Key methods and terminology Flashcards

Question

What are two-stage fractional powers?

Answer 1

With fractional powers where the numerator is not 1, split the fraction into a root and a power and do them in that order: root first, then power for example: 64^(5/6) = sixth root of 64 to the power of 5 = 2^5 = 32

Answer 2

- Difference Of Two Squares

Answer 3

1. *root* a X *root* b = *root* a X b 2. *root* a / *root* b = *root* a / b 3. *root* a + *root* b = **DO NOTHING! IT ISN’T *root* a + b** 4. (a + *root* b)^2 = (a + *root* b) X (a + *root* b) 5. (a + *root* b) (a - *root* b) = a^2 - b 6. ***rationalise the denominator***

Answer 4

It is when you get rid of the root on the bottom of the fraction: a / *root* b = a / *root* b X *root*b / *root*b = a X *root*b / b

Answer 5

1 - get rid of any fractions 2 - multiply out any brackets 3 - collect all the letter terms on one side, and the number terms on the other 4 - combine all the like terms 5 - divide both sides by the number attached to the letter (you may have to factorise to do this as you may need to divide by the brackets) 6 - if you had the letter to the power of a number, root by that number to end up with a positive/negative result

Answer 6

- rearrange into the format ax^2 + bx + c = 0 - find 2 numbers that multiply to ac and add/subtract to b - rewrite the equation where bx is replaced by the 2 numbers that you found in step 2 (don’t forget x) - place these 4 values into a grid and factorise - factorise to find what each bracket is - then solve the equation by setting each bracket equal to 0 to see the value of x

Answer 7

- rearrange the quadratic into the format ax^2 + bx + c = 0 - write out the initial bracket: (x + b/2)^2 - multiply out the brackets and compare to the original to find out what you need to add or subtract to complete the square - add or subtract the adjusting number to make it match the original **if a does not equal 1, you need to take out the factor of a first, and then multiply it back in later**

Answer 8

- find the common difference (this tells you what to multiply n by) - work out what to add or subtract to get to the original sequence using n (for example: 3n = 3,6,9) - write as one equation

Answer 9

- find the difference between each term - find the second difference between the first differences - divide this value by 2 to find what to give the coefficient of n^2 - subtract the n^2 term from the sequence to find the linear sequence - find the rule for the nth term of the linear sequence and add this to the n^2 term

Answer 10

arithmetic - where you add or subtract the same amount each time geometric - where you multiply or divide by the same amount each time

Answer 11

where you add together the two previous terms to find the next term

Answer 12

YOU MUST FLIP THE INEQUALITY SIGN

Answer 13

Use an open circle for greater than or less than, and a coloured in circle for greater than/equal to and less than/equal to

Answer 14

if x^2 > a^2 then x > a OR x > -a if x^2 < a^2 then -a < x < a

Answer 15

- convert each inequality to an equation (put an equals in place of inequality sign) - draw the graph for each equation (if it is > or < then draw a dotted line, but if it is ≥ or ≤ draw a solid line) - work out which side of the line you want (substitute a point into the inequality to see if it is on the correct side of the line) - shade the region this gives you

Answer 16

- make the equations have one coefficient of the same letter equal (you may need to multiply or divide on equation) - subtract the equations (if both are positive or negative) or add the equations (if one is negative and one is positive) to get one letter and coefficient on its own) - then solve the equation normally to find what one of the values are - substitute this back into one of the equations to find the other unknown value

Answer 17

- rearrange the quadratic equation so that it tells you what one of the unknowns are: y = 3x^2 - 3 - substitute this into the other equation to get a new equation - rearrange this to get a quadratic equation and solve - substitute the first value back into one of the equations (pick the easiest) - then try with the second value in the same equation - you should have two pairs of answers, and write these clearly, using and to link the pairs

Answer 18

- any even number can be written as 2n - any odd number can be written as 2n + 1 - consecutive numbers can be written as: n, n + 1, n + 2… - the sum, difference and product of integers is always an integer

Answer 19

Use an example that the statement does not work for (disproof by counter example)

