GCSE Mathematics - Pearson Edexcel - COMPLETED Flashcards
FLASHCARDS MADE MYSELF WITH AID OF SOURCE BOOK: GCSE (9-1) MATHEMATICS REVISION GUIDE HIGHER BY PEARSON EDEXCEL
1
Q
What are the factors of a number?
A
The factors of a number are any numbers that divide into it exactly
2
Q
How many factors does a prime number have?
A
A prime number has only 2 factors (1 and itself)
3
Q
List the prime numbers up to 20:
A
2, 3, 5, 7, 11, 13, 17, 19
4
Q
What is a prime factor?
A
A number is a prime factor if it is both prime and a factor of another number (REMEMBER PRIME FACTORISATION TREES - THE NUMBERS WHICH YOU CIRCLE)
5
Q
How can you find prime factors?
A
Using a prime factorisation tree (also known as a prime factor tree)
6
Q
Let’s say that you just used a prime factor tree to find the prime factors of 84. They are 2, 2, 3 and 7. How would you write 84 as a product of its prime factors in the answer box?
A
2^2 X 3 X 7
7
Q
What does HCF stand for?
A
Highest Common Factor
8
Q
What is the HCF of 2 numbers?
A
The highest common factor of 2 numbers is the highest number that is a factor of both numbers
9
Q
What does LCM stand for?
A
Lowest Common Multiple
10
Q
What is the LCM of 2 numbers?
A
The lowest common multiple of 2 numbers is the lowest number that is a multiple of both numbers
11
Q
What do indices include (as a category)?
A
Square roots, cube roots and powers
12
Q
List the 3 laws of indices:
A
- x^a X x^b = x^a + b
- x^a / x^b = x^a - b
- (x^a)^b = x^ab
13
Q
True or false: The cube root of a positive number is always positive, and the cube root of a negative number is always negative
A
True. The cube root of a positive number is always positive, and the cube root of a negative number is always negative
14
Q
x^0 =
A
1 - Anything raised to the power of 0 is equal to 1
15
Q
x^1 =
A
x - Anything raised to the power of 1 is equal to itself
16
Q
List the square numbers up to 15^2:
A
1 X 1 = 1
2 X 2 = 4
3 X 3 = 9
4 X 4 = 16
5 X 5 = 25
6 X 6 = 36
7 X 7 = 49
8 X 8 = 64
9 X 9 = 81
10 X 10 = 100
11 X 11 = 121
12 X 12 = 144
13 X 13 = 169
14 X 14 = 196
15 X 15 = 225
17
Q
List the cube numbers of 1, 2, 3, 4, 5 and 10:
A
1 X 1 X 1 = 1
2 X 2 X 2 = 8
3 X 3 X 3 = 27
4 X 4 X 4 = 64
5 X 5 X 5 = 125
10 X 10 X 10 = 1000
18
Q
True or False: A negative power always has a negative answer
A
False. A negative power can still have a positive answer
19
Q
What is the index law for negative powers?
A
x^-y = 1 / y^2
20
Q
What is the index law for powers of fractions?
A
(x / y)^n = x^n / y^n
20
Q
What is the index law for reciprocals?
A
x^-1 = 1 / x. This means that x^-1 is the reciprocal of x. You can find the reciprocal of a fraction by “turning it upside down” - switching the numerator and denominator
21
Q
What do fractional powers represent?
A
Roots. For example, x^1/2 = the square root of x (49^1/2 = 7). Or, x^1/3 = the cube root of x (27^1/3 = 3). Or, x^1/4 = the 4th root of x (16^1/4 = 2)
22
Q
What always happens if we raise a whole number to a power less than one (a fractional power)?
A
It gets smaller
23
Q
If the number on the right is 5 or more, we round…
A
Up
24
If the number on the right is less than 5, we round...
Down
25
True or false: When rounding significant figures, the leading zeros in decimals are included
False. Leading zeros in decimals are not counted as significant
26
Explain the steps you would go through to add or subtract fractions:
1. Add or subtract the whole numbers
2. Write the fractions as fractions with the same denominator
3. Add or subtract the fractions
4. If you have an improper fraction then convert to a mixed number and add/subtract
27
Explain the steps you would go through to divide fractions:
1. Convert any mixed numbers to improper fractions
2. Turn the second fraction "upside down" - switch the numerator and denominator - and change the / to a X
3. Multiply the numerators and multiply the denominators, cancelling where possible
4. Convert any improper fractions to mixed numbers
28
Explain the steps you would go through to multiply fractions:
1. Convert any mixed numbers to improper fractions
2. Multiply the numerators and multiply the denominators, cancelling where possible
3. Convert any improper fractions to mixed numbers
29
What are terminating decimals?
Decimals which can be written exactly. They can be written as a fraction with the denominator 10, 100, 1000 and so on
30
What are recurring decimals?
Decimals that have one digit or group of digits repeated forever. You can use dots above the numbers to show the recurring digit (single dot above the digit) or recurring group of digits (two dots on the first and last digits of the group)
31
How would you check if a fraction will produce a terminating or a recurring decimal?
Write the fraction in its simplest form, then find the prim factors of the denominator. If the prime factors are only 2 and 5 it is a terminating decimal. If the prime factors are other than just these it is a recurring decimal
32
How can you estimate the answer to a calculation?
Round each number to 1 significant figure and carry out the calculation with these numbers
33
What does ≈ mean?
Approximately equal to
34
Explain the steps you would go through to convert a recurring decimal into a fraction:
1. Write the recurring decimal as n
2. Multiply by 10, 100, 1000 (depending on the number of recurring digits)
3. Subtract to remove the recurring part
4. Divide by 9, 99 or 999 to write as a fraction (depending on the number of recurring digits)
5. Simplify the fraction
35
What are upper and lower bounds a measure of?
Accuracy
36
To find the overall upper bound of a + b, we use the equation...?
Overall upper bound of a + b = upper bound of a + upper bound of b
37
To find the overall lower bound of a + b, we use the equation...?
