[Mathematics Unit 3A] Course Notes

Trigonometry

Pythagoras:

Sine Rule:

Cosine Rule: and

Area of Triangle: or or

Parity of Sine and Cosine and

Indices

Indices come in the form:

  • Where the 'a' is known as the base; and
  • the 'b' is known as the exponent

[Mathematics Unit 3A] Course Notes

Law 1:

Law 2:

Law 3:

Law 4:

Law 5:

Law 6:

Law 7:

Law 8:

[Mathematics Unit 3A] Course Notes

Tips for difficult indices questions:

  • Try to convert all numbers in the question to the same base...

(e.g.

  • Try to change all the negative exponents to positive ones by using law 4...

(e.g. )

  • Watch out for indices such as this one that uses law 1 and 6 together...

(e.g.

  • Remember that if there is a square root in the question, there may be 2 solutions...

(e.g. )

Sets, Probability and Tree Diagrams

Sets Symbols:

  • - denotes is an element of.
  • - denotes union (i.e. two sets together)
  • - denotes the intersection (i.e. must be in both sets)
  • - denotes no members of a set
  • - denotes is not an element of
  • - denotes the complementary of set A (i.e. everything not in set A)

Probability Rules:

Mutually Exclusive Events:

  • Where one event does not affect the outcome of another
  • To show that A and B are mutually exclusive then .
  • Also

Independent Events:

  • Where one event does affect the outcome of another
  • To show that A and B are independent show that P(A|B)=P(A), P(B|A)=P(B) or P(AB)=P(A)xP(B)

Counting

  • Events were units can be repeated:
  • How many combinations can a 3 digit lock have where repeated digits (0-9) are allowed:
  • A 10 question test comprises of a response of either yes or no, assuming all questions are answered, how many possibilities are there to answer the test:
  • Events were units cannot be repeated:
  • Gary has 10 CDs arranged on a shelf, how many different ways can the CDs be arranged if there are no restrictions: 10!
  • Events were items are grouped together: treat the block of items as one (also times the answer by possibilities! when items within block can be rearranged)
  • 3 CDs are AC/DC and Gary would like them to be placed together with the other 7 regular CDs: 8! (10 CDs sorted into 8 groups) x 3! (because CDs can be arranged within group)
  • Having a unit first in an order: subtract one from the number of events and multiply them both together with factorials on both
  • Gary wants his favorite CD first on the shelf: 1! X 9!
  • Addition principle: if there are a ways event A can occur and b ways event B can occur then there are a + b ways A or B can occur:
  • A two digit code number is to be made by using 2 different digits from 1-6 or 1 different digits from 6-8, how many codes possible: (6x5)+(3x2)
  • 3 letters from A-F (not repeated) or 3 digits from 1-7 (repeated): (6x5x4)+(7x7x7)
  • Telephone numbers have 8 digits, first must be 9 or 6, second digit cannot be zero: 2x9x10^6
  • NOT mutually exclusive:
  • Three digits from 1-5 or three digits from 3-6, how many codes possible: (5x4x3)+(4x3)+6 (because there are 3 numbers repeated and two ways can be included: 3x2 = 6)
  • Probability:
  • A four digit number is to be made by 2-7 what is the probability that it will be odd: ½
  • Less than 4000: (2!x5x4x3)/(6x5x4x3) = 1/3

Applying 2 rules but only one may be in function at a time:

  • e.g. Even OR more than 7000
    Find the n(Even)+n(>7000)-n(even AND 7000)
  • Odd or less than 4000 = ½ +1/3 – 1/6 = 2/3

Functions

Transformations:

Parabola

Variable / What it does / Condition / Effect
A / Multiplies all y values by a / a > 1 / Skinner
0< a < 1 / Fatter
0 > a > -1 / Fatter/Upside Down
a < -1 / Skinnier/Upside Down
B / Multiplies all x values by 1/b / b > 1 / Skinnier
0 < b < 1 / Fatter
0 > b > -1 / Fatter
b < -1 / Skinnier
C / Shifts the function left and right / c > 0 / Moves right
c < 0 / Moves Left
D / Shifts the function up and down / d > 0 / Moves Up
d < 0 / Moves Down
Hyperbola