Answer 20

Takes an input, processes it and outputs a value

Answer 21

Put the numbers into the function and find the result

Answer 22

- it is a composite function - composite functions are written as fg(x) which means “do g first, then do f using the result of g” - to find a composite function, rewritten fg(x) as f(g(x) then replace g(x) with the expression it represents: Given f(x) = 3 x + 2 and g(x) = x + 5, find fg(x) = f(g(x)) = f(x+5) = 3(x+5) + 2 = 3x + 15 + 2 = 3x + 17

Answer 23

- an inverse function reverse f(x) - write out the equation x =f(y) - rearrange to make y the subject - finally, replace y with f^-1(x)

Answer 24

y = mx + c m = gradient c = y-intercept

Answer 25

change in y / change in x

Answer 26

Add the x-coordinates and divide by 2 Add the y-coordinates and divide by 2

Answer 27

- find the difference between the coordinates of two points (A and B) - Now look at the ratio - the ratio tells you the third point (C) is a fraction of the way from A to B - find the fraction of each distance (difference between points) - now add these to the coordinates of the first point (A) to find C

Answer 28

Their gradients - in the equations y=mx+c, the m will be the same

Answer 29

Their gradients are the negative reciprocal of each other

Answer 30

The equation for a circle with centre (0,0) and radius (r) is: x^2 + y^2 = r^2 If you know the radius, substitute it in for r

Answer 31

= 1/x graphs y = A/x or a y = A These are all the same basic shape where the two halves of the graphs do not touch and do. it exist for x = 0. They are all symmetrical about the lines y = x and y = -x

Answer 32

y = k^x or y = k^-x - They are exponential graphs, they are always about the x axis and go through the point (0,1) - If k > 1, and the power is positive, the graph curves upwards - if k is between 0 and 1, or the power is negative, the graph is flipped horizontally - the graphs will start to increase very little and then increase rapidly

Answer 33

y = ax^3 + bx^2 + cx + d (b,c,d can be zero) - cubic graphs have a wiggle in the middle (this can be flat or more pronounced) - in x^3 it goes up from the bottom left - in -x^3 it goes down from the top left

Answer 34

y = SIN x y = COS x sin graphs - waves (bounce between +1 and -1) cos graphs - buckets (between +1 and -1) sin graphs have a peak and a trough cos graphs start at the top, dip and come back up again

Answer 35

Plot both graphs and see where the graphs cross each other

Answer 36

y = f(x) + a

Answer 37

y = f(x - a)

Answer 38

y = -f(x) y = f(-x)

Answer 39

- at any point, gradient = speed - the steeper the graph, the faster it is going - flat sections are where it has stopped - if the gradient is negative, it is coming back

Answer 40

- at any point, gradient = acceleration - negative slope is deceleration (slowing down) - flat sections are steady velocity - area under graph = distance travelled

Answer 41

- Draw a straight line between the two points (these are given in the question) - find the gradient of the straight line

Answer 42

- Draw a tangent that touches the curve at the given point - the gradient of the tangent is the same as the rate at the chosen point

Answer 43

1) write the ratios as equations 2) turn the ratios into fractions 3) solve the equations simultaneously **example:** The ratio of male to female pupils on a trip is 5:3, four male teachers and nine female tea chefs are going. the ratio of males to females is 4:3. How many female pupils are going? *let m be the number of male pupils and f be the number of female pupils* m:f = 5:3 (m+4);(f+9) = 4:3 m/f = 5/3 and m+4/f+9 = 4/3 3 m = 5f and 3m + 12 = 4f + 36 f = 24 24 female pupils are going on the trip

Answer 44

y is directly proportional to x y = kx

Answer 45

y is inversely proportional to x y = k/x

Answer 46

direct proportion: divide for one, then times for all inverse proportion: times for one, then divide for all

Answer 47

of means times (multiply)

Answer 48

increase: convert the percentage to a decimal and add to 1 decrease: convert the percentage to a decimal and subtract from 1

Answer 49

divide x by y, and multiply 100

Answer 50

percentage change = (new - old) / old X 100

Answer 51

1) write the amount as a percentage of the original 2) divide to find 1% of the original value 3) multiply by 100 to give the original value

Answer 52

1cm =0.394 in 10mm 1m = 100cm 1km = 1000m 1kg = 1000g 1 tonne = 1000kg 1 litre = 1000ml 1 litre = 1000cm^3 1cm^3 =1 ml