Overall lower bound of a + b = lower bound of a + lower bound of b
38
To find the overall upper bound of a - b, we use the equation...?
Overall upper bound of a - b = upper bound of a - lower bound of b
39
To find the overall lower bound of a - b, we use the equation...?
Overall lower bound of a - b = lower bound of a - upper bound of b
40
To find the overall upper bound of a X b, we use the equation...?
Overall upper bound of a X b = upper bound of a X upper bound of b
41
To find the overall lower bound of a X b, we use the equation...?
Overall lower bound of a X b = lower bound of a X lower bound of b
42
To find the overall upper bound of a / b, we use the equation...?
Overall upper bound of a / b = upper bound of a / lower bound of b
43
To find the overall lower bound of a / b, we use the equation...?
Overall lower bound of a / b = lower bound of a / upper bound of b
44
What are the 2 most important rules to remember when dealing with surds?
Square root of ab = square root of a X square root of b
AND
Square root of a/b = square root of a / square root of b
45
What are you being asked to do if a surds question asks you to rationalise the denominator?
Make the denominator a whole number. You can do this by multiplying the top and bottom of the fraction by the surd part of the denominator
46
How would you count possibilities using calculation?
Multiply the number of choices for each option. For example:
I have a 4 letter password I am trying to guess: p7m$
The first character must be a letter - 26 possibilities
The second character must be a digit 0 - 9 - 10 possibilities
The third character must be a letter - 26 possibilities
The fourth character must be one of $, *, _ or # - 4 possibilities
So the total number of possibilities for my password is 26 X 10 X 26 X 4 = 27,040 possible combinations
47
How many sets of brackets will you need to factorise the expression x^2 + bx + c?
2 brackets
48
When factorising x^2 + bx + c, when will both factors be positive?
When b and c are both positive
49
When factorising x^2 + bx + c, when will the larger factor be positive and the smaller number be negative?
When b is positive and c is negative
50
When factorising x^2 + bx + c, when will the larger factor be negative and the smaller factor be positive?
When b and c are both negative
51
When factorising x^2 + bx + c, when will both factors be negative?
When b is negative and c is positive
52
What is factorising (and what do you need to look for)?
Factorising is the opposite of expanding brackets. You look for the largest factor you can take out of every term in the expression
53
How would you factorise the expression (something)^2 - (something else)^2 - WHAT FORMULA WOULD YOU USE?
a^2 - b^2 = (a + b)(a - b)
EXAMPLE:
x^2 - 36 = x^2 - (6)^2
= (x + 6)(x - 6)
54
When factorising x^2 + bx + c, you are looking for 2 numbers which...?
SUM to b and MULTIPLY to make c
55
What is a formula?
A mathematical rule
56
What is the plural form of formula?
Formulae
57
What does BIDMAS stand for?
Brackets
Indices
Division
Multiplication
Addition
Subtraction
58
What is an arithmetic (also known as a linear) sequence?
A sequence of numbers where the difference between the consecutive terms is constant
59
Explain the steps you would take to find the nth term of a linear sequence:
1. Write in the difference between each consecutive term in the sequence
2. Work backwards to find the zero term in each sequence (ie. if the sequence starts with 1 and the difference between all the following numbers is 4, -4 from one to find the zero term is -3)
3. Write down the nth term - remember nth term = difference X n + zero term
60
What is the formula for the nth term?
nth term = difference X n + zero term
61
What is the rule for a Fibonacci sequence?
The rule for generating a Fibonacci sequence is 'add two consecutive terms to get the next term'. EXAMPLE: 2, 3, 5, 8, 13, 21...
62
What is a quadratic sequence?
A sequence where the nth term contains an n^2 term (and no higher power of n)
63
You can also write the nth term in shorthand as...
u↓n
64
You can write the rule for the nth term of a quadratic sequence in shorthand as...
u↓n = an^2 + bn + c where a, b and c are numbers and a is not 0
65
True or false: The second differences in quadratic sequences are constant
True. EXAMPLE:
SEQUENCE: 2, 10, 24, 44, 70, 102
1ST DIFF: +8, +14, +20, +26, +32
2ND DIFF: +6, +6, +6, +6
So the second differences in quadratic sequences are constant
66
Explain how you would find the nth term of a quadratic sequence:
1. Write out the first set of differences between each consecutive value in the sequence - NOT CONSTANT
2. Next write out the differences between the first set of differences - THIS SHOULD BE CONSTANT - THIS IS THE NUMBER YOU CARRY FORWARD
3. The coefficient of n^2 in the nth term is always half the second difference, so halve your number to find the value of a
4. Draw out a table to compare the values of n (1 counting up) and the nth term (your sequence in order)
5. You now know the value of a - add another row for an^2 and multiply a by each number for n
6. Subtract this new sequence from each term in your original sequence to form an arithmetic sequence with a constant difference
7. The nth term of your arithmetic sequence is the second half of the nth term of your quadratic sequence (ie. if a was 0.5^2 and the arithmetic sequence had an nth term of 0.5n, your overall expression is 0.5^2 + 0.5n)
67
If an equation is in the form y = mx + c, its graph will be a ........... line
If an equation is in the form y = mx + c, its graph will be a straight line
68
In the equation y = mx + c, what does the m tell you?
The gradient
69
In the equation y = mx + c, what does the c tell you?
The y-intercept (at what y-coordinate the line intersects with the y-axis)
70
How would you go about plotting a straight line graph?
Draw a table of values for x and y, use substitution to figure out the values and plot the (x, y) coordinates on the graph - they should form a straight line graph
71
What is the equation for calculating gradient (m)?
m = y^2 - y^1 / x^2 - x^1
72
Explain how you would find the equation of a straight line graph given one point and the gradient:
1. Substitute the gradient for m into y = mx + c
2. Substitute the x and y values given in the equation (for y and x respectively)
3. Solve the equation to find c
4. Write out the finished equation
73
Explain how you would find the equation of a straight line graph given 2 points:
1. Draw a sketch showing the 2 points
2. Work out the gradient of the line by drawing a triangle
3. Now that you know the gradient and at least one of the points you can use the other method.
4. Substitute m into the form y = mx + c
5. Choosing one of the (x,y) points you have, substitute this into the equation y = mx + c
6. Solve the equation to find c
7. Write out the finished equation
74
If the line slopes down, the gradient is...
