Exam Revision Flashcards
Revise for exams
1
Q
N = ?
A
the set of positive integers and zero
2
Q
Z = ?
A
the set of all integers
3
Q
Z+ = ?
A
the set of positive integers without zero
4
Q
Q = ?
A
the set of all rational numbers
5
Q
Q+ = ?
A
the set of positive rational numbers without zero
6
Q
R = ?
A
every number
7
Q
R+ = ?
A
any number greater than zero
8
Q
Irrational = ?
A
infinite numbers
9
Q
How to solve simultaneous equations
A
SSLE or graphing intersection
10
Q
Solution of quadratic equations ?
A
If x^2 = k, then = +/- SqRt (k)
= x-axis
= solutions of polynomials
11
Q
Does x,y satisfy the equation? how does this work
A
substitue x and y into the equation and then compare answer to equation on the graph –> is there a point with those values
12
Q
how to find y-intercept
A
let x = 0
13
Q
how to find x-intercept
A
let y = 0
14
Q
Finding the quadratic equation from a graph
A
Factorised form = x-intercept + point –> y = (x-a)(x-b)
Expanded form = any 3 points –> y = ax^2 + bx + c
–> substitute x-intercept values for a + b
–> add lettered variable “k” after y
–> solve for “k”
–> rearrange equation to y = k(x-a)(x-b)
15
Q
Finding the quadratic equation in expanded form
A
3 points enter coordinates in SPREADSHEET label columns menu-->Stats-->stat calcs--> option 6 Sub in
16
Q
finding the equation of a line
A
the gradient + point on the line
two points on the line
17
Q
R- = ?
A
all numbers less than zero
18
Q
Z- = ?
A
negative integers without zero
19
Q
representing number of elements in a set
A
n(A) = 6
20
Q
what is a subset
A
If p and q are two set and all of the elements of p are in q then
–> (2,4,6)
21
Q
Intersection
A
elements that are COMMON in both sets
22
Q
Union
A
all elements in both sets
23
Q
Disjointed sets
A
neither sets have elements in common
24
Q
How do you writie set notation
A
- X < 3 = (-infinity, 3]
25
what is the compliment
the compliment of A is everything not in A
26
how many mm-->cm-->m-->km
mm x10 = cm x100 = m x1000 = km
27
Area conversions mm^2 --> cm^2 --> m^2 --> ha
x100 --> x10,000 --> x10,000 --> x100
28
Volume conversions mm^3 --> cm^3 --> m^3 --> km^3
x10 = x100 = 1000
29
How to solve right angle triangle with 1 angle + 1 side
If side = 20cm and angle = 59 degrees
| then --> nSolve (Sin/Tan/Cos (59) = 20/x,x)
30
Finding the area of a triangle
If we have Height + Base = 1/2bh
| If we have 2 side + included anngle = 1/2(a)(b)Sin(C)
31
Cosine rule - what and when
Used for NON RIGHT-ANGLE TRIANGLES
When 2 side + included angle = A^2 formula
When 3 sides = Cos(A) formula
32
Sine rule - what and when
Used for NON RIGHT-ANGLE TRIANGLES
When 2 angles + 1 side --> a/SinA formula to find side
When 2 sides + non-included angle --> SinA/a formula to find angle
33
common difference
d = U1 + 1 - Un
34
Arithmatic mean
b = A+C/2
35
Common ratio
Un + 1 / Un
36
Geometric mean
b^2 = (a)(c)
37
what is a relation
a finite number of ordered pairs
| or - any set of powers which connect two variables (e.g: x and y)
38
what is a function
a relation in which no two different ordered pairs have the same x co-ordinate
39
how to test for a function
if we draw a vertical line on the graph the relation is a function if each line cuts the graph no more than once
40
what is a domain
the set of x-values in the relation
41
what is a range
the set of y-values in the relation
42
Domain and range notation
Domain x E (-1,infinity)
| Range y E (-3, infinity)
43
equation of an exponential function
y = ka^x + c
44
Key features that must be shown when sketching all graphs
1. x-intercept + y-intercept (co-ordinates)
2. turning points
3. asymptotes --> horizontal + vertical
4. state any restrictions on domain and range
45
Catagorical data
divided into catagories
| e.g: male + female
46
numerical data
data has a numerical value
47
Discrete data
an exact value and is often the result of counting
48
Continuous data
can take any numerical value within a range and is often a result of measuring
49
Measures of spread
- range
- IQR
- Standard deviation
range = max value - min value
IQR = Upper quartile - lower quartile
Standard deviation = use calc
50
how to make Line of Best Fit by eye
1. calculate mean of x and mean of y
2. plot the mean point of x-mean and x-y on a scatter plot
3. draw a line through mean point which best fits the trend of the data
51
Null Hypothesis
```
H0 = x and y are independant
H1 = x and y are not independant
```
52
Expected values
Var = stat exp.matrix
53
When do you reject or accept the hypothesis?