Answer 53

1 yard = 3 feet 1 gallon = 8 pints 1 stones = 14 pounds 1 pound = 16 ounces 1 foot = 12 inches

Answer 54

1kg = 2.2 pounds 1 foot = 30cm 1 gallon = 4.5 litres 1 mile =1.6km

Answer 55

100cm X 100cm = 10000cm^2

Answer 56

speed = distance / time

Answer 57

density = mass / volume

Answer 58

pressure = force / area

Answer 59

- angles in a triangle add up to 180 degrees - angles on a straight line add up to 180 degrees - angles in a quadrilateral add up to 360 degrees - angles round a point add up to 360 degrees - isosceles triangles have 2 sides the same and 2 angles the same

Answer 60

vertically opposite angles are equal

Answer 61

- it is angles found in a Z shape - alternate angles are the same

Answer 62

- they are angles within a C or U shape - co interior (allied) angles add up to 180 degrees

Answer 63

- these are angles found in an F shape - corresponding angles are the same

Answer 64

sum of exterior angles = 360 degrees

Answer 65

sum of interior angles = (n - 2) X 180

Answer 66

interior angle = 180 degrees - exterior angle

Answer 67

exterior angle = 360 / n

Answer 68

A tangent is a line that just touches a single point on the circumference of a circle. A tangent always makes an angle of exactly 90 degrees with the radius it meets at this point

Answer 69

A tangent is a line that just touches a single point on the circumference of a circle. A tangent always makes an angle of exactly 90 degrees with the radius it meets at this point

Answer 70

unlike other isosceles triangles they don’t have the marks on the sides to say they are the same, the fact that they are both radii is enough to make it an isosceles triangle within the circle

Answer 71

A chord is any line drawn across a circle. No matter where you draw a chord, the line that cuts it exactly in half (at 90 degrees) will go through the centre of the circle

Answer 72

The angle subtended at the centre of a circle is exactly double the angle subtended at the circumference of the circle from the same two points (two ends of the same chord) *angle subtended at is the angle made at*

Answer 73

A triangle drawn from the two ends of a diameter will always make an angle of 90 degrees where it hits the circumference of the circle, no matter where it hits.

Answer 74

All triangles drawn from a chord will have the same angle where they touch the circumference. Also, the two angles on opposite sides of the chord add up to 180 degrees

Answer 75

A cyclical quadrilateral is a 4 sided shape with every corner touching the circle. Both pairs of opposite angles add up to 180 degrees.

Answer 76

Two tangents drawn from an outside point are always equal in length, creating two congruent right angled triangles

Answer 77

The angle between a tangent and a chord is always equal to the angle in the opposite segment (the angle made at the circumference by two lines drawn from the ends of the chord)

Answer 78

1) SSS - three sides are the same 2) AAS - two angles and a corresponding side match up 3) SAS - two sides and the angle between them match up 4) RHS - a right angle, the hypotenuse and one other side all match up

Answer 79

1) all the angles match up 2) all three sides are proportional 3) any two sides are proportional and the angle between them is the same

Answer 80

congruence - same shape, same size and same angles similarity - same shape, different size

Answer 81

- specify it is a translation - give the vector that it moves by

Answer 82

- specify it is a rotation - give the angle of rotation - the direction of rotation - the centre of rotation

Answer 83

- specify it is a reflection - give the equation of the mirror line

Answer 84

- specify it is an enlargement - give the scale factor - the centre of enlargement

Answer 85

scale factor = new length / old length

Answer 86

- if the scale factor is bigger than 1, the shape gets bigger - if the scale factor is smaller than 1, it gets smaller - if the scale factor is negative, the shape will be on the other side of the enlargement centre - the scale factor also tells you the relative distance of old points and new points from the centre of enlargement

Answer 87

area of sector = x /360 X area of full circle

Answer 88

length of arc = (x / 360) X circumference of full circle

Answer 89

1) find the area of the sector 2) find the area of the triangle, and subtract it from the sectors’ area

Answer 90

Surface area of sphere = 4πr^2

Answer 91

surface area of cone = πrl + πr^2 where r is the radius, and l is the slanted height of the cone

