Maths Revision Notes 2011

8/3/2019 Maths Revision Notes 2011

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MATHS REVISION NOTES 2011:

PROBABILITY:

-Probability: is to do with the likelihood of an event occurring.-Relative Frequency: when an experiment is carried out with many trials, the relative frequencyshould tend toward a constant value [long term relative frequency]

-Random Number Generator [Die]: Ran# x 6 + 1

Theoretical Probability:-P(event) = successful outcomes

Possible outcomes

-Note: We must consider whether the outcomes are equally likely.

-Example: a deck of cards [52] are shuffled and a card is selected at random.-Q. What is:

a) P(spade)= 13/52 or

b) P(Black ace)= 2/52 or 1/26

c) P(King, Queen or Jack)= 52-12

= 40

= 40/52 or 10/13

-If I picked ten cards at random, how many spades would I expect?

x 10= 2.5

Using Diagrams to list outcomes:-A good way to see all possible outcomes.

Example: Tables or a Probability Tree [refer to exercise book for examples]

The Multiplicative Rule [Not equally likely outcomes]-If events are independent [one doesnt affect the other] we can use the multiplicative rule [multiplythe possibilities]

-We can use this alongside a tree diagram:

A car is driving along Highway 2. It goes through three traffic lights. The probability that a light is

green is 0.8, 0.7 at the second and 0.9 at the third. What is the probability that the car has to stop at

two lights?


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Outcomes: Probability:1. GGG 1. 0.504

2. GGR 2. 0.056

3. GRG 3. 0.216

4. GRR 4. 0.024

5. RGG 5. 0.126

6. RGR 6. 0.014

7. RRG 7. 0.054

8. RRR 8. 0.006

P(green AND green AND green)= 0.504P(stop at 2 lights)= P(exactly 2 red)

P(exactly two red)= P(GRR) or P(RGR) or P(RRG)

*or= addition

0.024+0.014+0.054= 0.092

P(exactly two red)= 0.092

*Generally [in most cases]

AND= x [independent events]

OR = + [mutually exclusive events]


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MEASUREMENT: Metric conversions:

Divide by 1000 Multiply by 1000

Multiply by 1000 Divide by 1000

1m= 100cm 1cm= 10mm 1tonne= 1000kg

Area:-Units: mm2, cm2, m2, km2

-Conversions:

1m2= 10,000cm2

1cm2= 100mm2

1ha= 100m2 x 100m2 OR 10,000m2

Areas of simple polygons:-Rectangles: a= b x h

height

base

-Triangle: a= b x h

= b x h 2

Height

m= metre

g= gramL= litre

milli

(m)

m= metre

g= gram

L= litre

Kilo

(k)


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-Parallelogram: a= b x h

Height

Base

-Trapezium: A (a+b2) x h

a

h

b

Composite Areas:-A complex area which consists of the sum/difference of the areas of simple shapes.

Example: [refer to exercise book]

Circles:Diameter radius

-Circumference: the whole outer part of a circle [perimeter]

-d= 2r

C = [pi]D

- = 3.141592654 [approx.] or 22/7- Pi is irrational; cant be expressed properly as a fraction.

Finding the circumference:C= d or 2r

Area of a circle:-


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Volume:-Units: mm3, cm3, m3

-Prism: a 3D shape which has a constant cross sectional area [CSA] or i.e, looks the same when its

sliced. Examples of prisms:

Cuboid

Cylinder

Triangular Prism

Trapezoid

Volume of a prism:-V= Cross Sectional Area x Depth.

Example:

V= (6+4) x 10

2

= 150m2

4m

3m

6m

10m

20m [diameter] V= x 102

= 314.15 x 10

= 3142m3

10m


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Liquid Volume:- 1cm3= 1mL of water, weighs 1g

- 1000cm3= 1L of water, weighs 1kg

Example: What does 1m3

of water weigh?

1m3= 1,000,000cm3

Weighs 1,000,000g = 1000kg

= 1 tonne

Surface Area:-Is the entire area of the outside of a 3D shape.