Negative
75
True or false: Parallel lines have the same gradient
True. Parallel lines have the same gradient
75
If the line sloped up, the gradient is...
Positive
76
What does perpendicular mean?
At right angles
77
If a line has gradient m then any line perpendicular to it will have gradient of:
- 1 / m
78
What is a line segment?
A short section of a straight line
79
What contextual information will you need to know in order to find the midpoint of a line segment?
The coordinates of each end of the segment
80
What is the equation for finding the coordinates of the midpoint of a line?
Coordinates of midpoint = (average of x-coordinates, average of y-coordinates)
81
What type of graphs do quadratic equations have?
Curved graphs
82
What is the turning point of a graph?
The point where the direction of the curve changes
83
What are cubic graphs?
Graphs that contain an x^3 term and no higher powers
84
What are reciprocal graphs?
Graphs of the form y = k / x where k is a number
85
What does a distance-time graph show?
How distance changes with time
86
What does a horizontal line on a distance-time graph mean?
A horizontal line means no movement
87
What do straight lines on distance-time graphs mean?
That the subject was travelling at a constant speed
88
What does the gradient of a distance-time graph tell you?
The rate of change of distance with time (THIS IS ALSO CALLED SPEED)
89
What form are quadratic equations written in?
ax^2 + bx + c where a, b and c are numbers
90
Explain how you would solve a quadratic equation:
1. Rearrange into the form ax^2 + bx + c
2. Factorise the left hand side
3. Set each factor equal to zero and solve to find 2 values of x
91
What is the quadratic formula?
The solutions of the quadratic equation ax^2 + bx + c = 0 where a != o are given by:
x = -b +- square root of (b^2 - 4ac) / (2a)
92
How many solutions can a quadratic equation have?
A quadratic equation can have 2 solutions, 1 solution or no solutions
93
For quadratic equations in the form ax^2 + bx + c, if b^2 - 4ac is negative how many solutions are there?
No solutions - you cannot calculate the square root of a negative number
94
For quadratic equations in the form ax^2 + bx + c, if b^2 - 4ac is equal to zero how many solutions are there?
One solution
95
For quadratic equations in the form ax^2 + bx + c, if b^2 - 4ac is greater than zero how many solutions are there?
Two different solutions
96
What is the equation for completed square form?
If a quadratic expression is written in the form (x + p)^2 + q it is in completed square form - YOU CAN SOLVE QUADRATIC EQUATIONS WHICH DON'T HAVE INTEGER ANSWERS BY COMPLETING THE SQUARE
97
What are the 2 identities which you can use to save time when you are completing the square?
1. x^2 + 2bx + c ≡ (x + b)^2 - b^2 + c
2. x^2 - 2bx + c ≡ (x - b)^2 - b^2 + c
98
Explain the algebraic solution steps for solving simultaneous equations:
1. Number each equation
2. If necessary, multiply the equations so that the coefficients of one unknown are the same
3. Add or subtract the equations to eliminate that unknown
4. Once one unknown is found use substitution to find the other
5. Check the answer by substituting both values into the other equation
99
When graphing simultaneous equations, the coordinates of the point of intersection tell you what?
The solution to the simultaneous equations (the point gives you both an x and a y value)
100
What is the equation of a circle?
x^2 + y^2 = r^2
101
What is a tangent to a circle?
A line which just touches the circle once - it is always perpendicular to the radius of a circle
102
What is the angle between the tangent and the radius?
90 degrees
103
What does an inequality tell you?
When one value or expression is bigger or smaller than another value. You can represent inequalities on a number line
104
When drawing inequalities you mark a small circle at the end of your arrow mark - if the circle is open (empty) that means that...?
An open circle represents < or > and means that the number at the end of the arrow is NOT included
105
When drawing inequalities you mark a small circle at the end of your arrow mark - if the circle is closed (shaded in) that means that...?
A closed circle represents <= or >= and means that the number at the end of the arrow IS included
106
What is the rule for multiplying or dividing both sides of an inequality by a negative number?
If you multiply or divide both sides of an inequality by a negative number you have to reverse the inequality sign. EXAMPLE: -5x > 4 (/ -5)
x < - 4/5
107
What are integers?
Positive or negative whole numbers, including 0
108
What does the graph y = x^2 look like?
A downwards sloping curve
109
What is the difference between the trig graph of sin x and cos x?
They are the same shape, but y = sin x is translated 90 degrees to the right. So cos x is symmetrical about the y-axis (does not cross it) and sin x is symmetrical about the line x = 90 degrees (crosses the y-axis)
110
How often does the graph y = tan x repeat?
Every 180 degrees
111
Where are the asymptotes on the graph y = tan x?
There are asymptotes at -90 degrees, 90 degrees, 270 degrees, etc. The graph gets closer to these asymptotes but never reaches them
112
y = sin x is the same as...
y = cos (x - 90)
113
Explain the function of y = f(x) + a:
y = f(x) + a
Translation: ( O / a ) - WRITTEN AS A VECTOR
f(x) + a → move up a units
f(x) - a → move down a units
114
Explain the function of y = f(x + a):
y = f(x + a)
Translation: ( -a / O ) - WRITTEN AS A VECTOR
f(x + a) → move left a units
f(x - a) → move right a units
115
Explain the function of y = -f(x):
y = -f(x)
Reflection in the x-axis
116
Explain the function of y = f(-x)
y = f(-x)
Reflection in the y-axis
117
How do you find the turning point of a graph?
By writing the function in completed square form
118
What is the turning point of graph y = (x - a)^2 + b?
