O level Maths Notes: Algebraic Fractions
Summary
Algebraic fractions involve polynomials, usually
expressed in terms of a polynomial divided by
other polynomials in – they may need to be added,
subtracted, multiplied or divided. The rules are the
same as for ordinary fractions:
To add or subtract, we make a common
denominator then subtract the numerators.
To multiply, we just “multiply across”, so that the
end result is
To divide, turn the dividing, second fraction,
upside down, and multiply.
Adding Algebraic Fractions
The new denominator is
In the diagram above the arrows indicate
multiplication
To illustrate, I will expand the brackets in the
numerator and add
The numerator
can be factorised into
There is a common factor
in
numerator and denominator which can be
cancelled.
Subtracting Algebraic Fractions
The new denominator is
In the diagram above the arrows indicate
multiplication
To illustrate, I will expand the brackets in the
numerator and subtract
There is no common factor.
Multiplying Algebraic Fractions
Just multiply across
There are no common factors so we cannot cancel.
Dividing Algebraic Fractions
Turn the second, dividing fraction upside down
and multiply as above:
O Level Maths Notes: Circles, Sectors, Arcs,
Segments and Perimeters
Length of Arc and Area of Sector
Area and Circumference of Circle
Area of Segment
O Level Maths Notes: Displacement, Velocity and
Acceleration
The relationship between displacement, velocity
and acceleration are much more general than is
given by the suvat equations, or the simple
equations of motion. These equations only apply
for constant acceleration, but in fact the
acceleration is not constant in general. The general
relationships are illustrated below.
So for example suppose we have the velocity and
we want to find the acceleration. We differentiate.
If we have the velocity and we want to find the
displacement, we integrate.
Example:
a) Find the acceleration when
b) Find the displacement s as a function of time if
a)
so when
b}
We are told that
when
so
O Level Maths Notes: Indices
This is a summary of the rules of indices
since is not defined
(1) and
(2)
(3) and
(4)
When evaluating an expression like (4), find the
root first because the may turn out to be a very
large number.
(5) and
(7) and
(6)
(8)
The rules may be used as in the following
examples:
Express 81 as a power of
Simplify
We apply in succession rules (5) and (6) above:
Arrange in order:
In order the list is
O Level Maths Notes: Similar Shapes - Ratios of
Length, Areas and Volumes
It is important to note the units
of length:
of length:
of volume:
We can relate these directly to the scale factors
between the lengths, areas and volumes of
similarly objects – that is, mathematically similar,
with only a scale factor to separate them. As
shown on the diagram below, if the scale factor for
lengths is then the scale factor for areas is and
the scale factor for volumes is
All the length of the sides of the solid on the right
are twice the lengths of corresponding sides on the
left.
If the area of the cuboid on the left is
volume is
the area of the cuboid on the right is
the volume of the cuboid on the right is
and the
All the length of the sides of the solid on the right
are three times the lengths of corresponding sides
on the left.
If the area of the cuboid on the right is
the volume is
and
the area of the cuboid on the left is
the volume of the cuboid on the left is
O Level Maths Notes: The Cosine Rule
If we have a triangle, and we need to calculate
angles and/or sides, we can only use simple
trigonometry and Pythagoras theorem if the
triangle is right angled. If the triangle is not right
angled, then we must use general formulae which
apply to any triangle. One of these is the Cosine
Rule. With the sides and angles of a triangle
labelled as below, small letters labelling sides
opposite large letters labelling angles, we may use
the formula:
We may use the formula in two ways:
By labelling the sides and angles as required we
can find the formula above to find a side when we
have two sides and the angle between the two
sides, or we have three sides and we want to find
an angle.
By rearranging the formula to make
the subject
–
and labelling the sides as required we
can find the angle
Example:
to 2dp.
Example:
degrees.
