maths standard 2019 hsc

maths standard 2019 hsc

scatterplots

a graph of points where each point represents a value of two different variables (bivariate data)

correlation

two variables plotted on a scatterplot that shows a linear association (correlated) and is also a measure of strength of the linear association between the two variables

causality in correlation

leads to correlation and is where the 2 variables aren't affected by external factors (e.g. height and weight is affected by fat, growth)

Pearson's correlation coefficient (Pearson's r)

A statistical estimate which is designed to summarise the relationship between two variables using a single number (e.g. 0.3 is weak, 0.31-0.5 is moderate, 0.51-0.8 is strong and 0.81-1.00 is very strong/perfect)

how to calculate pearson's correlation (r) on calculator

given a table of data with 2 variables, say x and y
press:
1. Mode, 1, 1
2. Enter data (x and y): 190, 2NDF, STO, 192, M+
4. Repeat until all numbers are in
5. ALPHA, r, \=
r \= x

line of best fit

a line drawn in a scatter plot to fit most of the dots and shows the relationship between the two sets of data

straight line

y\=mx+c

How to draw the line of best fit

1. draw your data points onto a number plane
2. draw a line through as many points as you can, make sure its equal points on each side of the line
3. do an equation y-y1\=m(x - x1) and substitute points into it (find gradient first)

least squares regression line

the line with the smallest sum of squared residuals as small as possible or as close as possibly to the line

least squares regression formula

m \= r x (standard deviation of y-scores ÷ standard deviation of x-scores)
where r is pearsons correlation

how to graph a least squares regression line

1. find the mean (x) and standard deviation (ox) of each x and y value
2. find pearsons correlation coefficient
3. graph the points
4. use the equation m \= r x (sd of y ÷ sd of x)

maximum heart rate (MHR)

Males is 220 - age
Female is 226 - age

heart range formula

Recommended exercise percentage in decimal form x beats/min (e.g. 0.55 x 201 b/m)

target heart rate (THR)

THR \= I x (MHR - RHR) + RHR
Where THR is target heart rate
I is intensity exercise (rate) as a decimal
MHR is maximum heart rate
and RHR is resting heart rate

Energy units

1 kilojoule \= 1000 joules
1 megajoule \=

0 joules \= 1000 kJ
1 calorie \= 4.184 kJ

watt (W) units

1 W \= 1000 MW \= 1 000 000 milliwatts (uW)
1 kW \= 1000 W

domestic rate electricity

electricity used during the day

off-peak rate electricity

electricity used during late evening, early morning, night

running costs formula for electrical appliances

power used (kWh) x electricity cost ($/kWh)

standby power

power that is always provided, even when a computer is powered off. It is why you have to unplug a computer when working inside it.

energy efficient housing: orientation

living areas face north to capture winter sun, house axis run along east to west

energy efficient housing: insulation

roof insulation, light-coloured roofing and double-glazed windows reduce energy costs for heating and cooling

energy efficient housing: windows/skylights

full-length windows facing north to allow winter sun

energy efficient housing: ventilation

cross-ventilation allows airflow for cooling

energy efficient housing: landscaping/shading

plants and trees can shade walls and windows

energy efficient housing: building materials

choose materials that can absorb and store heat

solving an equation

1. keep the equation balanced by performing the same operation on both sides
2. aim to have pronumerals (for example, x) and numbers on separate sides

gradient formula

m \= rise / run

the terms in a linear function, y\= mx + c

the number in front of the x is called the coefficient of x whilst the number on its own is called a constant term

direct linear variation

If y varies as x, or y is directly proportional to x, then y \= kx, where k is a constant.

how to solve a linear variation equation

1. identify the 2 variables (say x and y) and form an equation "y \= kx"
2. substitute values for x and y to find k, the constant of variation
3. rewrite y \= kx using the value of k
4. substitute a value for x or y into y\=kx to solve the problem

break-even point

the point at which the costs of producing a product equal the revenue made from selling the product

how to solve a simultaneous equation

you are given two equations with two unknown variables in it (say x and y) and you are to find the answer that satisfies both
example:
4x - 2y \= -12
3x + y \= 1

1. rearrange any of the 2 equations above to make x or y the subject
2. using the 2nd equation it'll be y \= 3x - 1
3. substitute that formula into the y variable in the other equation
" 4x - 2 (3x-1) \= -12 " so you only have to find the x variable
4. solving the x variable, substitute into any of the above formulas to find y

simple interest

1 \= p x r x n
p is principal
r is rate p.a.
n is number of periods
interest is always rounded down

financial compound interest

FV \= PV (1 + r)n
FV is the future value
PV is the initial (current) value
r is the rate
n is the number of periods
payments is always rounded up

how to find the rate (r) in a compound interest formula

example:
21,440 \= 16,000 (1+r)6

1. do the same operations of a normal equation (division and minus) until you're left with (1 + r)6
2. Calculate 6 -\> 2NDF -\> yx -\> 1.34 (21,440 divided by 16,000) to get rid of 6 on both sides until you're left with the answer
3. minus 1 from both sides

formula is "power number (n), 2NDF, yx, (FV divided by PV)"
always round down for future value

inflation

a general increase in prices and fall in the purchasing value of money.

appreciation

An increase in the value of a currency

presented value (PV) is always rounded...

...UP

future value (FV) is always rounded...

