Paper 1
Things to Learn Off
Algebra –
Difference and Sum of Cubes
𝑥 3 − 𝑦 3 = 𝑥 − 𝑦 𝑥 2 + 𝑥𝑦 + 𝑦 2
𝑥 3 + 𝑦 3 = 𝑥 + 𝑦 𝑥 2 − 𝑥𝑦 + 𝑦 2
Forming a Quadratic
𝑥 2 − sum of roots 𝑥 + (product of roots) = 0
Nature or Roots
Real
𝑏 2 − 4𝑎𝑐 ≥ 0
Equal
𝑏 2 − 4𝑎𝑐 = 0
No Real Roots
𝑏 2 − 4𝑎𝑐 < 0
Factor Theorem (understand NOT prove)
If 𝑥 − 𝑘 is a factor of 𝑓(𝑥) then 𝑥 = 𝑘 is a root.
Binomial Theorem
𝑛
𝑛 𝑛−𝑟 𝑟
𝑥+𝑦 𝑛 =෍
𝑥
𝑦
𝑟
the most
important topic
and the largest.
Every topic
contains
Algebra.
𝑟=0
Topics
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Substitution
Rearrange (Manipulate) Formula
Expand Expressions (including large powers
using Binomial)
Factorising Algebraic Expressions & Simplify
Simplify Algebraic Fractions
Form Algebraic Expressions (Word Problems)
Rationalise the Denominator
Algebraic Identities
Nature of Roots
Use the Factor Theorem
Quadratic Factor of a Cubic Function
Prove Abstract Inequalities
Form Quadratic/ Cubic Equations Given Roots
Above formula (in tables) is for the 𝑟 + 1 term.
Complex
Numbers –
De Moivre’s Proof by Induction
De Moivre’s to Prove Trigonometric Identities
The use of
imaginary
numbers.
𝑖 = −1
Polar Form
𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃
If 𝑧 = 𝑥 + 𝑖𝑦
•
then the modulus 𝑟 = 𝑧 = 𝑥 2 + 𝑦 2
and the argument 𝜃 is found using the tan ratio.
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Conjugate Roots Theorem
Let 𝑎𝑧 2 + 𝑏𝑧 + 𝑐 = 0 be a quadratic equation.
If 𝑧1 is a root then so is its conjugate 𝑧1ҧ
Not valid for 𝑖’s in the coefficients of 𝑧.
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Number &
Arithmetic –
Proof By Contradiction ( 𝟐 is irrational)
Construction of 𝟐 and 𝟑
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Very small but
practical
section.
Commutative → 𝒂 + 𝒃 = 𝒃 + 𝒂
Associative → 𝒂 + 𝒃 + 𝒄 = 𝒂 + 𝒃 + 𝒄
Distributive → 𝒂 𝒃 + 𝒄 = 𝒂𝒃 + 𝒂𝒄
Lots of Algebra
and
Trigonometry
Error
Error = Accurate − Observed
Error
Relative Error =
Accurate Value
Error
Percentage Error =
× 100
Accurate Value
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Addition, Subtraction, Multiplication in
Rectangular form.
Conjugate of a Complex Number
Division using the Complex Conjugate
Argand Diagram – Plotting and Reading
Transformations of Complex Numbers
Modulus of a Complex Number
Polar Form – Finding Modulus (Distance to
Origin) and Argument (Angle)
Multiplication and Division in Polar form.
Solve quadratic equation with complex roots.