O Level Maths Notes: The Sine Rule
If we have a triangle, and we need to calculate
angles and/or sides, we can only use simple
trigonometry and Pythagoras theorem if the
triangle is right angled. If the triangle is not right
angled, then we must use general formulae which
apply to any triangle. One of these is the Sine
Rule. With the sides and angles of a triangle
labelled as below, small letters labelling sides
opposite large letters labelling angles, we may use
the formula
We can use the Sine Rule in two ways:
To find an angle given two sides and an angle, or
to find a side given a side and two angles. Even
though the Sine Rule has three terms, we only ever
use two of them at a time.
In the triangle above we need to find the angle C:
We don't have or so don't include the
We use the equation
term .
In the diagram below we need to find the side a.
We don't have or so don't include the
term.
We use the equation
to 2 dp.
O Level Maths Notes: Finding the Mode, Median
and Mean From Frequency Tables
Suppose we start from a frequency table of lengths
versus frequencies.
Length
Frequency
0-10
2
10-20
16
20-30
46
30-40
35
40-50
7
Finding the Mode
The mode is just the most common length interval,
with the highest frequency: 20-30.
To find the median interval we insert an extra
column, a cumulative frequency column.
To find the numbers in the cumulative frequency
column we find the totals of the frequency column
as we go down.
Length
Frequency
Cumulative
Frequency
0-10
2
2
10-20
16
2+16=18
20-30
46
2+16+46=64
30-40
35
2+16+46+35=99
40-50
7
2+16+46+35+7=106
Finding the Median
The median will lie in the length interval in which
the cumulative frequency just passes the halfway
point:
The total frequency is 106 so the halfway point is
The cumulative frequency just passes this point in
the row where the cumulative frequency is 64, so
the median class interval is 20-30
Note: It does not matter how the they label the
length intervals:
The calculations are the same as long as the
distribution is continuous – if the table involves
lengths heights etc.
Finding the Mean
We insert two extra columns – the midpoint and
midpoint*frequency columns.
Length Frequency Midpoint Midpoint*Frequency
0-10
2
5
10
10-20
16
15
240
20-30
46
25
1150
30-40
35
35
1225
40-50
7
45
245
Total
106
2870
The Mean is now the
Note: It does not matter how the they label the
length intervals:
The calculations are the same as long as the
distribution is continuous – if the table involves
lengths heights etc.
O Level Maths Notes: Inverting Functions
A function allows us to find
for any value of
We just substitute the value for into
If
then we find
when
by calculating
Inverting a function allows us to find x for any
value of
if this is possible.
The Procedure is:
1. Write
2. Rearrange the equation to make the subject.
3. Swap and over.
4. Replace by
5. The graph of
may be sketched by
reflecting the graph of
in the line
Example: The graph of
is shown below.
Make the subject:
Swap and
Replace by
O Level Maths Notes: Inverting Functions
A function allows us to find
for any value of
We just substitute the value for into
If
then we find
when
by calculating
Inverting a function allows us to find x for any
value of
if this is possible.
The Procedure is:
1. Write
2. Rearrange the equation to make the subject.
3. Swap and over.
4. Replace by
5. The graph of
may be sketched by
reflecting the graph of
in the line
Example: The graph of
is shown below.
Make the subject:
Swap and
Replace by
O Level Maths Notes: Simple Trigonometry –
Finding Sides of Right Angled Triangles
Simple trigonometry can be used with right angled
triangles to a side given a side and one of the
interior angles other than the right angle,
With the sides of the triangle as labelled above, we
can use one of the formulae:
Example: If we have and and want to find
For the triangle above, we have and and want to
find This means we use
Example: If we have and and want to find
For the triangle above, we have and and want to
find This means we use
Example: If we have and and want to find
For the triangle above, we have and and want to
find This means we use
O Level Maths Notes: Vectors 1
A vector is the from one point to another. The
vectors and
shown below are the same. Both
vectors go 1 to the right and 3 up. We can
represent this as
Vectors may be multiplied and added in a natural
way.
The angle between the x axis and the vector is
The magnitude or modulus or length of the vector
is
O Level Maths Notes: Curve Sketching and
Solving Graphical Inequalities
To solve the equation
could turn out to be
quite a tricky problem. To solve exactly you may
have to factorise the cubic equation
There is a simple to solve the inequality
approximately. You can sketch the curve
and find those values for which
This will be
the solution set of the inequality. The points to be
plotted are shown in the table below.