Answer 92

area of cylinder = 2πrh + 2πr^2 where r is the radius, and h is the height of the cylinder

Answer 93

volume of sphere = 4/3 π r^3

Answer 94

volume of pyramid = 1/3 X base area X vertical height

Answer 95

volume of cone = 1/3 X πr^2 X vertical height

Answer 96

= volume of original cone - volume of removed cone

Answer 97

- “a fixed distance from a given point” - this is a circle - “a fixed distance from a given line” - tic tac shape - “equidistant from two given lines” - “equidistant from two given points”

Answer 98

1) FROM - put your pencil on the diagram at the point you are going from 2) NORTH LINE - at the point you are going from, draw a north line (it may already be there) 3) CLOCKWISE - draw in the angle clockwise from the nor5h line to the line joining the two points

Answer 99

Use pythagoras’ theorem: - sketch the right angled triangle - find the lengths of shorter sides by subtracting coordinates - use Pythagoras’ to find the length of the hypotenuse (this will be the distance)

Answer 100

sin = opposite / hypotenuse cos = adjacent / hypotenuse tan = opposite / adjacent

Answer 101

sin 0 = 0 sin 30 = 1/2 sin 60 = *root*3 / 2 sin 45 = 1 / *root*2 sin 90 = 1

Answer 102

cos 0 = 1 cos 30 = *root*3 / 2 cos 60 = 1 / 2 cos 45 = 1 / *root*2 cos 60 = 1/2 cos 90 = 0

Answer 103

tan 0 = 0 tan 30 = 1 / *root*3 tan 45 = 1 tan 60 = *root*3 tan 90 = **undefined**

Answer 104

label the angles a lowercase “A,B,C” and the opposite length a capital “a,b,c”

Answer 105

a / sinA = b / sinB = c / sinC

Answer 106

a^2 = b^2 + c^2 - 2 bc cosA

Answer 107

Area of triangle: 1/2 ab sinC - you can use this when you know two sides and the angle between them

Answer 108

1)you can have column vectors: (2) (5) - column vectors are how far something moves on the x above how far on the y axis. It is not written with a line between, and the one thing is written in one set of big brackets 2) it can be represented by a letter, if handwritten it is underlined “_a_”, or if typed, it is in bold “**a**” 3) it can also be written in as the two points and an arrow to show the direction the vector is moving in

Answer 109

- vectors have direction (the way it moves) and magnitude (the size)

Answer 110

multiplying a vector by a positive number changes the vectors’ size but not its direction. It scales the vector. If the number is negative, then the direction gets switched.

Answer 111

- You can describe movements between points by adding and subtracting known vectors - “_a_ + _b_” means “go along _a_ then _b_” - to add or subtract column vectors, add/subtract the top to the top and the bottom to the bottom.

Answer 112

- you can use vectors to show that points lie on a straight line - you need to show that the vectors along es h part of the line point in the same direction - they are scalar multiples of each other - this means you can write them as fractions of each other

Answer 113

Ratios are used in vectors to tell you the lengths of different sections of a straight line. If you know the vector along part of that line, you can use this to find other vectors along the line.

Answer 114

- they are always between 0 and 1 - a probability of zero means it will never happen, and one will mean it definitely will happen

Answer 115

probability = number of ways for something to happen / total number of possible outcomes

Answer 116

- It will always add to one - P(event happens) + P(event doesn’t happen) = 1

Answer 117

The number of ways to carry out a combination of activities equals the number of ways to carry out activity multiplied together.

Answer 118

relative frequency = frequency / number of times you tried the experiment

Answer 119

expected frequency of a result = probability X number of trials

Answer 120

- two events are independent if one event happening does not affect the probability of the other happening - if one affect does affect the probability of the other, the events are dependent

Answer 121

If two events (A and B) are independent, then: P(A and B) = P(A) X P(B) It is the probability of both events happening

Answer 122

for two events (A and B): P(A or B) = P(A) + P(B) - P(A and B) it is the probability of at least one of the events happening

Answer 123

It is two events that cannot happen at the same time P(A or B) = P(A) + P(B)

Answer 124

- on any set of branches which meet at a point, the probabilities add up to 1 - multiply along the branches to get the end probabilities - the end probabilities should add to 1 - to answer any question, add up the relevant end probabilities