The graph of y = (x - a)^2 + b has a turning point at (a, b)
119
What are the roots of a function f(x)?
The roots of a function f(x) are the values of x for which f(x) = 0. This means that the roots of the function are the x-values at the points where y = f(x) crosses the x-axis
120
What should a sketch of a graph show?
All the major features, usually including turning points and places where the graph crosses the axis
121
When sketching cubic graphs, what are the 2 key things to remember?
1. If one factor is x then the curve will pass through the origin
2. If one factor is squared then the curve will just touch the x-axis at the corresponding point
122
What do we use iteration for?
Finding exact answers to more complicated questions, like those involving cubes or square roots. You can use iteration to find numerical answers to a given degree of accuracy
123
There is not only one iterative formula →
If you have to use an iteration formula to find the root or solve an equation, you will usually be given it in the exam. You will also usually be told what value of x↓O to use
124
What do we use an iteration formula to find?
The roots
125
What is the golden rule when simplifying algebraic fractions?
If the top or bottom of the fraction has more than one term, you will need to factorise before simplifying
126
Explain how to add or subtract algebraic fractions with different denominators:
1. Find a common denominator
2. Add or subtract the numerators
3. Simplify if possible
127
Explain how to multiply algebraic fractions:
1. Multiply the numerators AND multiply the denominators
2. Simplify if possible
128
Explain how to divide algebraic fractions:
1. Change the second fraction to its reciprocal
2. Change / to X
3. Multiply the fractions and simplify
129
How do you remove fractions from an equation?
Multiply everything by the LCM of the denominators
130
When expanding quadratics, what does FOIL stand for?
First Outer Inner Last
131
In f(x), what does f stand for?
Function
132
In f(x), what does x represent?
Input
133
How can you also write the composite function fg(x)?
f[g(x)]
134
What is a composite function?
A composite function is created if you apply 2 functions one after the other - it is a single function which has the same effect as the 2 combined functions
135
What is the inverse of a function?
For a function f, the inverse of f is the function that UNDOES f. Written as f^-1, if you apply f then f^-1 you will end up back where you started
136
How do you write the inverse of a function?
f^-1
137
How do you find the inverse of a function given in the form f(x)?
1. Write the function in the form y = ...
2. Rearrange to make x the subject
3. Swap any y's for x's and rewrite as f^-1(x) = ...
138
What do you always have to use to prove something about numbers is true/false?
Algebra
139
Algebraic proof - how to represent an even number?
2n
140
Algebraic proof - how to represent an odd number?
2n + 1 OR 2n - 1
141
Algebraic proof - how to represent a multiple of 3?
3n
142
Algebraic proof - how to represent consecutive numbers?
n, n + 1, n + 2, n + 3...
143
Algebraic proof - how to represent consecutive even numbers?
2n, 2n + 2, 2n + 4, 2n + 6...
144
Algebraic proof - how to represent consecutive odd numbers?
2n + 1, 2n + 3, 2n + 5, 2n + 7...
145
Algebraic proof - how to represent consecutive square numbers?
n^2, (n + 1)^2, (n + 2)^2, (n + 3)^2...
146
What is an exponential function?
A function of the form f(x) = ka^x
147
What does the shape of an exponential graph depend on?
Whether x is positive or negative
148
What do exponential graphs generally represent in questions?
Growth or decay
149
How can you estimate the gradient of a curve?
You can estimate the gradient of a curve at a given point by drawing a tangent to the curve at that point. You can then draw a large triangle with that tangent line as the hypotenuse and calculate the gradient by dividing the height of the triangle by the base of the triangle
150
What is velocity?
Speed in a certain direction
151
What is the equation for acceleration?
Acceleration = change in velocity / change in time
152
How do you interpret distance-time graphs?
The gradient of a velocity-time graph tells you the acceleration. The area underneath a velocity-time graph tells you the distance travelled - you can treat the area under the graph as shapes to find the area of
153
What is the most common unit for acceleration?
m/s^2
154
How would you go about calculating areas under curves?
You can estimate the area under a curve by drawing trapeziums in EQUAL INTERVALS underneath the graph. Then find the areas of each of these trapeziums and add them together to find an estimate
155
You can solve lots of percentage problems by working out what 1% represents. Explain how to find the percentage of an amount:
1. Divide the percentage by 100
2. Multiply by the amount
156
Explain how to write 1 quantity as a percentage of another:
1. Divide the first quantity by the second quantity
2. Multiply your answer by 100
157
How to convert a percentage to a fraction?
Write the percentage as a fraction with denominator 100
158
How to convert a fraction to a decimal?
Divide the numerator by the denominator
159
How to convert a decimal to a percentage?
X 100
160
How to convert a percentage to a decimal?
/ 100
161
What are ratios used for?
To compare quantities
162
What does it mean to say two quantities are in direct proportion?
Double one, double the other - both quantities increase at the same rate
163
What does it mean to say two quantities are in inverse proportion?
Double one, halve the other - one quantity increases at the same rate as another quantity decreases
164
Explain how to calculate a percentage increase or decrease:
1. Work out the amount of the increase or decrease
2. Write this as a percentage of the original amount
165
How would you go about using a multiplier to solve a reverse percentage?
If you are given the final amount you need to find the original amount - DIVIDE BY THE MULTIPLIER
166
What happens if you leave your money in a savings account?
It will earn compound interest
167
What is the equation for calculating a repeated percentage change?
Final amount = (starting amount) X (multiplier)^n
where n is the number of times the change is made
168
What is the equation for speed?
Average speed = total distance travelled / total time taken
169
What is the equation for time?
Time = distance / average speed
170
What is the equation for distance?
Distance = average speed X time
171
What are the 3 most common units of speed?
1. m/s
2. km/h
3. mph
172
How to convert minutes to hours?
Divide by 60
173
How to convert hours to minutes?
Multiply by 60
174
How many seconds in an hour?
60 X 60 = 3600 seconds
175
What is the density of a material?