-3
-2
-1
0
1
2
3
-27
-8
-1
0
1
8
27
-9
-4
-1
0
-1
-4
-9
15
10
5
0
-5
-10 -15
-21
-2
3
0
-6
-6
3
The curve is sketched below.
To solve the inequality we find the intersection of
the curve with the line and read off the –
values for those points of intersection.
The line
intersects the curve at the points where
and
The curve is less than, or below the line
and
for
O Level Maths Notes: Simplifying Surds
We ALWAYS have to leave expressions in the
simplest form. Any expression involving a root is
called a surd, and we often have to write an answer
in surd form which implicitly means simplified
surd form:
or
where
are whole
numbers or fractions and and are as small as
possible.
For example:
Simplify
We can write 75 as the product of a number, 3, and
a perfect square, 25. Then we take the 25 outside
the square root but then we have to square root it:
This is the general procedure. We take out the
largest perfect square which is a factor of the
number to be rooted. The perfect square can be
either a whole number or a fraction.
Example:
Simplify
Example
Simplify
Example: Expand and simplify
Now add up all the terms to obtain
O Level Maths Notes: Vectors 2
ABCD is a parallelogram,
and
Because vectors can be moved and is parallel to,
and the same length as we can write
Similarly, since is parallel to, and the same
length as so we can write
Vectors add in a natural way so
and
Suppose now that we find the midpoint of AC and
call it M and we find the midpoint of BC and call
it N. Because M and N are the midpoints of AC
and BC respectively,
and
Suppose we want to find the vector vec NM.
We can go from N to C and then from C to M.
From N to C is and from C to M is
is since
is against the direction of
Parallel Vectors
Any vector that is a multiple of is parallel to
so is parallel to
O Level Maths Notes: Factorising General
Quadratics and Differences of Squares
Any expression of the form
cannot always
be factorised as with simple quadratics, by finding
factors of c. Instead there is a slightly more
complicated procedure, best illustrated by an
example:
Factorise
1. Take out any common factor. Every term in
the above expression has a factor 3, so we may
write the expression as
2. Multiply the coefficient of by the constant
term:
Find the two factors of this
product which add to give the coefficient of
which in this case is -1.
3. Rewrite the term in brackets using these two
factors:
4. Take out common factors for each pair:
5. Factorise completely:
Example
Factorise
1. Take out common factors:
2. Multiply the coefficient of by the constant
term:F
ind the two factors of this
product which add to give the coefficient of
which in this case is -7: -1 and -6.
3. Rewrite the term in brackets using these two
factors:
4. Take out common factors for each pair:
5. Factorise completely:
Differences of Squares
Any expression of the form
almost instantly:
can be factorised
Example:
Example:
O Level Maths Notes: Graphs and Relations of
Trigonometric Functions
O Level Maths Notes: Percentages
Percentage questions typically take one of three
forms:
If a quantity increases (decreases) by x% what is
the new quantity?
What is A as a percentage of B? or find the
percentage increase (decrease).
If a price increases by x% and is now £A find the
price before the increase.
I will illustrate these in turn.
If the price of a car is £6000 and it increases by
15% what is the new quantity?
Find 15% of
The new price is £6000+£900=£6900.
If the price of a car is £6000 and it increases by
£1200 what is the new percentage increase?
Use the formula %
The new price is £7200. The percentage change is
%.
If the price of a car increases by 10% and the new
price is £13200 what was the original price?
Construct a Ratio Table and scale down from the
new price to the original. Since the price increased
by 10% the new price is 110% of the original. We
have to scale down from 110% to 100%.
%
£
110
13200
100
12000
The scale factor from 110 to 100 is and this is
the scale factor from the new to the original price:
the original price is
O Level Maths Notes: Constructions – Bisecting
Angles and Lines
Bisecting an Angle
To bisect the angle ABC draw arcs of equal length
centred at B. Draw arcs of equal length centred at
E and F to cross at G. The line BG bisects the
angle.