Conjugate Roots Theorem
Complex Identities
Types of Number
• (𝑁, 𝑍, 𝑄, 𝑅, 𝑅/𝑄, 𝐶)
Commutative/ Associative/ Distributive Rules
Order of Operations
Percentages/ Fractions/ Decimals
Factors and Multiples
Product of Primes
Ratio and Proportion
Order of Magnitude
Topics
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Solve Linear Equations
Solve Equations with Surds
Solve Indices Equations (Rules in Tables)
Solve Logarithmic Equations (Rules in Tables)
Solve Irrational Equations
Solve Quadratic Equations
Solve Cubic Equations
Solve Inequalities
Solve Compound Inequalities
Solve Rational Inequalities
Solve Modulus Equations
Solve Modulus Inequalities
Solve Simultaneous Equations
(2 variable – linear)
(2 variable – linear & non linear)
(3 variable – linear)
• De Moivre’s Theorem to find Higher Powers
𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃 then
𝑧 3 = 𝑟 3 cos 3𝜃 + 𝑖 sin 3𝜃
• De Moivre’s Theorem to find Roots
𝑧 3 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃 then
1
1
1
𝑧 = 𝑟 3 cos 𝜃 + 2𝜋𝑛 + 𝑖 sin 𝜃 + 2𝜋𝑛
3
3
𝑛 = 0, 𝑛 = 1, 𝑛 = 2
• Roots of Unity
• Proof by Induction of De Moivre’s
• De Moivre’s to Prove Trigonometric Identities
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Estimation and Rounding
Scientific Notation
Construction of 2 and 3
Proof by Contradiction ( 2 is irrational)
Accumulated Error
Percentage Error
Tolerance
Average Rates of Change
Metric vs Imperial
Paper 1
Things to Learn Off
Topics
Patterns
(Sequences
and Series) –
Sum of a Geometric Series By Induction
Sum to Infinity if Geometric Series (using Limits)
General Sequence Notation
𝑇𝑛 = 4𝑛 + 3
General Series Notation
Adds formulae
to the practical
methods we
used in Junior
Cert questions.
Lots of little
things to learn
and lots of
Algebra.
Proof by
Induction can
be very tricky
and requires
lots of practice.
Financial
Maths –
Small but tricky
section.
Need good
Sequences and
Series and
Algebra skills.
Proof by Induction
𝑛=1
𝑛=𝑘
𝑛 =𝑘+1
Arithmetic Sequence
𝑑 = 𝑇3 − 𝑇2 = 𝑇2 − 𝑇1
𝑇𝑛 = 𝑆𝑛 − 𝑆𝑛−1
Geometric Sequence
𝑇3
𝑟=
𝑇2
𝑇𝑛 = 𝑆𝑛 − 𝑆𝑛−1
Quadratic Sequence – General Term
A quadratic sequence will be in the form
𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐
2nd Difference = 2𝑎
Generate Amortisation Formula using Sum of
Geometric Series
% Profit (Mark-Up)
Profit/Loss
=
× 100
Cost Price
% Profit (Margin)
Profit
=
× 100
Selling Price
Tax
Gross Tax = Standard Tax + Higher Tax
Tax Payable = Gross Tax – Tax Credit
Net Income = Gross Income – Tax Payable
Area and
Volume –
Trapezoidal Rule (only new learning from JC)
Generally a part
of other Q’s.
Area ≈
h
First + Last + 2(Rest)
2
Topics
4
Proof By Induction:
෍ 𝑇𝑟 = 𝑇1 + 𝑇2 + 𝑇3 + 𝑇4
3 Types
𝑟=1
Arithmetic Sequences and Series
(formula in tables)
𝑇𝑛 = 𝑎 + 𝑛 − 1 𝑑
𝑛
𝑆𝑛 = 2𝑎 + 𝑛 − 1 𝑑
2
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(Divisibility)
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(Series)
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(Inequalities)
Reoccurring Sequences
Geometric Sequences and Series
(formula in tables)
𝑇𝑛 = 𝑎𝑟 𝑛−1
𝑎 1 − 𝑟𝑛
𝑆𝑛 =
1−𝑟
𝑎
𝑆∞ =
for 𝑟 < 1
1.54ሶ 7ሶ = 1.54747474747 …
= 1+
5
47
47
+
+
+⋯
10 1000 100000
1−𝑟
Quadratic Sequences and Series (see left)
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Compound Interest (formula in tables)
Depreciation (formula in tables)
Present Value (rearrange compound formula)
Loan Repayments (can use amortisation
formula)
Investments (savings)
Present Value of Loan Repayments
𝐴
𝐴
𝐴
𝐴
=
+
+
+ ⋯+
1+𝑖 1
1+𝑖 2
1+𝑖 3
1+𝑖
Future Value of Instalment Savings
= 𝑃 1 + 𝑖 𝑡 + 𝑃 1 + 𝑖 𝑡−1 + ⋯ + 𝑃 1 + 𝑖
𝑡
1
Nets of Prisms, Cylinders and Cones
Perimeter of circle, triangle, rectangle, square,
parallelogram, trapezium, sectors of discs, and
figures made from combinations of these.