-Measured in square units.

-It is helpful to consider the net of the shape.

Example:

4cm

6cm 5cm

2 x 5 x 6= 60 Top & Base

2 x 5 x 4= 40 Sides2 x 4 x 6= 48 Front & Back

= 148cm2

[surface area]

5cm

8cm

2 x r2= 157.08 Top & Base8 x x 10= 251.33 Curved Surface

= 408.4cm2 [surface area]

- 2r2 + 2rh= 2r (r+h)


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NUMBER: Calculations and Estimations:

-Example: 36.3 + 21.85

3.009

Approx: 35+20 = 55 = approx. 183 3

Calculator: (36.3+21.85)3.009= 19.32 [2dp]

Significant Figures [sf]:-Used to round [approximate] numbers.

-Much for very large and very small numbers.

-take from left hand side [highest place value]

-if the number is decimal, the zero before the first NON-ZERO digit doesnt count.-if the zero is between the two non-zero digits, they count.

-fill the 0s where they affect place value.-then round normally, like decimal places.

Standard From [scientific notation]:-used to write really big and really small numbers.

*Big numbers:

Example: 37,000= 37 x 1000

= 3.7 x 10 [37] x 10 x 10 x 10 [1000]

= 3.7 x 104

5,000,000= 500 x 10,000

= 5 x 106

6,800,000,000= 6.8 x 109

-Numbers in standard form always have ONE non-zero digit to the left of the d.p.

*Small numbers:

-Work the opposite BUT the powers are negative.

Example: 0.0000358= 3.58 x 10-5

4.8 x 10-7

= 0.00000048

On Your Calculator:-EXP= x10^

-This is the standard form button. It means x10^ [times 10 to the power of]

Example: 3.72 x 106= 3.72 6

= 3720000

EXP


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Fractions:- 1 x 2 = 2 multiply the numerators and denominators.

5 3 15

-3 1 x 4 3 = 25 x 19 = 475 convert mixed numbers into improper fractions, then multiply8 4 8 4 32 numerators and denominators.

- 7 3 4 = 38 x 1 = 19 in the case, covert mixed number to improper. Then flip the right side

5 5 4 10 fraction. Then cross cancel.

- 1 1 3= 4 3= 16- 9 = 7 In this case, covert mixed to improper. Multiply two denominators to3 4 3 4 12 12 12 get final denominator. Then cross multiply for two numerators. Then

subtract two numerators for final numerator.

- 3 + 2 = 3 + 2 = 5 Find a common denominator.

8 4 8 8 8

Percentages of an amount:-Convert the percentage to a fraction or decimal.

Example: 15.5% of $318.95

0.155 x 318.95= $49.44

-Round to the nearest cent unless told otherwise.

-Unless you have an integer dollar value, round to 2dp.

An amount as a percentage of another:Example: In a test score you score 17/26. What % did you get right?

17 x 100= 65.4% [1dp]

26

Percentage Increase:-When we increase by x% it is the same as adding x% to the original [100%]

Example: increase $35 by 32%132% of $35= 1.32 x 35= $46.50

Percentage Decrease:-In this case we subtract x% from the original. We use the result as the percentage and then we

solve.

Example: Decrease $35 by 32%100%-32%=68%

68% of $35= 0.68 x 35

= $23.80


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Calculating a percentage change:Example: A house appreciates in value from $345,000 to $360,000

Nb: appreciate- increase in value

Depreciate- decrease in value

Q. What is the percentage value?

% change= actual change x 100

Original amount

360,000-345,000 x 100 = 4.347% increase

345,000

Working out an original amount after an increase/decrease:Example: after a 5.8% increase in house values, a house is valued at $375,000. What was its original

value before the increase?

Let x be the original value.

1.058 times x= $375,000

X= $375,0001.058

X= 354,422.344

= $354,000 [4sf]

GST:-tax at 15% on all goods and services.

-Usually GST is included in the price.

Prices excluding GST:Example: a builder quotes a price of $3650+GST to build a deck. What will you actually pay?

