulating algebraic expressions
and rearranging formulae. The ability to sketch graphs and recognise the graphs
of the equations you have worked with at GCSE is also important. This may be
daunting to many students but if you are likely to achieve a grade A* or a high
grade A at GCSE Maths and you enjoy the challenge of a more demanding Maths
then Further Maths might be for you !
Harder Equations
Exercise N:
Using all your algebraic skills to solve the following equations:
1.
(t + 4)N = 4
2.
Mu…S
3.
4u = 64
4.
3(2t + 3) = 5 − 4(3 − t)
5.
√t − 9 = 5
u
u
=5
6.
u…d
7.
√4t Y = 50
8.
d(u…M)W
9.
2usM = 8
N
=4
− t N = 3t N
_
10. t sW = 4
19
Graphs of Quadratic Functions
A quadratic function is usually written in the form a x 2 + b x + c where a, b
and c are constants. All quadratics are parabolas and look like:
a<0
Maximum
point
a>0
Minimum
point
NOTICE – the graphs are symmetrical and have one turning point
(a maximum or a minimum)
• We can find the point of intersection with the y-axis by substituting x=0
into the equation.
For the equation y = x 2 + 2 x – 8
at x = 0 we obtain y = −8.
• We can find any points of intersection with the x-axis by substituting
y = 0 into the equation.
For the equation y = x 2 + 2 x – 8
if y = 0 we obtain
which can be factorised
0 = x2 + 2 x – 8
0 = (x + 4)( x – 2)
and solved
to give
x = - 4 and x = 2
the two points on the x -axis.
The picture below is y = x 2 + 2 x – 8 showing the x axis intercepts of -4 and 2
as well as being able to see that the y axis intercept is -8.
20
Graphs of Quadratic Functions
Exercise O
Sketch the following curves, showing where they intersect with each axis.
1.
y = x2 + 5 x + 6
2.
y = x2 + x – 2
y
y
x
3.
x
y = 25 – x 2
4. y = - x 2 + 5 x - 6
y
y
x
x
21
Changing the Subject of a Formula
Examples
Make t the subject of the following formulae :
1. L + qt N = ˆ
subtracting L from both sides
qt N = ˆ − L
tN =
‰sQ
t={
2.
dividing by q
k
‰sQ
square rooting
k
q(L + t) = y
expanding the brackets
subtracting qL from both sides
qL + qt = y
qt = y − qL
t=
3.
|skQ
k
dividing by q
ty − 3 = 4y − 5t
ty + 5t = 4y + 3
t(y + 5) = 4y + 3
t=
Y|…M
|…d
Ensure all t terms are on the same side
Factorise
Divide by y + 5
Exercise P
Make t the subject of the following formulae :
1. Lt + O = ~
5.
u
Q
=
T
X
u
2. y(t − Š ) = †
3. o − Lt = m
4.
6. q(‹ − t) = ˆ
7. †t N + Š = ‹
8. L − t N = O
9. ty + 4 = 5t + 7
10. ty − † = †t + ‡
22
|
+† =‡
Answers
1.
Exercise A :
53
3.
66
3.
4.
6-7
4.
5.
46
5.
6.
7.
8.
9.
10.
4-6
46
40
29
211
1.
2.
3.
Simplify:
4.
5.
1.
3a
6.
2.
3.
4.
5.
8a5b4
2a3
3a3b2
3a2c4
2
1.
2.
3.
4.
5.
6.
1.
2.
3.
4.
5.
6.
1.
43
2.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Exercise B:
3
3
5
10
5
6
3
5
Exercise D:
g
g
=
gwb
gw www
1
64
1
o
4
‡Y
3
4t d
Exercise E:
3
1
1
8
y16
5
2
27
64
Exercise F:
F
√6
6
3
2
8√5
20√6
144
5√3
10√7
10. 8√2
11. 6
12. L
~
Exercise C:
25
64
81
9p2
x 6y15
Rewrite as :
t2/3 7. m3/4
23
1.
2.
3.
4.
5.
6.
Exercise G:
2 x2 – 2 x
-15 x + 6 x 2
x2 + 5 x + 6
x 2 + x – 12
6 x 2 + 11 x – 10
2 - 7 x + 3 x2
7.
49 x 2 – 1
1.
2.
3.
Exercise H:
H
(x + 3)( x - 3)
(k + 10)(k - 10)
(4 x + 3)(4 x -3)
4.
(4y + 5 x)(4y – 5 x)
1.
2.
3.
4.
5.
6.
1.
False
False
7. True
8. False
4.
5.
True
False
9. False
2.
3.
or x = -1
x = 2.5 or x = -2.5
x = 2 or x =-1.5
2.
1.
2.
3.
4.
1.
2.
3.
4.
5.
24
gM
9.
10.
4.
(x + 4)( x + 2)
(z + 8)(z + 2)
(y + 7)(y + 1)
(2 x + 1)( x + 2)
(3 x – 9)( x + 4)
(5 x + 1)( x – 2)
(3y – 2)(y – 4)
(7 x + 1)( x + 1)
(2 x + 5)(2 x – 5)
5 x (x + 2)
(½x – 3)(½ x + 3)
(3t + 2)(t – 6)
Exercise M:
True
6. False
gg
x = 3.5 or x = -2.5
3.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
x=
7.
8.
1.
Exercise I:
Exercise J:
x = 5 or x = 1
x = 5or x = -1
x = 2.5 or x = -3.5
x = 2.5 or x = -2.5
x = 3 or x = -1/3
x=
sg
d
or x = -1
Exercise K:
K
1 ± √41
2
−9 ± √33
8
−12 ± √24
12
9 ± √17
8
Exercise L:
L
x = 2, y = -3
x = 6, y = 1
x = 4, y = -3
x = -1, y = -2
Exercise N:
x = -2 and x = -6
x = 4.5
x=3
2√10
x = -8
x = 34
Nw
6. x = M
7. x = ±5
8. x = 5 ±
9. x =6
g
10. x = gP
Exercise O :
1.
2.
3.
4.
Exercise P:
1. t =
XsT
Q
5. t =
QT
9. t =
M
X
|sd
2. t =
6. t =
…Ž|
|
k‘s‰
10. t =
k
…’
3. t =
s
Q
7. t = {
|s
25
‘sŽ

4. t = y(‡ − †)
8. t = √L − O