Trapezoidal rule to approximate area.
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Profit/ Loss
Discount
Selling Price
Mark Up and Margin
Income Tax
Net Pay
Costing & Materials
Use of Geometric Formula
𝑎 1 − 𝑟𝑛
𝑆𝑛 =
1−𝑟
Surface area and volume of the following solid
figures: rectangular block, cylinder, right cone,
triangular-based prism (right angle, isosceles and
equilateral), sphere, hemisphere, and solids made
from combinations of these.
Paper 1
Functions and
Graphs –
Need excellent
algebra skills.
Generally asked
through real
world scenarios.
Often linked with
Calculus below.
Things to Learn Off
Injective Functions (one to one)
Every output is the image of a unique input.
Horizontal Line Test – any horizontal line drawn
will never cut the graph at more than one point.
Surjective Functions (onto)
Every element in the co-domain is an output.
Horizontal Line Test – every horizontal line
intersects the graph of 𝑓 at at least one point.
Bijective Functions (one to one & onto)
Both injective and surjective.
Trigonometric Functions
𝑓 𝑥 = 𝑎sin 𝑛𝑥
𝑔(𝑥) = 𝑎cos 𝑛𝑥
𝐑𝐚𝐧𝐠𝐞 −𝑎, 𝑎 𝐏𝐞𝐫𝐢𝐨𝐝 =
2𝜋
𝑛
Vertical Asymptote
Bottom = 0
Horizontal Asymptote
𝑦 = lim 𝑓 𝑥
Topics
•
continuation of
functions and
graphs.
All about ‘Rates of
Change’.
Calculus
Integration –
The reverse of
differentiation.
Differentiation from 1st Principles
𝑓 𝑥 =
𝑓 𝑥+ℎ =
𝑓 𝑥+ℎ −𝑓 𝑥 =
𝑓 𝑥+ℎ −𝑓 𝑥
=
ℎ
𝑓 𝑥+ℎ −𝑓 𝑥
lim
=
ℎ→0
ℎ
Differentiation Rules are in the Tables
Notation
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Linear Functions
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Domain, Codomain, Range
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Quadratic Functions
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Vertical Line Test for a Function
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Cubic Functions
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Forming Polynomial Function given the roots.
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Exponential Functions
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Composite Functions
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Logarithmic Functions
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Inverse Functions
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Trigonometric Functions
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Injective, Surjective, Bijective Functions
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Modular Functions
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Limit of a Function
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Continuity
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Asymptotes
𝑏
𝑎
Integration Rules are in the Tables
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Differentiation by Rule
Product Rule (in tables)
Quotient Rule (in tables)
Chain Rule
Differentiation from 1st Principles
Trigonometric Differentiation
Logarithmic Differentiation
Exponential Differentiation
2nd and 3rd Derivatives
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Idea that integration is the reverse of
differentiation
Definite vs Indefinite Integrals (constant of
integration 𝑐)
Integrate functions of the form:
• 𝑥 𝑎 , 𝑎 𝑥 , sin 𝑎𝑥 , cos 𝑎𝑥.
Find areas bounded by curves.
Find the average value of a function.
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Graphs of the 1st and 2nd Derivatives of
Functions (see differentiation)
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Average Value
1
න 𝑓(𝑥) 𝑑𝑥
𝑏−𝑎
Sketching the Graphs and Transformations of:
•
𝑥→∞
Calculus
Differentiation
- Very much a
Topics
Completing the Square
Differentiate to find:
• Slopes of tangents (lines) to functions.
• Gradient
• Tangent to a Circle.
• Stationary/ Turning Points (Max and Min).
• Points of Inflection.
• Increasing and decreasing functions.
• Rate of Change.