1.15 x 3650= $4197.50

Prices including GST:Example: tourists can claim back GST on souvenirs they have purchased. If they spend $780, how

much can they claim?

Pre GST value: 780 1.15= $678.26

GST: 780-678.26= $101.74

-Pre GST Post GST: multiply by 1.15

-Post GST Pre GST: divide by 1.15

-Pre GST: excluding GST

-Post GST: including GST


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Simple Interest:-Interest is only paid on the principal amount [original investment]

Example: $300 is invested for 3 years in an account paying 3.8% p.a [simple interest]

300 x 0.038 x 3 = Interest

= $34.20

-I= p x r x t = prt

100 100

P= principal

R= rate

T= time [years]

I= Interest

Compound Interest:-Interest is paid on the principal and continues to be paid on the interest.Example: $300 is invested for 3 years at 3.8% p.a

YEAR START END

1 $300 300x1.038= $311.40

2 $311.40 311.40x1.038= $323.23

3 $323.23 323.23x1.038= $335.52

I= $35.52

300 x 1.0383= $335.52

-Formula: P(1 + R )T

100

A= the final amount.

Ratio:-A ratio represents the proportion of one amount to another.

Example- 3:5, 1:50, 3:50,000

Ratios and Fractions:Example: The ratio of boys to girls in a school is 2:3

What fraction are boys?

2:3 2+3=5 students in total.

2= 40%

5

If there are 850 students, how many are girls?

3 of 850= 3 x 850

5 5

=510 girls.


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Proportional Rates:Example: if half of half a candle takes half an hour to burn, how long will a box of 12 candles take?

of a candle: of a candle which takes an hour to burn.

Whole Candle: 4 x = 2 hours to burn.

12 candles: 2 x 12= 24 hours to burn.

Exchange Rates:Example: an iPod in the USA costs $180USD. In the UK, the same iPod costs 125GBP. The rates are

as follows:

CURRENCY: 1NZD:

US Dollar 0.78

GB Pound 0.475

Which is cheapest?

180 USD= $230.77 NZD

0.78

125 GBP= $263.16 NZD

0.475

The US iPod is cheaper by $32.39 NZD

- NZ Currency Foreign Currency: multiply by the exchange rate.- Foreign Currency NZ Currency: divide by the exchange rate.

ALGEBRA:

Powers:Example: 35[power/index form]= 3 x 3 x 3 x 3 x 3 [expanded form]

35= 243

-The 3 is the BASE.

-The 5 is the INDEX or POWER

-The 243 is the RESULT or EVALUATED.

-Power, exponent, index are all the same.

Roots:-= means the number that we square to get 64

-3= means the number we CUBE to get 27

-n= the number that we multiply by itself n times to get a

To type in a calculator: 3 [press shift] x 27

-Anything to the power of 0 equals 1.


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Formulae and Expressions:-Expression: an algebraic statement involving letters.

Example: 3x+2, n2, 3n2-7

-Formula: like an expressions but contains a = sign. The letters are called VARIABLES.Example: f=maf is the subject of the formula.ma= the formula is written in terms of m & a

Rules of Indices:-Multiplying:

xa + xb= xa+b

If we multiply indices, they have to have the same base and we add the powers.

-Dividing:x

a x

b= x

a-b

When dividing, they have to have the same base. We subtract the powers.

-Power raised to a power:

(xa)b= xa x b

When raising a power, we multiply the two powers together.

Nb: x0=1

Anything raised to the power of zero will always be 1.

-Roots with Indices:When we take a square root of an index number, we half the power.

Expanding Brackets:-We multiply the number [letter] with its sign outside of the bracket by everything inside the

bracket.

Examples:

3(a+5)= 3 x a + 3 x 5

-3x2(5xy-2x4z)= -15x3y + 6x6z

3(x+2) + 5(x+4)

= 3x + 6 + 5x + 20

=26+8x

5(3x-4) 8(2x+2)= 15x- 20 16x 16= -x-36


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Factorising:-Opposite of expanding- put brackets in.

-Look for the highest common factor.