Variable / What it does / Condition / Effect
A / Multiplies all y values by a / a > 1 / Spreads Away From Origin
0 < a < 1 / Closes In Towards The Origin
0 > a > -1 / Closes In Towards The Origin/Upside Down
a < -1 / Spreads Away From Origin/Upside Down
B / Multiplies all x values by 1/b / b > 1 / Closes In Towards The Origin
0 < b < 1 / Spreads Away From Origin
0 > b > -1 / Spreads Away From Origin/Upside Down
b < -1 / Closes In Towards The Origin/Upside Down
C / Shifts the function left and right / c > 0 / Moves right
c < 0 / Moves Left
D / Shifts the function up and down / d > 0 / Moves Up
d < 0 / Moves Down
Cubic Function

Variable / What it does / Condition / Effect
A / Multiplies all y values by a / a > 1 / Skinner
0 < a < 1 / Fatter
0 > a > -1 / Fatter/Upside Down
a < -1 / Skinnier/Upside Down
B / Multiplies all x values by 1/b / b > 1 / Skinnier
0 < b < 1 / Fatter
0 > b > -1 / Fatter/Upside Down
b < -1 / Skinnier
C / Shifts the function left and right / c > 0 / Moves right
c < 0 / Moves Left
D / Shifts the function up and down / d > 0 / Moves Up
d < 0 / Moves Down
Exponential Function

Variable / What it does / Condition / Effect
A / Multiplies all y values by a / a > 1 / 'a' is the y-intercept (Shift Up)
0 < a < 1 / 'a' is the y-intercept (Shift Down)
0 > a > -1 / 'a' is the y-intercept (Shift Down /Upside Down)
a < -1 / 'a' is the y-intercept (Shift Down Further/Upside Down)
B / Multiplies all x values by 1/b / b > 1 / Skinnier
0 < b < 1 / Fatter
0 > b > -1 / Fatter/Reflect in y-axis
b < -1 / Skinnier/ Reflect in y-axis
C / Shifts the function left and right / c > 0 / Moves right
c < 0 / Moves Left
D / Shifts the function up and down / d > 0 / Moves Up
d < 0 / Moves Down
Square Root Function

Variable / What it does / Condition / Effect
A / Multiplies all y values by a / a > 1 / Fatter
0 < a < 1 / Skinnier
0 > a > -1 / Skinnier/Upside Down
a < -1 / Fatter/Upside Down
B / Multiplies all x values by 1/b / b > 1 / Fatter
0 < b < 1 / Skinnier
0 > b > -1 / Skinnier/Reflect in y-axis
b < -1 / Fatter/Reflect in y-axis
C / Shifts the function left and right / c > 0 / Moves right
c < 0 / Moves Left
D / Shifts the function up and down / d > 0 / Moves Up
d < 0 / Moves Down

Domain and Range:

  • Domain refers to what spread of x-values are included in a particular function and is set out as follows:
  • Range refers to what spread of y-values are included in a particular function and is set out as follows:
  • Be sure to include asymptotes (an imaginary line on the x or y axis that values of the graph never intersect with)
  • Domain and Range for Un-Transformed Functions:
  • Parabola:
  • Hyperbola:
  • Cubic:
  • Exponential:
  • Square Root:

Proportions:

  • Direct Proportion: if x and y are in direct proportion, then the division of x and y will be constant ‘c’.
  • Inverse Proportion: if x and y are in direct proportion, then the product of x and y will be constant ‘c’.

Central Tendency and Dispersion

Measures of Central Tendency:

  • Mean: add all scores, divide by the number of scores.
  • Standard Deviation: how spread the data is from the mean.
  • Mode: most frequent number in a dataset.
  • Max: highest value in the dataset
  • Min: lowest value in the dataset.
  • Range: is equal to Max - Min.
  • Median: the middle score, the first 50% of the data and is also known as Q2. To determine the median in any given dataset, arrange the values in ascending order, and the position of the median in this order is denoted by the formula:
  • If the position is not a whole number using the formula (e.g. 2.5), then take the AVERAGE of the 2nd and 3rd numbers in the dataset to determine the median value.
  • Q1: Represents the first 25% of the data and its position can be determined using a formula similar to that above: .
  • Q3: Represents the first 75% of the data and its position can be determined using a formula similar to that above: .
  • Interquartile Range: is equal to Q3 – Q1 or the width of the box in the box plot diagram.