Its mass per unit volume
176
What is the equation for density?
Density = mass / volume
177
What is the equation for volume?
Volume = mass / density
178
What is the equation for mass?
Mass = density X volume
179
What are the 2 most common units of density?
1. g / cm^3 (grams per cubic metre)
2. kg / cm^3 (kilograms per cubic metre)
180
What are compound measures?
Compound measures are made up of 2 or more other measurements. Speed is a compound measure because it is calculated using distance and time. Density is a compound measure because it is calculated using mass and volume
181
What is the equation for pressure?
Pressure = Force / Area
182
What is the equation for force?
Pressure X Area
183
What is the equation for area?
Area = Force / Pressure
184
What is pressure a measure of?
Pressure is a measure of the force applied over a given area
185
What are the 2 most common units of pressure?
N/cm^2 and N/m^2
186
How can you tell if a compound measure is calculating a rate?
It is calculating a rate if the bottom (denominator) compound measure is time. EXAMPLES:
Speed = Distance / Time
Rate of Flow = Volume / Time
Rate of Climb = Height / Time
Rate of Pay = Salary / Time
187
Which symbol is used to show proportion?
∝
188
Are your quantities directly or inversely proportional if you can write an equation y = kx where k is a number?
They are directly proportional
189
What does the graph showing x and y are directly proportional look like?
The graph of x against y is a straight line passing diagonally through the origin
190
What is the expression for x and y to show they are directly proportional?
y ∝ x
191
What is the expression for x and y to show they are inversely proportion?
y ∝ 1 / x → y is directly proportional to the reciprocal of x
192
What does the graph showing x and y are inversely proportional look like?
A reciprocal graph - line slopes in direction of origin but never touches either axis
193
What is the equation for inverse proportionality involving k where k is a number?
y = k / x where k is a number
194
What is k called in proportionality?
The constant of proportionality
195
Explain how you can find a proportionality formula - assume that you know the type of proportionality and are given 2 corresponding values for the variables:
1. Write down the formula using k for the constant of proportionality
2. Substitute the values you are given into the formula
3. Solve to find the value of k
4. Write down the formula with the value of k
196
Describe the relationship between corresponding angles:
Corresponding angles are equal
197
What are parallel lines on angle diagrams marked with?
Arrows
198
Describe the relationship between alternate angles:
Alternate angles are equal
199
Describe the relationship between vertically opposite angles:
Vertically opposite angles are equal
200
What do co-interior angles add up to?
Co-interior angles add up to 180 degrees
201
What do the angles in a triangle add up to?
180 degrees
202
What do the angles in a quadrilateral add up to?
360 degrees
203
What is the exterior angle of a triangle equal to?
The sum of the interior angles at the other 2 vertices
204
Describe the relationship between the opposite angles of a parallelogram:
Opposite angles of a parallelogram are equal
205
What do angles on a straight line add up to?
Angles on a straight line add up to 180 degrees
206
What do angles around a point add up to?
Angles around a point add up to 360 degrees
207
Describe the relationship between opposite angles:
Opposite angles are equal
208
Describe relationship between corresponding angles:
Corresponding angles are equal
209
What is the relationship between the base angles of an isosceles triangle?
The base angles of an isosceles triangle are equal
210
What is the equation for calculating the sum of the interior angles of a polygon?
Sum of interior angles = 180 X (number of sides - 2)
211
What is the sum of the exterior angles of a polygon?
Exterior angles of a polygon sum to 360 degrees
212
True or false: In a regular polygon the sides are not equal
False. In a regular polygon all of the sides are equal and all the angles are equal
213
What is the equation for finding one exterior angle of a polygon?
360 / n where n is the number of sides
214
What is pythagoras' theorem?
a^2 + b^2 = c^2
215
When do we use pythagoras' theorem?
1. If it is a right-angled triangle
2. Lengths of 2 sides are known
3. Length of third side unknown
216
How many degrees in a right angle?
90 degrees
217
What is the acronym for remembering trigonometric ratios?
SOH CAH TOA
218
Trigonometric ratio → sin x =
opposite angle / hypotenuse
219
Trigonometric ratio → cos x =
adjacent angle / hypotenuse
220
Trigonometric ratio → tan x =
opposite angle / adjacent angle
221
What is the exact value of sin 0?
0
222
What is the exact value of cos 0?
0
223
What is the exact value of tan 0?
0
224
What is the exact value of sin 90?
1
225
What is the exact value of cos 90?
0
226
Why is there no exact value of tan 90?
It is undefined - if you enter it into your calculator you will receive an error message
227
What is the exact value of sin 30?
1 / 2
228
What is the exact value of cos 30?
Square root of 3 / 2
229
What is the exact value of tan 30?
1 / Square root of 3
230
What is the exact value of sin 60?
Square root of 3 / 2
231
What is the exact value of cos 60?
1 / 2
232
What is the exact value of tan 60?
Square root of 3
233
What is the exact value of sin 45?
1 / Square root of 2
234
What is the exact value of cos 45?
1 / Square root of 2
235
What is the exact value of tan 45?
1
236
What is the equation for the area of a triangle?
Area of a triangle = 1/2 X base X height
237
What is the equation for the area of a parallelogram?
Area of a parallelogram = base X height
238
What is the equation for the area of a trapezium?
Area of a trapezium = 1/2 X (a (top width) + b (base width)) X height
239
How many mm^2 in 1 cm^2?
100mm^2
240
How many mm^3 in 1 cm^3?
1000mm^3
241
How many cm^2 in 1 m^2?
10 000cm^2
242
How many m^2 in 1 km^2?
1 000 000m^2
243
How many mm^2 in 1 cm^2?
100mm^2
244
How many mm^3 in 1 cm^3?
1000mm^3
245
How many cm^3 in 1 m^3?
1 000 000 cm^3
246
How many cm^3 in 1 litre?
1 litre = 1000 cm^3
247
How many cm^3 in 1 ml?
1 ml = 1 cm^3
248
What do you multiply by to convert metres into cm?