Examples:

3x + 6= 3(x+2)

8abc + c= c(8ab+1)

-6x + 2= -2(3x-1)

Solving Equations:Example: 3x 7= -2x + 5

3x + 2x= 5 + 7

5x = 12

X = 12/5

-When changing sides, change operations.-Put xs on the side with the biggest x.

Equations with brackets:-Expand the brackets, then simplify.

Examples: 3(x+3)= 18

3x + 9 = 18

3x = 18-9

3x = 9

X = 9/3X = 3

2(5x-2)-3= 8

10x 4 3= 810x = 8+4+3

10x = 15

X = 15/10

X = 3/2

Equations with Fractions:-If there is a fraction on both sides, cross multiply [the whole side must be over a commondenominator]

Examples: 3x = 8

5

3x= 8 x 5

3x= 40

X= 40/3


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2x = 5x

5 3

2x X 3= 5x X 5

6x = 25x

= 25x-6x

0 = 19x0/19 = x

X = 0

Multiplication and Division:-Turn a division problem into a multiplication problem, then simplify.

-Turn the reciprocal of the second fraction and solve.

Example: 3xy 9= 2 xy

Z 2z 3

Quadratic Expressions:-A quadratic expression is a polynomical [expression with powers of x] where the highest power is 2.

Example: 3x2+5, x2+3x+2, x2-8

Expanding Quadratic expressions:-Make sure that everything in the first brackets are multiplied by everything in the second brackets.

-Expand, then simplify.

Examples:

(x + 1) (x + 2)= x2

+ 2x + x + 2= x2 + 3x + 2

(x 5) (x + 2)= x2 + 2x 5x -10= x2 -3x-10

-Shortcut: if there is not a number in front of the xs in the brackets [i.e. their co-efficients are 1] wecan get the x co-efficient in the expansion by adding the numbers and the constant by multiplying.

Special Cases- Perfect Squares:-When we have a bracket squared.

-Also called, Difference of Two Squares

Example: (x + 5)2 = (x + 5) (x + 5)

= x2

+ 10x + 25

How to factorise quadratic equations:-look for a common factor first.

-if there is no x term- is it the difference of 2 squares ?

-if theres no constant term, is it straightforward common factor?


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How to Solve a Quadratic Equation:-When solving, we make the equation equal zero.

-Then factorise.

-Make each bracket equal to zero and solve.

Examples: (x + 2) (x - 3)= 0So either x + 2= 0 or x 3= 0

X = -2 OR x = +3

Solution: x= -2 or +3

(x + 5)2= 0

Solution: x= -5

X( x + 3) = 0

Solution: x= -3 or 0

(2x + 1) (3x 2)= 0X= -1/2 or 2/3

Factorising Quadratics with no common factor:2x2 + 7x + 5

(2x + 5) (x + 1)

Multiply = 7x

-The numbers with x should add up to the middle number in the equation.

GEOMETRIC REASONING:

Angle Properties:

1.

a + b + c= 180o

Adjacent Angles on a Straight Line Add to 180o[adj


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3.

a + b + c= 180o

Interior angles of a triangle add to 180o[int


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8.

y=z w=x

Alternate angles on parallel lines are equal [alt.


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3. Chord- is a line segment that joins two points on a curve. In geometry, a chord is often used to

describe a line segment joining two endpoints that lie on a circle.

4. Arc- part of the circumference of a circle.

5. Sector- part of the circle bounded by an arc and two radii.

6. Segment- part of the circle bound by an arc and chord.

7. Circumference- total distance of a circle [perimeter]

8. Tangent- a line which touches the circle.

Major= large part of circle.

Minor= small part of circle.

Examples: major arc, minor arc,

major segment, minor segment,

major sector, minor sector etc


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SEQUENCES, PATTERNS AND GRAPHS: Simultaneous Equations:

-A linear equation has only one solution.

Example: 3x-5= -23x = -2+5

3x = 3

X = 3/3

X = 1

-A pair of simultaneous equations has a pair of solutions [linear simultaneous equations]

-There are only two variables.