• Related Rates of Change.
• Distance→ Speed → Acceleration
Integrate to find:
• Areas Bounded by Curves.
• The Average Value of a Function.
• Anti-Derivatives.
• The function given its slope.
• Acceleration → Speed → Distance
Paper 2
Things to Learn Off
Things to Learn Off
Topics
Probability –
Both Statistics
and Probability
tend to have
less Algebra
than some of
the other topics
(though not
always).
The downside is
that they are
both large
sections with a
good bit to
learn off.
There is a bit of
crossover with
inferential
statistics being
as much
probability as
statistics.
Margin of Error for a Population Proportion (95%)
In OR events we ADD the probabilities
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 𝑜𝑟 𝐵
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
In AND events we MULTIPLY the probabilities
𝑃 𝐴 ∩ 𝐵 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴). 𝑃(𝐵
Mutually Exclusive events have no outcomes in
common.
Events that CANNOT occur at the same time.
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
Independent events are where the outcome of the
1st does NOT affect the outcome of the second.
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴). 𝑃(𝐵
Conditional Probability
𝑃 𝐴∩𝐵
𝑃 𝐴𝐵 =
𝑃(𝐵)
Bernoulli Trial
There are two outcomes: success or failure
The trials are independent.
The probability of success does not change from
one trial to another.
𝑛 𝑟 𝑛−𝑟
𝑝 𝑞
𝑟
𝑝 = probability of success
𝑞 = probability of failure
𝑛 = number of trials
𝑟 = number of desired outcomes
Expected Value E(X) is the average outcome of an
event
𝐸 𝑋 = σ 𝑥. 𝑃(𝑥)
To find expected value we multiply every possible
outcome by the probability for that outcome and
then add all these values together.
= ±1.96
𝑝ො 1 − 𝑝ො
𝑛
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Sample Spaces – Listing Outcomes
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Fundamental Principal of Counting
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Arrangements (Permutations)
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Combinations
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Probability
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Relative Frequency
Confidence Interval of Sample Mean at 5% Level of
Significance
𝜎
𝜎
𝑥ҧ − 1.96
< 𝜇 < 𝑥ҧ + 1.96
𝑛
𝑛
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Expected Frequency
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Sets and Venn Diagrams
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AND/ OR
Central Limit Theorem
When we are dealing with a simple random sample
from a population we must adjust the standard
deviation and z-score. Applies when:
𝑛 > 30 (for any population, normal or otherwise).
𝑛 ≤ 30 (if the underlying population is normal).
The mean of the sampling distribution of the
sample mean is equal to the population mean.
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Mutually Exclusive
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Expected Value
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Fair Games
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Bias
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Independent
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Conditional Probability
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Probability of 2 Events
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Bernoulli Trial
95% Confidence Interval for a Proportion
𝑝ො 1 − 𝑝ො
𝑝ො 1 − 𝑝ො
𝑝ො − 1.96
< 𝑝 < 𝑝ො + 1.96
𝑛
𝑛
Margin of Error for a Sample Means (95%)
𝜎
𝐸 = +1.96
𝑛
𝜇 𝑥ҧ = 𝜇
The standard deviation of the sampling
distribution of the sample mean (the standard
error of the mean) is
•
Sampling Distributions
𝜎
𝜎 𝑥ҧ =
𝑛
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The Normal Distribution & z-scores
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Margin of Error
In the above case the corresponding z-score for the
sample mean is
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Confidence Interval
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Hypothesis Testing
𝑥ҧ − 𝜇
𝑧=
𝜎
𝑛
The above is a joint Statistics/ Probability section.
Paper 2
Things to Learn Off
Things to Learn Off
Topics
Statistics –
Population – is the entire group being studied
Census – is a survey of the whole population
Sample – is a group that is selected from the
population
Parameter – is a numerical measurement
describing some characteristic of a population.
Statistic – is a numerical measurement describing
some characteristic of a sample.
Simple random sample – selected a sample of size
n, in such a way that every sample of size n has an
equal chance of being selected.