X 100
249
What do you multiply by to convert m^2 into cm^2?
X 100^2
250
What do you multiply by to convert m^3 into cm^3?
X 100^3
251
What is a prism?
A 3D solid with a constant cross-section
252
What is the formula for calculating the volume of a prism?
Volume of a prism = areas of cross-section X length
253
How do you work out the surface area of a 3D shape?
Add together the areas of all the faces
254
What is the formula for the circumference of a circle?
πd OR 2πr
255
What is the formula for the area of a circle?
πr^2
256
What is the formula for the volume of a cylinder?
πr^2h
257
What is the formula for the surface area of a cylinder?
2πr^2 + 2πrh
258
Each pair of radii divides a circle into 2 sectors - what are these called?
Major sector and minor sector (depending on which is smaller/larger)
259
How can you find the area of a sector?
You can find the area of a sector by working out what fraction it is of the whole number. EXAMPLE: For a sector with angle x and radius r: sector = x / 360 of the whole circle so
area of sector = x / 360 X πr^2
arc length = x / 360 X 2πr
260
What is the equation for the area of a sector?
x / 360 X πr^2 where x is the angle between the 2 radius lines
261
What is the equation for arc length of a sector?
x / 360 X 2πr where x is the angle between the 2 radius lines
262
What is the formula for the volume of a cone?
1/3πr^2h
263
What is the formula for the volume of a sphere?
4/3πr^3
264
What is the formula for the volume of a pyramid?
1/3Ah where A is the area of the base
265
What is the formula for the curved surface area of a cone?
πrl
266
What is the formula for the surface area of a sphere?
4πr^2
267
What are plans and elevations?
2D drawings of 3D shapes as seen from different directions - a plan is the view from above, the front elevation is the view from the front, the side elevation is the view from the side, etc.
268
When looking at the plan and elevations of a 3D shape, you notice that the views from 2 directions are rectangles. What can you deduce about the shape?
It is a prism
269
To describe a translation you need to give...
A vector
270
To describe a reflection you need to give...
The equation of the mirror line
271
To describe a rotation you need to give...
The direction, the angle of turn, the centre of rotation and the word 'rotation'
272
What does it mean to say you have performed a transformation on a shape but it is still congruent to the original shape?
The shape is exactly the same shape and size - lengths of sides and angles have not changed
273
To describe an enlargement you need to give...
The scale factor and the centre of enlargement
274
What does the scale factor of an enlargement tell you?
How much each length is multiplied by
275
What is the equation for calculating the scale factor of an enlargement?
Scale factor = enlarged length / original length
276
How can you find the centre of enlargement?
Lines drawn through corresponding points on object (A) and object (B) meet at the centre of enlargement
277
If the scale factor of shape A is between 0 and 1, how will shape B be enlarged?
Shape B will become smaller than shape A
278
What happens when you enlarge Shape A by a negative scale factor to create Shape B?
Shape B will be on the other side of the centre of enlargement and will be upside down
279
Where are bearings always measured from?
Bearings are always measured clockwise from North
280
How many figures must a bearing have?
3 - if the answer is less than 100 add a place value 0 (e.g. 048 degrees)
281
What is the acronym for remembering the points on a compass?
Never ↑
Eat →
Shredded ↓
Wheat ←
282
What is the shortest distance from a point to a line?
A perpendicular line
283
Explain the 3 methods you may be asked to use to construct a perpendicular bisector:
METHOD 1 - CONSTRUCTING A PEREPENDICULAR BISECTOR AT LINE AB PASSING THROUGH POINT P: Use a compass to mark 2 points on the line equal distance from P. Keep the compasses the same and draw 2 arcs with their centres at these points
METHOD 2 - CONSTRUCTING A PERPENDICULAR BISECTOR ST LINE AB PASSING THROUGH POINT P: Use a compass to mark 2 points an equal distance from P. Then widen the compass and draw arcs with their centres at these 2 points
METHOD 3 - CONSTRUCT THE PERPENDICULAR BISECTOR OF LINE AB: Use compass to draw intersecting arcs with centres at A and B
284
Explain how to construct a triangle with sides of specified lengths (no other information provided):
Draw and label one side (to scale) with a ruler. Then use your compasses to find the other vertex
285
Explain how to construct the bisector of a given angle (say PQR)
Mark points on each arm equal distance from Q. Then use arcs to find a third point an equal distance from these 2 points
286
Explain how to construct a 45 degree angle at a straight line (say P is the end of the line to construct the angle from):
Construct the perpendicular bisector of the line. Mark the midpoint as M. Then set your compasses to length PM. Draw an arc on your bisector and join this point to P with a ruler
287
Explain how to construct a 60 degree angle at a straight line (say P is the end of the line to construct the angle from):
Construct an equilateral triangle (all sides the same length) including point P - each angle within an equilateral triangle is 60 degrees
288
What is the plural of locus?
Loci
289
What is a locus?
A set of points which satisfy a condition. You can construct loci using a ruler and compasses. A set of points can lie inside a region rather than on a line or curve
290
What makes 2 triangles congruent?
2 triangles are congruent if they have exactly the same shape and size
291
To prove 2 triangles are congruent, you have to show that at least 1 of 4 conditions is true. Name these 4 conditions:
1. SSS (3 sides are equal)
2. AAS (2 angles and a corresponding side are equal)
3. SAS (2 sides and the included angle are equal)
4. RHS (right angle, hypotenuse and a side are equal)
292
What makes 2 shapes similar?
Shapes are similar if 1 shape is an enlargement of the other
293
For triangles to be similar, they must satisfy 3 conditions. Name these 3 conditions:
1. All 3 pairs of angles are equal
2. All 3 pairs of sides are in the same ratio
3. 2 sides are in the same ratio and the included angle is equal
294
What defines the relationship between similar shapes?
A scale factor
294
A and B are similar shapes. B is an enlargement of A by scale factor k. What is the equation for calculating the enlarged surface area of a shape when enlarged by linear scale factor k?