Constructing the Perpendicular Bisector to a
Line
To bisect the above draw arcs centred of equal
radius at each end to cross at B and C. Draw arcs
centred at B and C to cross at E and F. The line EF
bisects the original line.
Constructing an Angle of Sixty Degrees to a
Line
Given the line AB, from A draw an arc to cross
AB at D, and then draw an arc of equal radius so
that the two arcs cross at a point – call this C and
draw the line AC. The angle CAD is 60 degrees.
O Level Maths Notes: Angles in Polygons
Shape
Picture Numbe Interio Sum Exterio
r of
r
of r Angle
Sides Angle Interio
r
Angles
Triangle
3
60
180
120
Square
4
90
360
90
Pentago
n
5
108
540
72
Hexago
n
6
120
720
60
Octagon
8
135
1080
45
n- agon
To see why the angles in a triangle add to
notice that a triangle has three sides so n=1 and the
angles add to 180, a square can be cut into two
triangles, each with internal angles 180, so all the
internal angles of both triangles sum to 360.
Consider the pentagon below. It is cut into three
triangle, each with internal angles that sum to 360,
so the internal angles of a pentagon sum to 3*180
=540.
O Level Maths Notes: Direct and Inverse
Proportion
Direct Proportion
If two quantities and are in direct proportion then
they increase together by the same ratio. If one
doubles so does the other, and if one increases by a
factor of 10, so does the other. We can write down
the relationship between these two quantities in the
form of an equation,
where is the constant of
proportionality. We may have to find but once we
have found it, then we can find for any given
value of or for any given value of
Direct proportion is illustrated on the graph above.
For this direct proportion relation
the gradient
of the graph. We can also find if we are told
specific values of and If we are told that
when
then
Now we know that
we can:
Find if
Find if
Inverse Proportion
Two quantities and are inversely proportional if
their product is a constant
– we can also write
this equation as
The graph of two quantities in
inverse proportion is given below.
We can find k if we are told specific values of and
If we are told that
when
then
Now we know that
we can:
Find if
Find if
O Level Maths Notes: Formulae
Volumes and Surface Areas
Volume of Cone
Curved Surface Area of Cone
is the slant height.
where l
Volume of Sphere
Surface Area of Sphere
Probability
If two events and are independent with
probabilities and respectively then the
probability of and both happening,
If the probability of an event happening is and
then are occasions in which may happen then the
expected number of occurrences of is
Integration and Differentiation
Trigonometry
Geometry
If
at a point
tangent at
is
them the equation of the
and the equation of the
normal is
A circle with equation
and radius
has centre
O Level Maths Notes: Problem Solving With
Algebra
Problem solving typically involves using algebra
to set up an equation or system of equations which
must be solved to find the possible solutions. The
simplest might be to find the value of given the
sides of a polygon in terms of together with a
value for the perimeter.
Example: The perimeter of the trapezium below
is found by adding up the lengths of all the sides:
Suppose we know that
We have in terms of
so we can equate 99 and
to obtain the equation
and we solve this equation to obtain
Suppose we are told instead that the area is 66.
The formula for the area of a trapezium is
so given the expressions for the lengths of the
sides on the diagram above we can find the area in
terms of
We equate 66 and
We solve this equation:
or
Only is possible since
values for the lengths.
means negative
O Level Maths Notes: Solving Linear equations
The simplest linear equations are very easy to
solve if you need to solve the equation to find
just make the subject:
Add 3 to both sides
Divide by 9
Slightly more complicated equations have two
terms involving These may both on the same
sides, or one term on each side.
If they are the same same we collect like terms
then solve as above:
Collect like terms to give
Now solve as in above. First subtract 7.
Divide by 7
If the x terms are on opposite sides then we have
to move them to the same side. YOU MUST
MAKE SURE THAT IF A TERM CHANGES
SIDE IT CHANGES SIGN!
Example
If the equation has fractions the best strategy is to
clear all the fractions.
Example:
To clear all the fractions we multiply by the
product of the denominators. In this case we
multiply by
Expand the brackets
Now divide by
Example
The product of the denominators is 2*5=10.