Stratified random sample – first divide population
into subgroups so that individuals within each
subgroup share characteristics. Then a sample
random sample is drawn from each group. Eg. We
might first divide population by gender.
Systematic random sample – We select the sample
based on random starting point and select a fixed
periodic interval. Eg Select every 5th entry.
Cluster sample – population is divided by sections
or clusters. Then some of those clusters are
randomly selected and all members from those
clusters are chose. Eg. We want a sample of
students. We get a list of schools and then select a
school and use those students.
Quota sample – Non probability sampling. We
select to fill a quota of a certain type of subgroup.
Eg Selecting men between age 30 and 40.
Convenience sample – selecting group of people as
it was easy to contact them. Eg. Selecting 20 people
by taking the first 20 names on a register.
Descriptive Statistics – summarise and present
data so that people can easily understand.
Inferential Statistics – predict or forecast based on
responses of a sample group.
Control Group - The control group in an
experiment is the group who does not receive any
treatment and is used as a benchmark against
which other test results are measured.
Explanatory variable – controlled variable
Response variable – the effect being observed
Commentating on Graphs
•
Continuation of
much of the
Junior Cert
course where
students
analyse and
graph data.
Long questions
generally
combine the
topics of
Statistics and
Probability, in
particular areas
concerning the
normal
distribution and
z-scores,
confidence
intervals,
margin of error
and hypothesis
testing.
• Quote the range of the data.
• Can we estimate the mean? Is the mode
obvious?
• Comment on standard deviation of the data.
Large standard deviation means the data is well
spread. Low standard deviation gives more of a
cluster. Are there any outliers.
• We use a scatter graph when we have data that
can be paired together (bivariate data). An
example would be heights and weights or age
and salary. We measure how well they are
related through Correlation. A line of best fit is
one that comes as close as possible to the points.
We can find the equation of this line by selecting
two points on the graph and using co-ordinate
geometry.
Describing the distribution
•
Normal Distribution, symmetrical, bell curve
•
Skewed left or negatively skewed (your left
foot)
•
Skewed right or positively skewed (your right
foot)
The Empirical rule states that in any Normal
distribution:
68% of the population lie within one standard
deviation of the mean 𝑥ҧ − 𝜎, 𝑥ҧ + 𝜎
95% of the population lie within two standard
deviations of the mean 𝑥ҧ − 2𝜎, 𝑥ҧ + 2𝜎
99.7% of the population lie within three standard
deviations of the mean 𝑥ҧ − 3𝜎, 𝑥ҧ + 3𝜎
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Central Tendencies
• Mean
• Mode
• Median
Mean of a Frequency Distribution
Grouped Frequency Distributions (MidInterval)
Measures of Spread
• Range
• Interquartile Range
• Standard Deviation
• Outliers
Describe Shape of Distribution
Relative Standing – Percentiles
Stem and Leaf Diagram
Bar Chart
Pie Chart
Line Graph
Histogram
Scatter Plot
Correlation Coefficient
Line of Best Fit
Types of Surveys (Advantages/ Disadvantages)
Misuse of Statistics
The Normal Distribution
Margin of Error
Confidence Interval
Hypothesis Testing
Types of Data
• Numerical
• Categorical
• Discrete
• Continuous
• Nominal
• Ordinal
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•
Primary
Secondary
•
•
Univariate
Bivariate
Paper 2
Things to Learn Off
The Line –
Line in the form 𝒂𝒙 + 𝒃𝒚 + 𝒄 = 𝟎
Continues
learning from
the Junior Cert
adding more
formulae, most
of which are in
the tables.
Need good
Algebra and
Geometry!
Things to Learn Off
Perpendicular Slope
Turn the slope upside down and change the sign.
3
If a line has a slope of the perpendicular slope
We can find:
where it crosses the 𝒙 axis by letting
𝑦=0
where it crosses the 𝒚 axis by letting
𝑥=0
(also do this to draw a line)
5
5
is −
3
To prove slopes perpendicular 𝑚1 𝑚2 = −1
Parallel Slope
If lines are parallel then the slopes are equal
the slope 𝒎, using
𝑎
−
𝑏
if a point is on the line by subbing the
values of the point 𝑥1 , 𝑦1 in for 𝑥 and 𝑦.