Enlarged surface area = k^2 X original surface area
295
A and B are similar shapes. B is an enlargement of A by scale factor k. What is the equation for calculating the enlarged volume of a shape when enlarged by linear scale factor k?
Enlarged volume = k^3 X original surface area
296
A and B are similar shapes. B is an enlargement of A by scale factor k. What is the equation for calculating the enlarged mass of a shape when enlarged by linear scale factor k?
Enlarged mass = k^3 X original mass
297
What information do you need to know about a triangle to apply the sine rule?
You need to know a side length and the OPPOSITE angle
298
What is the sine rule in the form used to calculate the length of an unknown side?
a / sinA = b / sinB = c / sinC
299
What is the sine rule in the form used to calculate the size of an unknown angle?
sinA / a = sinB / b = sinC / c
300
What information do you need to know about a triangle to apply the cosine rule?
You use the cosine rule when you know either:
1. 2 sides and the included angle (SAS)
OR
2. 3 sides and you want to work out an angle (SSS)
301
What is the cosine rule in the form used to calculate the length of an unknown side? WHERE WE KNOW 2 SIDES AND THE INCLUDED ANGLE
a^2 = b^2 + c^2 - 2bccosA
302
What is the cosine rule in the form used to calculate the length of an unknown angle? WHERE WE KNOW 3 SIDES
cosA = (b^2 + c^2 - a^2) / 2bc
303
When you know the lengths of 2 sides and the angle between them, the area of any triangle can be found using which formula?
1/2absinC
304
What does a chord do?
A chord divides a circle into 2 segments
305
What is the equation for the area of the minor segment?
Area of minor segment = area of whole sector - area of triangle
306
What is the formula for Pythagoras in 3D?
a^2 + b^2 + c^2 = d^2 where d is a diagonal going from one base corner to the opposite top corner
307
In a circle, the angle between a radius and a tangent is...
90 degrees
308
True or False: 2 tangents which meet at a point outside the circle are the same length
True
309
In a circle, a triangle which has 1 vertex at the centre of a circle and 2 vertices on the circumference is an...
Isosceles triangle - each short side of the triangle is a radius so they are the same length
310
How many degrees is the angle in a semicircle?
90 degrees
311
Name the 6 key circle theorems:
1. Opposite angles of a cyclic quadrilateral add up to 180 degrees
2. The perpendicular from a chord to the centre of the circle bisects the chord
3. The angle in a semicircle is 90 degrees
4. Angles in the same segment are equal
5. The angle at the centre of the circle is twice the angle on the circumference
6. The angle between a tangent and a chord is equal to the angle in the alternate segment
312
Angles in the same segment are .....
equal
313
Opposite angles of a cyclic quadrilateral add up to ...........
180 degrees
314
The angle at the centre of a circle is ...... the angle on the circumference
twice
315
The perpendicular from a chord to the centre of the circle ...... the chord
bisects
316
What is alternate segment theorem?
The angle between a tangent and a chord is equal to the angle in the alternate segment (this is a circle theorem)
317
What does a vector have?
A vector has magnitude (size) and a direction
318
Can you multiply a vector by a number?
Yes. The new vector has a different length but the same direction. EXAMPLE: LINE A IS A VECTOR:
line a is 15cm long and points North
line 3a will be 45cm long and still points North
line 1/2a will be 7.5cm long and will still point North
319
True or False: If b is a vector then -b is a vector with the same length but the opposite direction
True
320
Explain how to multiply a column vector:
If you multiply a column vector you have to multiply both parts.
EXAMPLE:
2 X (p) = (2p)
(q) (2q)
321
Explain how to add 2 column vectors:
If you add 2 column vectors you add the top numbers and you add the bottom numbers.
EXAMPLE:
(c) + (e) = (c + e)
(d) + (f) = (d + f)
322
How can you tell if 2 vectors are parallel?
If 1 vector can be written as a multiple of the other then the vectors are parallel
323
What does it mean to say 3 points are collinear?
If 3 points lie on the same straight line then they are collinear
324
What are the 3 types of average you may be asked to calculate?
Mean, median and mode
325
What is the mean?
The mean is the average of a set of values
326
What is the median?
The median is the middle value of a set of values
327
What is the mode?
The mode is the most frequent value of a set of values
328
What is the range?
The range is the difference between the highest value and the lowest value in a set of values
329
How do you find the range of a set of values?
Highest value - lowest value
330
Explain how to find the mean of a set of values:
1. Add up all of the values
2. Divide by the total number of values
3. Do not round your answer
331
Explain how to find the median of a set of values:
1. Write the values in order of size - smallest first
2. Count the number of values
3. If the number of values is odd - the middle number is the median
4. If the number of values is even - the median is half way between the 2 middle values
332
Explain how to find the mode of a set of values:
1. Look for the most common value
2. If there is 1 value with the highest frequency - this is the mode
3. If there is more than 1 value with the highest frequency - these values are all the modes
4. If all values have the same frequency - there is no mode
333
What is the equation for finding the sum of values using their mean?
Sum of values = mean X number of values
334
Explain how to calculate mean from a frequency table:
1. The class (ie. number of pets owned) will be represented by x
2. The frequency will be represented by f
3. Add a new column for f X x
4. Calculate the total of column f X x
5. Calculate the total of column f
6. Mean = (total of column f X x) / (total of column f)
335
Explain how to calculate an estimate for mean from a frequency table:
1. Add extra columns to the table for midpoint of class (x) and midpoint X frequency (f X x)
2. Estimate of mean = (total of f X x column) / (total of frequency column)
336
Explain how to work out the interquartile range:
1. Count thee total number of values, n
2. Check that the data is arranged in order of size
3. Find the (n + 1) / 4 th data value - THIS IS THE LQ
4. Find the 3(n + 1) / 4 th data value - THIS IS THE UQ
5. UQ - LQ = IQ range
337
What is the equation for interquartile range?