Answer 125

- the conditional probability of A given B is the probability of event A happening given that event B happens - a lot of questions that say “without replacement” is most likely to be a conditional probability question - if events A and B are independent then P(A given B) = P(A) and P(B given A) = P(B)

Answer 126

P(A and B) = P(A) X P(B given A)

Answer 127

- they use sets: these are collections of things “called elements” - sets can be written in different ways, but always be in a pair of curly brackets: {} - n(A) means the number of elements in set A - on a venn diagram, each set is represented by a circle containing the elements of the set or the number of elements in the set - this ξ means universal set

Answer 128

- P(A U B) means everything in set A or B - P(A ∩ B) means everything in set A and B - P(A’) means everything not in set A

Answer 129

To find out about a population or amount

Answer 130

- this means every member of the population has an equal chance of being in it

Answer 131

you increase the reliability

Answer 132

1) assign a number to every number of the population 2) create a list of random numbers - by using a computer, calculator or picking out of a bag 3) match the random numbers to members of the population

Answer 133

consider: - when, where, and how the sample is taken - how many members are included 1) if certain groups are excluded, it is not random. This can lead to BIAS from things like age, gender, different interests

Answer 134

Quantitative data - measures quantities using numbers. For example heights of people Qualitative data - it is descriptive, it uses words rather than numbers - for example pet names

Answer 135

discrete data - it is discrete if the numbers can only take exact values, like the number of customers in a shop in one day continuous data - if the number can take any value in a range, it is continuous data. Like heights and weights are continuous measurements

Answer 136

mode - most common median - middle value (when data is ordered in size) mean - total of items / number of items range - difference between the lowest and highest value

Answer 137

- add a 3rd column and enter the mid-interval. slur for each class - add a 4th column to show ‘frequency X mid-interval values’ for each class - You need to use this to find the modal class and the class containing the median, not exact values, The range can also be estimated too (using the class boundaries)

Answer 138

Find the difference between the highest and lowest class boundsriesv

Answer 139

upper quartile - 75% of the way into the data lower quartile - 25% of the way into the data median - 50% of the way into the data interquartile range - difference between the upper and lower quartile (and contains the middle 50% of values)

Answer 140

- draw on the lower and upper quartile values and draw the box - draw the median line (this should be within the box) - mark on the minimum and maximum points and draw on horizontal lines to connect them to the box

Answer 141

It just means adding it up as you go along - the total frequency so far

Answer 142

- add a “cumulative frequency” column to the table - and fill it with the running total of the frequency column - plot points on the graph using the highest value of each class and the cumulative frequency - join the points with a smooth. urge or straight lines *make sure the cumulative frequency goes up the y-axis, and the x-axis is the collection name (like height)*

Answer 143

median - go halfway up the side, across to the curve, then down and read the bottom scale lower/upper quartiles - go 1/4 and 3/4 up the side, across to the curve, then down and read off the bottom scale interquartile range - the distance between the lower and upper quartiles

Answer 144

frequency density = frequency / class width

Answer 145

frequency = frequency density X class width frequency = area of bar

Answer 146

strong correlation - when the points make a fairly straight line weak correlation - when the points don’t line up properly, but you can still draw a line of best fit through them positive correlation - when the points move from the bottom left upwards negative correlation - when the points move from the top left downwards no correlation - when the points do not have any pattern (strong or weak, positive or negative)

Answer 147

- draw a line of best fit (crossing through as many points as possible), and continue until the end of the graph - use the extended line to make an estimate for the value you have been asked about

Answer 148

- with time series, a basic pattern often repeats itself - seasonality - the time taken for the pattern to repeat itself (measured from peak to peak, or trough to trough) is called a period - you can also look at the overall trend - look at the peaks and troughs

Answer 149

-proportion - the total of everything = 360 degrees

Answer 150

- grouped data - the frequency of each class is plotted against the mid-interval value and the points are joined with straight lines

Answer 151

Spreading the data

Answer 152

- You can read the median and work out the IQR and range. Remember to say what these values mean in context of the data - A larger spread means that the values are less consistent, and larger variation

Answer 153

- analyse the shape - compare the centres - examine the spread - consider overlapping sreas