Parallel Lines
If 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 is a line then a parallel line
can be written 𝑎𝑥 + 𝑏𝑦 + 𝑘 = 0
A perpendicular line can be written
𝑏𝑥 − 𝑎𝑦 + 𝑘 = 0
Line in the form 𝒚 = 𝒎𝒙 + 𝒄
𝑚 will be the slope
𝑟𝑖𝑠𝑒 3
𝑚=
=
𝑟𝑢𝑛 4
Lines Parallel to the Axes
𝑥 = 2 is a line parallel to the 𝑦-axis through 2 on
the 𝑥 axis
𝑦 = −1 is a line parallel to the 𝑥-axis through
− 1 on the 𝑦 axis
𝑐 the 𝑦-intercept
(the place where the line crosses 𝑦 axis)
The Circle –
Uses most of the
formulae from
the line adding
just the
‘Equation of a
Circle’ formulae.
Need good
Algebra and
Geometry!
Equation of circle with centre (𝒉, 𝒌) and radius 𝒓
𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2
Circles touch Internally
If circles touch internally the difference of their
radii will equal the distance between their
centres.
General Equation of a circle
𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0
1
2
Centre is (−𝑔, −𝑓) which is the coefficient of 𝑥
1
2
and the coefficient of 𝑦
Radius is
𝑔2
+
𝑓2
− 𝑐, providing
Circles touch Externally
𝑑 = 𝑟1 + 𝑟2
If circles touch externally the sum of their radii
will equal the distance between their centres.
𝑔2
+
𝑓2
−𝑐 >0
𝑑 = 𝑟1 − 𝑟2
Circles touching the x-axis
𝑔2 = 𝑐
Circles touching the y-axis
𝑓2 = 𝑐
Topics
From Junior Cert
• Distance between two points (in tables)
• Finding Slopes
• with slope formula (in tables)
• with 𝑦 = 𝑚𝑥 + 𝑐 (in tables)
• with rise over the run
• Equation of a Line (in tables)
• Intersection of 2 Lines
• Line intersecting axis
New For Leaving Cert
• Angle Between Lines (in tables)
• Area of a Triangle (in tables)
• Division of a Line in Ratio (in tables)
• Perpendicular Distance from a Point to a Line
(in tables)
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Translations, Symmetry
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Centroid
Circumcentre
Orthocentre
Incentre
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Real World Applications
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Standard Equation of a Circle (in tables)
General Equation of a Circle (in tables)
Touching Circles (Internally and Externally)
Intersection of Line and a Circle
Common Chord of a Circle
Circle intersecting x and y axis
Tangent to a Circle at a Point on the Circle
Tangents to a Circle Parallel to a Given Line
Tangent to a Circle at a Point NOT on the Circle
Equation of a Circle Given:
• 3 Points on the Circle
• 2 Points and a Tangent to the Circle at a
Given Point
• 2 Points and the Equation of the Line
containing the centre
• 1 point, the length of the radius and the
line containing the centre and the radius.
Paper 2
Things to Learn Off
Things to Learn Off
Topics
Solving Triangles (Small number of Tools)
8 Trigonometric Proofs
Pythagoras
𝑐 2 = 𝑎2 + 𝑏 2
•
cos2 𝐴 + sin2 𝐴 = 1
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Radian Measure
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Special Angles
Cosine Rule
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
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•
•
•
•
𝑎
𝑏
𝑐
=
=
sin 𝐴
sin 𝐵
sin 𝐶
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
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Trigonometric Ratios (in tables)
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Pythagoras Theorem (in tables)
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Area of a Triangle (in tables)
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Length of an Arc (in tables)
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Area of a Sector (in tables)
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3D Problems
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Graphing Trigonometric Functions
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8 Trigonometric Proofs (see left)
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Application of the 24 Trigonometric Identities
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Compound Angles
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Double and Half Angles
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Sum, Difference and Product
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Solving Trigonometric Equations (see left)
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Inverse Trigonometric Functions
Trigonometry –
Big topic that goes
beyond just
solving triangles.