Different ways to Measure Data:

  • Cropped or Trimmed Values: usually a reference of removing any outliers from the dataset and see how they affect things such as the mean, median and range.
  • Standardized Score: is equal to
  • Weighted Means: is equal to

Box Plots and Histograms

Commenting on Dispersion in Data:

  • Gaps: in a frequency table, when the data drops down to zero for a certain value.
  • Clusters: groups of similar numbers in the data
  • More Dense Regions: Very similar numbers all in a location
  • Less Dense Regions: Varied numbers in a location
  • Skewness:
  • If right whisker > left whisker: positively skewed data or skewed to the right
  • in this scenario, usually the mean > median
  • If left whisker > right whisker: negatively skewed data or skewed to the left
  • in this scenario, usually the mean < median
  • Outliers:
  • if a number is < (1.5 x IQR) - Q1 or > Q3 + (1.5 x IQR), it is considered an outlier
  • Class Intervals: e.g. 10-20, 20-30, 30-40.
  • Mid-point of above: you must use the mid-point (i.e. 15, 25, 35) and then determine Central Tendency
  • Class Intervals w/ Missing Digits: e.g. 100-109, 110-119
  • Draw the results in a frequency histogram based on the class intervals above with 99.5, 109.5, 119.5
  • Modal Class: Which class interval has the mode in it.
  • Median Class: Which class interval has the mode in it.

Box plot Notes:

  • When drawing a box plot, draw the whiskers without the outliers and represent the outliers as a dot on the plot unless otherwise stated.
  • Use a broken axis when the axis doesn’t start from 0.
  • Label axis and title.

Histogram Notes:

  • Ensure that class intervals are drawn correctly.
  • Use a broken axis when the axis doesn’t start from 0.
  • Label axis and title.

Sequences

  • Geometric Sequences: where r is a constant called the common ratio.
  • Arithmetic Sequences: where d is a constant called the common difference.
  • Neither: if both a Geometric and Arithmetic Sequence.
  • Growth/Decay: display percentage as a decimal and then add growth to 1 or subtract decay from 1 (e.g. ).
  • Compound Interest Financial Tab on Class Pad:
  • N – total number of compounds (years x number of times compounded per year)
  • I% - interest rate (as a whole number) and is equal to (interest rate p.a. / number of times compounded per year)
  • PV – principal value (write as negative if no PMT)
  • PMT – repayment (write as negative if no principal)
  • FV – future value (for loans, it is zero)
  • P/Y – payments per year (how many times you pay back)
  • C/Y – compounds per year (how many times compounded)

Sampling Methods and Capture-Recapture

  • Random Sampling on Class Pad: RandInt(x,y,z) where: x is lower value, y, is upper value and z is the number of random numbers to be produced.
  • Stratified Sampling: (note: must round the number to the nearest whole)
  • Capture Recapture:

Normal Distribution

  • Thirds Rule:
  • FROM: mean to SD 1 on either side = 68% rule (0.68 of the data)
  • FROM: mean to SD 2 on either side = 95% rule (0.95 of the data)
  • FROM: mean to SD 3 on either side = 99.7% rule (0.997 of the data)
  • Standardized Score: if z is a standard score, the following process turns it into the raw score:
  • Standard Formula for normal distribution: which reads: X is normally distributed with mean myew and standard deviation sigma.
  • IF mean and standard deviation are not stated, assume mean = 0 and standard deviation = 1.
  • NOTE: be careful of standard deviation being squared or not in the question.

Misc: Surds, Binomial Theorem, and Quadratic Formula

[Mathematics Unit 3A] Course Notes

Surds:

Law 1:

Law 2:

Law 3:

Law 4:

Law 5:

NOTE:

Binomial Theorem:

Quadratic Formula:

[Mathematics Unit 3A] Course Notes

Space for Extra Notes and Examples