IQR = UQ - LQ
338
What is a time series graph?
A line graph that shows how a variable changes over a period of time
339
What does it mean to say a scatter graph has correlation?
If the points are almost on a straight line then the scatter graph shows correlation. The better the straight line, the stronger the correlation
340
What is an outlier when analysing graphs (the anomaly)?
A value that does not fit the pattern of the data
341
What is a line of best fit?
A straight line drawn as close to as many of the points on a graph as possible which does NOT need to go through the origin
342
If you are predicting a value that is within the range of the data your prediction will be more accurate. What is this called?
Interpolation
343
If you are predicting a value that falls outside the range of the data your prediction will be less accurate. What is this called?
Extrapolation
344
What is sampling?
Sampling is when you survey a smaller group (a sample) of a larger population. You can use the data from the sample to make predictions about the wider population
345
Name 3 advantages of sampling:
1. It is cheaper to survey a sample than a whole population
2. It is quicker to collect data from a sample
3. It is easier to analyse data from a sample and calculate statistics
346
What happens in a random sample?
Every member of the population has an equal chance of being included in the sample. 2 ways of selecting a random sample are:
1. Put the names of every member of the population in a hat and select a sample at random
2. Assign a number to every member of the population and choose random numbers using a computer program or calculator
347
What happens in a stratified sample?
A stratified sample is one in which the population is split into groups. The number of members selected from each group for the sample is proportional to the size of that group. EXAMPLE:
If there are twice as many boys as girls in a population, you will also need twice as many boys as girls in a stratified sample
348
What is the equation for finding the sampling fraction for a stratified sample?
Sampling fraction = sample size / population size
You then multiply the sampling fraction by the size of each group to work out how many members to select from that group
349
What method can you use to estimate the size of a large animal population?
The Peterson capture-recapture method
350
Explain how to carry out the capture-recapture method to estimate the number of fish in a lake:
1. Catch a sample of fish from the lake and mark them
2. Return the marked fish to the lake
3. Later, catch a second sample of fish
4. Count how many of this sample are marked
5. Use the formula to estimate the population size, N
N = Mn / m where:
M = no. of fish marked then released
n = size of recapture sample
m = no. of marked fish in recapture sample
351
What is the formula used for estimating population size using capture-recapture?
N = Mn / m
where:
N = population size
M = no. of fish marked and then released
n = size of recapture sample
m = no. of marked fish in recapture sample
352
Explain how to draw a cumulative frequency graph:
1. Plot 0 at the beginning of the first class interval
2. Plot the cumulative frequency of each value at the UPPER end of its class interval
3. Join your points with a smooth curve
353
Explain how to read LQ, UQ and median from a cumulative frequency graph:
1. Look at the top of the graph and establish the total
2. Divide the total by 4, draw a line along to the graph from the y axis and down to the x axis - the number at the x axis is the LQ
3. Divide the total by 4, multiply it by 3, draw a line along to the graph from the y axis and down to the x axis - the number at the x axis is the UQ
4. Divide the total by 2, draw a line along to the graph from the y axis and down to the x axis - the number at the x axis is the median
354
What are histograms good for?
Representing grouped data with different class widths
355
What is the equation for frequency density?
Frequency density = frequency / class width
356
On a histogram, the vertical axis is labelled...
Frequency density
357
What does the area of each bar on a histogram represent?
Frequency
358
True or False: If you draw a histogram on the same graph as a frequency polygon, the frequency polygon joins together the midpoints of the tops of the bars
True - frequency polygon points are plotted at the midpoint of the class width on the x axis and at the frequency on the y axis
359
What is the equation for estimating the mean for data given in a frequency polygon?
Mean = (sum of (midpoint X frequency)) / Total frequency
360
For equally likely outcomes, the probability (P) that something will happen is represented by the equation:
Probability = number of successful outcomes / total number of possible outcomes
361
If you know the probability that an event WILL happen, you can calculate the probability that it won't happen using the equation:
P(Event doesn't happen) = 1 - P(Event happens)
362
What does it mean to say events are mutually exclusive?
They cannot both happen at the same time
363
What is the equation for finding the probability of mutually exclusive events (events which cannot both happen at the same time)?
P(A or B) = P(A) + P(B)
364
What does it mean to say events are independent?
Events are independent if the outcome of one does not affect the outcome of the other
365
What is the equation for finding the probability of independent events (events where the outcome of one does not affect the outcome of the other)?
P(A and B) = P(A) X P(B)
366
What does a sample space diagram show you?
All the possible outcomes of an event (shown in a table similar to a Punnett square)
367
What is the formula for estimating probability from a frequency table?
Probability = frequency of outcome / total frequency
368
Explain how to use a tree diagram to solve questions involving conditional probability:
At each branch the probabilities on either side add up to 1. The outcome of the first event can affect the probability of the second. You write the probability for each event on the corresponding branch. Once you have finished, multiply along the branches to find the probability of each outcome
369
Why do you need to pay attention to the words 'replace' or 'put back' in probability questions?
WITH replacement: probabilities stay the same
WITHOUT replacement: first probability stays the same while the others change
370
What is the symbol for the universal set on a Venn diagram?
ξ
371
What does A ∩ B mean on a Venn diagram?
∩ means 'and' or 'intersection'. A ∩ B would only include all the values in the intersection between A and B
372
What does A ∪ B mean on a Venn diagram?
∪ means 'or' or 'union'. A ∪ B would include all the values in A, B and the intersection between A and B
373
What does A' mean on a Venn diagram?
A' is the complement of A - A' means NOT A. It includes everything in and outside of the Venn diagram except for A and any sections overlapping with A
374
How do you express the probability event X occurs given that event Y has already occurred?
P(X|Y)
375
You can solve some conditional probability problems using a Venn diagram. If an event has already occurred, then the sample space for the other events is ..........
restricted
376
The x-intercept is the point where y = ....
The x-intercept is the point where y = 0
377
The y-intercept is the point where x = ....
The y-intercept is the point where x = 0