O Level Maths Notes: The Quadratic Formula
A quadratic equation is an equation of the form
where
We can solve equations of this
form by identifying a, b and c and substituting
them into the formula
to find
Example:
Solve
Then
or
It is very important to get the correct values of
For the following equations the values are given
Notice that if a minus sign appears in the formula
we have to solve then that value of
will be
negative.
Then for the first
equation
We may also solve quadratic equations by
sketching a quadratic graph or by completing the
square.
Example: Solve
Then
or
O Level Maths Notes: Compound Interest
People who form the healthy habit of saving
money pretty soon find themselves rich. This is
because money by itself earns interest, so that you
do not need to put money in the bank to save – you
can just leave the money there and it will grow all
by itself, ignoring the very real possibility of a
bank going bankrupt.
Suppose then that you put £100 in a bank at 10%
interest. This means that the amount of money in
the bank increases by 10% each year.
The amount at the end of the 1st year is £100 +10%
of £100 =£100 +£10=£110
The amount at the end of the 2nd year is £110
+10% of £110 =£110 +£11=£121
The amount at the end of the 3rd year is £121
+10% of £121 =£121 +£12.1=£133.1
The amount of money in the account each year is
shown on the graph below.
There is a formula to find the amount of money,
in the bank at the end of each year, after interest
has been added:
In this formula, is the original amount invested –
the Principle.
is the rate of interest
is the number of years since the investment was
made.
For this example, the amount of money in the
account at the end of the 20th year is
O Level Maths Notes: Functions – Domain,
Range/Codomain
A function
takes a number and returns the
value
corresponding to that number. We can
plot the point
as ordinary points. In fact we
can plot one point
for each allowed value of
and hence obtain the graph of
The graph of the
function
is shown below.
Some definitions are necessary:
Domain
The domain of a function is the set of values that
can take. We can read the domain off the – axis.
For the function
the domain is the set of all
real numbers but for
we may not have
because that would result in the division which is
not defined. If for any reason the possible set of
values of is restricted to certain values, for
example
then that is the domain.
Range or Codomain
The range or codomain of a function is the set of
values the function
may take. We read the
codomain off the – axis. For the function
the codomain is the set of all real numbers because
may take any value, or alternatively, the equation
has a solution for all values. For
we
may not have because
has no solutions for
If for any reason the possible set of values of is
restricted to certain values, for example
then
that may restrict the codomain: for
if
then
With
to evaluate when
we find
O Level Maths Notes: Simple Trigonometry –
Finding Angles in Right Angled Triangles
Simple trigonometry can be used with right angled
triangles to find an angle given two sides, or a side
given a side and one of the interior angles other
than the right angle,
With the sides of the triangle as labelled above, we
can use one of the formulae:
Example: Finding an angle using cos
For the triangle above, we have and and want to
find This means we use
Example: Finding an angle using tan
For the triangle above, we have and and want to
find This means we use
Example: Finding an angle using sin
For the triangle above, we have and and want to
find This means we use
O Level Maths Notes: Solving Inequalities
Simple inequalities are are very easy to solve. The
process in very similar to making the subject of an
equation, bearing in mind to have the inequality
instead of the equals sign throughout:
If the sign is replaced by any of
is identical.
the process
If we ever have to divide or multiply by a minus
number the
sign has to point the other way.
For example
The above equations are “one sided” inequalities
but there are also “two sided” inequalities such as
To solve these inequalities we have to make the
sole term of the central term
We add one to obtain
Then divide by 2 to obtain
It may be required, if is a whole number or
integer, to write down all the values of which
satisfy the inequality.
For the inequality
may only be equal to 2 or
4 since so is greater than 2 and so is less
than 5.
To solve
follow the same procedure as
above, obtaining
and note that now may be
equal to 2, so may be
To solve
follow the same procedure as
above, obtaining 2<=x<=5 and note that now
may be equal to 2 or 5, so may be
To solve
follow the same procedure as
above, obtaining
and note that now may not
be equal to 2 but may be equal to 5, so may be
O Level Maths Notes: Venn Diagrams 1
Venn diagrams are a means to display categories
of data graphically. They are more flexible than
contingency tables, allowing complex reasoning,
and have many applications in set theory.