-The pair of solutions will be true for both equations.

Example: 2x+y= 5 x+y= 2

x=3

Y= -1

2(3)+(-1)=5

(3)+(-1)=2

-There are 3 methods to solve simultaneous equations.

-2 are algebraic: elimination and substitution.

-The third is using graphs.

The Elimination Method:-We eliminate a variable by adding or subtracting the equations together.

-Sometimes, we use multiplication.

NB: if the two equations are + and +, subtract.

If they are + and -, add.

Example:

2x+y= 5X+y= 2

Equation -Equation

2x-x= xy-y= ---

5-2= 3

X=3

-Find the solution for one variable and substitute into either equation to find.

Substitute into x+y= 2

3+y= 2

Y= 2-3Y= -1


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So: x= 3 and y= -1

Sequence:-A sequence is made up of terms.

-When looking at a sequence, we try to establish a pattern.-The first thing to look at is the difference between the terms.

The Constant [first] difference:Example: 5, 8, 11, 14, 17-5 is the first term [t1] 11 is the third term [t3] etc-The pattern is to add 3 to the previous term to get the next term.

The General Term [tn]:-Called the nth term.

-Allows us to work out any number in the sequence.

Example: 5, 8, 11, 14, 17Expression: 3n+2

tn= (difference) x n + 2/0th

term

NB: substitute into the first term to find the 2 OR use the 0th term.

Example 2: 20, 18, 16, 14, 120

thterm: 22

50th

term= 22-2n

= 22-2x50= -78

Linear Sequence:-A linear sequence has a general term [tn] of the form: an+b

-It is called a linear sequence because when graphed, the points fall on a line. [refer to book for

graph]

-DO NOT join points together when graphing sequences.

The algebra behind it all [finding the nth term]:Example: tn 5, 8, 11, 14, 17-Linear sequence of form an+b as 1st difference is constant.

n= 1 t=5 tn= an+b

a + b= 5 .

n=2 t=8

2a + b= 8.

a= 3

b= 2


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Quadratic Sequences:-If the second difference is constant the sequence is quadratic.

-In the form of an2+bn+c

-Since we have 3 unknowns, we need 3 equations.

Example: 9, 15, 23, 33, 45The first difference between the numbers is not constant as it goes 6, 8, 10, 12.

The second difference between the numbers [6, 8, 10, 12] is constant as it goes +2 each time.

tn= an2+bn+c

n=1 a+b+c=9 n=2 4a+2b+c=12n=3 9a+3b+c=23

*NB: the a+b+c, 4a+2b+c and 9a+3b+c will always stay the same. It is the result that changes

according to the sequence.

-4a-a= 3a

2b-b= b

15-9=6

3a+b=6

-9a-4a= 5a

3b-2b= b

23-15= 85a+b= 8

-5a-3a= 2a

8-6=2

2a=2

a=1

Substitute into (3x1) + b= 6

b=3

Substitute into a+b+c=9

1+3+c=9

c=5

So:

a=1

b=3

c=5


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Straight Line Graphs:-Co-ordinate: (x, y)

-x= x ordinate y= y ordinate

-x axis [horizontal]

-y axis [vertical]-Axes intersect at (0,0) called the origin.

Drawing Graphs:Example: Sketch y=x+5

-This is a straight line as it is a linear equation because the highest power of x is 1.

Need to find 3 co-ordinates.

X axis Y axis Co-ordinate

-5 -5+5=0 (-5, 0)

O O+5-5 (0, 5)5 5+5=10 (5, 10)

Then plot these on a graph [see back of exercise book]

NB: in an equation [like y= x-2], x can be substituted for anything, unless stated otherwise.

y=mx+c:-a linear graph has the form y=mx+c

-m is the gradient [steepness of the line]

-c is the y intercept [where it crosses the y axis]

Gradient:-The gradient of a line is its steepness.

-It is defined as the change in y = m

the change in x

-m= rise

run

^1

(rise)

1

(run)

Gradient: 1/1

m= 1


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Maths Revision Notes 2011

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