If chi>k --> reject H0
| If chi accept H0
54
When are two events independent
If Pr (A ^ B) = Pr (A) x Pr (B)
55
When are two events dependant
```
If Pr (A then B) = Pr (A) x Pr (B given A has occured)
--> Pr (B/A) = Pr (B intersection A) / Pr (A)
```
56
What is Pr (AuB) equal to
Pr (AuB) = Pr (A) + Pr (B) - Pr (A ^ B)
57
Formula for conditional probability
Pr (A/B) = Pr (A ^ B) / Pr (B)
58
Logical propositions
can be true / false / indeterminate
59
Compound propositions
propositions that include 'and' + 'or'
60
What is a conjunction
symbol is ^ and the word used is 'and'
61
What is a disjunction
symbol is upside down ^ and the word used is 'or'
62
What is exclusive disjunction
True if --> x or y but not both
| symbol is upside down ^ with underline
63
What is a tautology
a tautology is a compound proposition if all the values in the truth table column are true
64
Logical contradiction
a logical contradiction is a compound proposition if all the values in the truth table column are false
65
What is logical equivalent
when both columns in a truth table are identical
66
Converse of a statement
q --> p
| p --> q (converse)
67
Inverse of a statement
q --> p
| -q --> -p (inverse)
68
Calculus: Average rate of change
gradient of a line between 2 points
| y2-y1 / x2-x1
69
Contrapositive of a statement
q --> p
| -p --> -q
70
Derivative function
gives the gradient of a function at a point
71
How to find the equation of a tangent
find a point
substitute x value into derivative to find gradient
nSolve for C
72
How to find equation of the normal (perpendicular) to an equation
find point
substitute x value into derivative to find gradient
reverse the gradient
Nsolve for C
73
when does a stationary point occur
when the gradient = 0
74
types of stationary points
maximum, minimum, inflexion
75
global maximums + global minimums
the maximum / minimum for the entire domain
76
local maximum + local minimum
occur at turning points
77
two ways to find stationary points
1. graph the function and find the max / min
| 2. find derivative of the function --> solutions of polynomials --> substitute x value into equation for y value
78
what does dy/dx give other than derivative
rate of change
| --> gives the rate of change in y with respect to x
79
how to apply differential calculus (derivative) and find the time rate of change?
1. find dy/dx
2. substitute value in for x
3. if dy/dx is > 0 then it is increasing
- -- if dy/dx is < 0 then it is decreasing
80
general rates of change
other rate problems can be treated the same way as those involving time
--> E.g: the cost of manufacturing x items has the cost function of C (x) dollars.
81
How to work out optimisation problems
1. draw a large clear diagram of the situation (shape)
- -> create a formula for the volume of the shape
- --> E.g: 4000 = x^2y
2. construct a formula with the variable to be optimised as the subject
- -> E.g: A(x) = x^2 + 16000x^-1
3. Find the derivative and fine the value of x where it is zero
- -> E.g: A'(x) = 2x - 16000x^-2
- -> E.g:
4. show using a sign diagram or a graphical test that you have a maximum or a minimum
82
What is the equation of normal distribution?
(x ~ N (u, 6^2)
u = mean
6^2 = standard deviation
83
What are the normal distribution boundaries
0. 13%
2. 15%
13. 59%
34. 13%
84
What are the mean boundaries for normal distribution
-36
-26
-6
0
+6
+26
+36