Typically each set is illustrated by a bubble,
allowing intersection, with one or more of the
other sets, and an intersection simultaneously of all
the sets. All the regions are a Venn diagram are
mutually exclusive and exhaustive, since each
element may only appear once on a Venn diagram.
This does not mean that the sets are mutually
exclusive since, for example, on a Venn diagram
illustrating which newspapers people read, with
each set representing a different newspaper, a
person may read more than one newspaper, hence
be in more than one set.
Example: 100 people were asked which
newspapers they read. The results showed that 30
read Daily Trash, 26 read The Honest Untruth, 21
read The Dirty Digger, 5 read both Daily Trash
and The Honest Untruth, 7 read both The Honest
Untruth and The Dirty Digger, 6 read both The
Dirty Digger and Daily Trash and 2 read all three.
We have to fill out the diagram above. It is best to
work out from the centre. 2 people read all three
newspapers, so the entry in the central region is 2.
Complete the regions in the order shown:1) 2) 3)
4) 5) 6). At the end, of the 100 people asked, 61
read a newspaper an 39 don't. This number, 39,
goes out side any set as shown.
O Level Maths Notes: Probability and Tree
Diagrams
The definition of independent events is that neither
can affect the other: if A and B are independent
then the probability of A happening does not
depend on whether B has happened or will happen,
and vice versa. There is an equation that we can
use when two events A and B are independent:
In plain English this says that if two events are
independent then to find the probability of them
both happening we multiply the individual
probabilities together.
The equation may be used in the following way.
Suppose John and Bill take their driving tests on
the same day. The probability that John will pass is
0.6 and the probability that Bill will pass is
0.3.Find the probability that
a)Both pass
b)Neither pass
c)Exactly one passes
d)At most one passes.
We start by drawing a probability tree:
a)The probability John Passes AND Bill Passes =
0.6*0.3=0.18
b)The prbability That John fails And Bill fails
=0.4*0.7=0.28
c) Exactly can pass in two ways:
John can pass AND Bill can fail =0.6*0.7=0.42
OR
John can fail AND Bill can pass =0.4*0.3=0.12
Because either the first OR the second can happen,
we add the two probabilities: 0.42+0.12=0.54
d)At least one passes mean that both can pass OR
exactly one can pass
ie 0.18+0.54=0.72
Example: A bag contains 4 red balls and 7 green
balls. Two balls are taken out one at a time and put
to one side. Find the probability that
a)Both are red
b)One is red
c)At least one is red
d)both are the same colour
e)Both are different colours
To start we have 4 red balls out of 11, so the
probability of picking a red ball is Now we take
the ball and put it aside. There are only 3 red balls
and 7 red balls out of 10. The probability of the
second ball being red is and the probability of the
second ball being green is This labels the top
half of the probability tree as shown.
The probability of the first ball being green is
Then this ball is put aside and there are now 4 red
balls and 6 green balls out of 10, so the probability
of the second ball being red is and the probability
of the second ball being green is This labels the
bottom half of the probability tree.
a)First Ball Red and Second Ball Red=
b)One of the two balls can be red in two ways:
The first ball is red and the second one is green =
The first ball can be green and the second ball can
be red =
Since we can have either the first OR the second
way round we add these two answers:
C)At least one is read mean both can be red, OR
exactly one can be red, so we add the answers to a)
and b)
d)Both have the same colour if they are both red
OR if they are both green
The probability of the first one being green and
second one being green is
Now we add this and the answer to a)
d) If they are different colours, they cannot be the
same colour so we can find 1 -the answer to d)
=
O Level Maths Notes: Solving Simultaneous
Equations Algebraically
Simultaneous equations involve at least two
unknown that must be found. If we have two
equations and two unknowns or three equations
and three unknowns then we can generally solve
the equations. Typically the two unknowns are
labelled and as in the following simultaneous
equations.