...DOWN

dividend

The portion of corporate profits paid out to stockholders

dividend yield

a stock's expected cash dividend divided by its current price and is expressed as a %

dividend yield formula

dividends per share/market price per share

market price...

is a price of a commodity (a raw material/product) that is sold in each market (Woolworths, NBA)

how to find total earnings from an investment/share

dividend + selling price - brokerage - total cost

depreciation

the loss of value in an item over time

the two methods of depreciating an item are...

straight-line depreciation and declining-balance rate

straight-line method of depreciation

the value of the item decreases by the same equal amount each period

declining balance rate method

the value of the item decreases by the same percentage each period

straight-line depreciation formula

S \= Vo - Dn
where S is salvage value (current/future)
Vo is initial value (current/original)
D is amount of depreciation per period
n is the number of periods

declining balance formula

S \= Vo - Dn

total depreciation formula

Total depreciation \= purchase price - value after n years

reducing balance method

is a method of depreciation that reduces the value of a fixed asset by the same percentage each year throughout its useful life. This is the more realistic method to use.

credit card

A plastic card used to make purchases now and pay for them later.

annuity

a series of equal regular deposits

To make upfront charges of a home loan into a percentage:

total charges / home loan x 100%

a loop is...

an edge that connects a vertex with itself

directed network

a network diagram that has arrows on the edges to indicate the direction the flow is going to

weighted network

a network with weights associated with the edges

degree of a vertex

the number of edges that connect to that vertex. an even number of edges (even degree), an odd number of edges (odd degree)

the sum of degrees of all vertices in a network is equal to

twice the number of edges because each edge belongs to two vertices

a walk is...

any route with repeated edges/vertices

a closed walk is...

a walk that starts and ends at the same vertex

a trail is...

a walk with no repeated edges

a circuit is...

a closed trail

a path is...

a walk with no repeated edges or vertices

a closed path is called a...

cycle

eulerian trail

A trail that uses all edges of a graph, only occurs with 2 odd vertices ONLY and must start and end at the odd vertex

eulerian circuit

a eulerian trail that is closed (starts and ends at the same vertex) and only exists when all vertices have an even no. of edges attached

two efficient methods for minimum spanning tree

Kruskal's and prim's algorithm

Kruskal's algorithm

taking edges from smallest to largest with no cycles/repeats/closed trails

prim's algorithm

starting at any vertex and selecting the smallest edge that connects to it

how to find the shortest path

1. Redraw the network with circles at each vertex, except at the starting vertex, S
2. For all vertices one edge away from S, write down the shortest distance inside the circle
3. For all vertices 2 edges away from S, write down the shortest distance inside the circle
4. Continue this process until finish vertex, F, is reached
5. The shortest path is then identified by starting at F and moving backwards to the vertex from which the shortest distance at F was obtained, and continuing until S is reached

how to draw a directed network from an activity table

- Label the first and last vertices START and FINISH
- Represent activities by arrows and start and end them at a vertex
- Connect activities that do not have prerequisites to the start box
- Begin activities with the same predecessor at the same vertex
- Ensure that any 2 vertices are NOT CONNECTED by multiple edges

dummy activity

an activity having no time that is inserted into a network to maintain the logic of the network and to eliminate repeated edges

how to draw a directed network from an activity table

1. have two vertices labelled start and finish
2. connect all the activities with no predecessor to the start
3. start drawing the activities with those predecessors from the already made ones

critical path

critical path (most essential) has the longest weight sum but is considered the smallest hours possible to complete a task

forward scanning

calculates the longest time to get to the next activity by using EST with a box with the numerator and denominator (EST is the top box)

how to draw a network for forward scanning

1. redraw the network with boxes at each vertex
2. at the start vertex, write 0 on the top box
3. work along each path or activity from the start (left to right), writing the total time of the path at each vertex (in the TOP BOX)
4. when two or more paths meet at a vertex, select the highest total number
5. continue until vertex F is reached

backward scanning

similar ot forward scanning but uses LST in the bottom box and starts at vertex F and works it's way back with the smallest time

identifying the critical path (analysis)

longest time path from start to finish

float time or slack time

amount of time the early start of a task may be delayed without delaying the finish date of the project (non-critical only)

float time formula

LSTnext - ESTnext if it is the only path leading to the next activity
LSTnext - ESTnext - activity time used (the activity on their weighted edge) (used when there are multiple paths going to the next task)

use this for each non-critical activity where "next" means the next activity

what's in a flow network?

-the start is called source (S) and the end is called the sink (T)
-the weights (edges) are called capacities and represent the amount of flow an edge can hold

inflow and outflow of a vertex

total capacity of all edges entering the vertex is INFLOW
total capacity of all edges exiting the vertex is OUTFLOW

double-check the directions of the weighted edges to see if it's going in or out of the vertex

maximum outflow of a vertex

the smallest out of either the inflow or outflow of a vertex

capacity of a cut

a cut is drawn through the edges of a flow network, not the vertices, and acts as roadblock to stop all flow from the source (S) to the sink (T). the capacity of a cut is the amount of flow blocked by the cut, found by summing the cut edges passing from source to sink

maximum flow - minimum cut theorem

the value of maximum flow \= minimum cut

how to find maximum flow

find the minimum edge of each path and add it up for the total

linear functions

straight line, y\=mx+b, & there is not exponent on "x" in the equation.

non-linear function

a function whose graph is a curve

quadratic function

y\=x^2

parabola equation

y\=ax^2+bx+c

reciprocal function

y \= k / x

shape of a hyperbola

has 2 separate branches in opposite quadrants on the number plane
- if k is positive the hyperbola is decreasing and its branches are in the 1st and 3rd quadrants
- if k is negative the hyperbola is increasing and its branches are in the 2nd and 4th quadrants
- the higher the value of k the further the hyperbola is from the x- and y-axes

inverse variation

y \= k/x, when two variables go in different directions (e.g. speed, the faster you go , the less time it takes)

ratio

A comparison of two quantities by division

Dividing a quantity in a given ratio

1. Find the total number of parts (by adding)
2. Find the size of one part (by dividing)
3. Find the size of each term of the ratio (by multiplying)

unit price: price of one item

unit price \= price / no. of items or units