Algebra needs to
be good and there
is lots to learn off
by heart.
Needed for
Complex numbers,
the Line and
Circle, Calculus
and some Area
and Volume
questions!
Sine Rule
𝑎
𝑏
=
sin 𝐴 sin 𝐵
•
Area of Triangle
1
𝑎𝑏 sin 𝐶
2
cos 𝐴 − 𝐵 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
cos 2𝐴 = cos2 𝐴 − sin2 𝐴
sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
tan 𝐴 + 𝐵 =
tan 𝐴+tan 𝐵
1−tan 𝐴 tan 𝐵
Signs of ratios (CAST) to find all solutions to
Trigonometric Equations
Tan, Sin or Cos
𝑂
𝑂
𝐴
𝑇=
𝑆=
𝐶=
𝐴
𝐻
𝐻
Radians
Degrees
Length of Arc
𝑟𝜃
2𝜋𝑟 ×
Area of Sector
1
𝜃𝑟 2
2
𝑟2 ×
𝜃
360
𝜃
360
𝜋
Degrees to radians ×
180
180
Radians to degrees ×
𝜋
Enlargements –
Small section with
some construction
or use of basic
formula involved.
The scale factor, k, is the number by which the
object is enlarged.
𝐒𝐜𝐚𝐥𝐞 𝐅𝐚𝐜𝐭𝐨𝐫 𝒌 =
Image Length
Object Length
𝐈𝐦𝐚𝐠𝐞 𝐀𝐫𝐞𝐚 = 𝑘 2 × Object Area
𝐈𝐦𝐚𝐠𝐞 𝐕𝐨𝐥𝐮𝐦𝐞 = 𝑘 3 × Object Volume
The centre of enlargement is the point from
which the enlargement is constructed.
A translation is when a point or shape is moved in a
straight line.
If we are given the object and the image we can
find the center of enlargement by drawing lines
through the corresponding vertices of the object
and image.
Central Symmetry is a reflection through a point
An axial symmetry is a reflection in a line or axis.
Paper 2
Things to Learn Off
Things to Learn Off
Geometry –
A theorem is a statement deduced from the axioms by logical
argument.
A proof is a series of logical steps which we use to prove a theorem.
The section that
can get you the
most ‘easy’ marks
through learned
constructions,
theorems and
definitions.
A good knowledge
of Geometry is
needed to solve
Trigonometry
problems and can
help in the Line
and Circle
sections.
Theorem 11
If three parallel lines cut off equal segments on some
transversal line, then they will cut off equal segments on any
other transversal.
Theorem 12
Let ∆ABC be a triangle. If a line l is parallel to BC and cuts
[AB] in the ratio s:t, then it also cuts [AC] in the same ratio.
Theorem 13
If two triangles ∆ABC and ∆A' B' C' are similar, then their
sides are proportional, in order.
Understand all Theorems 1 to 21 (sketching them out helps)
An axiom is a statement accepted without proof, as a basis
for argument.
Axiom 1 (Two Points Axiom).
There is exactly one line through any two given points.
The converse of a theorem is the reverse of a theorem.
Example: In an isosceles triangles the angles opposite the equal sides are equal.
Converse: If two angles are equal in a triangle then the triangle is isosceles. Converse is true.
Implies is a term we use in a proof when we can write down a fact we have proved from our
previous statements. The symbol for implies is ⇒
Is equivalent to means something has the same value or measure as, or corresponds to,
something else. For example $3 is equivalent to €2.
If and only if: I will give you €100 if and only if you eat this apple. This means that if you eat
this apple I’ll give you €100 and if I have given you €100 you have eaten the apple.
Proof by Contradiction is where we cannot directly prove a statement but we can prove that
the opposite statement is false.
A corollary is a statement that follows readily from a
previous theorem. Often a corollary is a statement of a
theorem in a more specific context.
Corollary 1. A diagonal divides a parallelogram into two
congruent triangles
Constructions 1 – 22
Proving Triangles are Congruent
SSS
SAS
ASA
Proving Triangles are Similar/ Equiangular
(Angles are the same)
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