(1)
(2)
The procedure for solving simultaneous equations
is:
1. Choose or and make the size of the
coefficients of or the same. In the above
equations the coefficients of are 2 and 3, and
the coefficients of are 1 and 2. We can make
the coefficients of the same by multiplying
(1) by 2, then both equations have The new
equations are
(3)
(2)
2. We can now eliminate the terms by
subtracting:
gives
3. Now find by substituting this value for back
into one of the equations (1) or (2) and solve
to find
Suppose we substitute
into
Example: Solve the simultaneous equations
(4)
(5)
We can make the coefficients the same size by
multiplying (4) by 2 and multiplying (5) by 3. This
will result in them being the same size but having
opposite sign. We do not subtract – we add to
eliminate the - terms.
(6)
(7)
(6)+(7) gives
Substitute
into (4) to obtain
O Level Maths Notes: Factorising General
Quadratics and Differences of Squares
Any expression of the form
cannot always
be factorised as with simple quadratics, by finding
factors of c. Instead there is a slightly more
complicated procedure, best illustrated by an
example:
Factorise
1. Take out any common factor. Every term in
the above expression has a factor 3, so we may
write the expression as
2. Multiply the coefficient of by the constant
term:
Find the two factors of this
product which add to give the coefficient of
which in this case is -1.
3. Rewrite the term in brackets using these two
factors:
4. Take out common factors for each pair:
5. Factorise completely:
Example
Factorise
1. Take out common factors:
2. Multiply the coefficient of by the constant
term:F
ind the two factors of this
product which add to give the coefficient of
which in this case is -7: -1 and -6.
3. Rewrite the term in brackets using these two
factors:
4. Take out common factors for each pair:
5. Factorise completely:
Differences of Squares
Any expression of the form
almost instantly:
Example:
can be factorised
Example:
O Level Maths Notes: Matching Graphs With
Equations
O Level Maths Notes: Maximising and
Minimising Expressions
Suppose we want to find the maximum distance
between two points. We might know where the
two point are, but it is in the nature of
measurements that they are never exact.
A is at 2 to the nearest whole number. This means
is must be closer to 2 than any other whole
number, but this means it can be anywhere
between 1.5 (halfway between 1 and 2) and 2.5
(halfway between 2 and 3), and B is at 5 to the
nearest whole number but this means it can be
anywhere between 4.5 and 5.5. From the diagram
above the
maximum possible distance between A and B is
5.5-1.5=4
minimum possible distance between A and B is
4.5-2.5=2
In general to find the maximum possible value of
we find
To find the minimum possible value of
we find
TO FIND THE MAXIMUM POSSIBLE VALUE
OF
WE DO NOT FIND
TO FIND THE MINIMUM POSSIBLE VALUE
OF
WE DO NOT FIND
The above may seem counter intuitive. So is this:
To find the maximum possible value of find
To find the minimum possible value of we find
Examples:
If x=2.5 to the nearest 0.1 and y is 3.4 to the
nearest 0.1 find the maximum and minimum
possible values of
the maximum and minimum possible values of
are 2.45 and 2.55 respectively and the maximum
and minimum possible values of are 3.35 and
3.45 respectively.
to 4 d.p.
To 4 d.p.
O Level Maths Notes: Sketching Inequalities and
Finding The Region Satisfied By Inequalities (1)
The only real way to solve multiple inequalities of
the form
etc is to put all the inequalities
on the same diagram and find the region satisfying
all the inequalities as a part of the plane.
Example: Find the region satisfying the three
inequalities
the letter
Label this region with
The line
is drawn. We want the region below
the line so we shaded the region we do not want –
above the line.
The line is drawn. We want the region above
the line so we shade the region we do not want –
below the line.
The line
is drawn. We want the region below
the line so we shade the region we do not want –
above the line.
All the lines are solid because points on the line
may satisfy the inequalities.
The region we want is the triangle in the middle,
labelled R.
O Level Maths Notes: Loci
The set of all points distance r from O is the circle,
centre O, radius r.
The set of points the same distance from B as from
C is the perpendicular bisector of the line BA.