Maths Year 1 Notes

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P → Q

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176 Terms

1

P → Q

P implies Q or if P is true then Q is true or P is sufficient for Q

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2

P ← Q

P is implied by Q or if Q is true then P is true or P is necessary for Q

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3

P ↔ Q

P is equivalent to Q or Q is true if and only if P is true

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4

x ∈ ( a , b )

a < x < b

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5

x ∈ [ a , b ]

a ≤ x ≤ b

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6

x ∈ [ a , b )

a ≤ x < b

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7

x ∈ ( a , b ]

a < x ≤ b

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8

x ∈ A ∪ B

Union of A and B where x can be in either A or B or both

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9

x ∈ A ∩ B

Intersection of A and B where x is in both A and B

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10

x ∈ ∅

No solutions to the inequality

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11

Symbol for an empty set

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12

a^n × a^m

a^(m+n)

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13

a^(1/n)

n√a

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14

a^m ÷ a^n

a^(m-n)

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15

a^(m/n)

n√a^m

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16

(a^m)^n

a^(m × n)

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17

a^n × b^n

(ab)^n

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18

a^0

1

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19

a^n ÷ b^n

(a/b)^n

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20

a^(-n)

1/a^n

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21

Rationalize the denominator

Multiply the top and bottom of the fraction by an appropriate expression to create a difference of two squares

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22

Positive parabola

a > 0

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23

Negative parabola

a < 0

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24

y-intercept

Value of c in quadratic equation, (0,c)

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25

Quadratic equation

ax² + bx + c

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26

Complete the square formula

a(x+p)²+q

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27

Turning point of graph

(-p,q) from completing the square

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28

Line of symmetry

x = -p = (x₁+x₂)/2

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29

Quadratic formula

(-b±√b²-4ac)/2a

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30

Discriminant

∆ = b² - 4ac

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31

Graph has two real roots

∆ > 0

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32

Graph has one real root

∆ = 0

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33

Graph has no real roots

∆ < 0

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34

Factor theorem

(ax+b) is a factor of a polynomial f(x) if and only if f(-b/a) = 0

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35

Polynomial

ax^k

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36

y = f(x) + c

Translation c up

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37

y = f(x+d)

Translation d to the left

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38

y = pf(x)

Vertical stretch, scale factor p relative to the x-axis

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39

y = f(qx)

Horizontal stretch, scale factor 1/q relative to the y-axis

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40

y = -f(x)

Reflection in the x-axis

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41

y = f(-x)

Reflection in the y-axis

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42

Direct proportion

y = kx

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43

Inverse proportion

y = k/x

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44

Distance between two points

√(x₂ - x₁) + (y₂ - y₁)

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45

Midpoint of two points

( (x₁+x₂)/2 , (y₁+y₂) )

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46

Gradient of a line

m = (y₂-y₁) / (x₂-x₁)

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47

Method to find equation of straight line

y - y₁ = m (x - x₁)

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48

Equation of a straight line

y = mx + c / ax + by + c = 0

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49

Parallel lines

Have the same gradient

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50

Perpendicular lines

m₁ m₂ = -1

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51

Equation of a circle

(x - a)² + (y - b)² = r²

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52

Tangent

Perpendicular to the radius

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53

Normal

Same direction as radius

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54

b^c = a

logb(a)

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55

log10(x)

log(x)

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56

loga(x^a)

x=a^(loga(x))

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57

loge(x) / natural logarithm

In(x)

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58

loga(x) + loga(y)

loga(xy)

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59

loga(x) - loga(y)

loga(x/y)

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60

loga(1/x)

-loga(x)

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61

loga(x^k)

kloga(x)

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62

loga(1)

0

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63

loga(a)

1

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64

Graph of y=a^x

y-intercept is (0,1) and x-axis is an asymptote

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65

Gradient of e^(kx)

ke^(kx)

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66

Ae^(kt) model

Initial value is A and the rate of change equals ky

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67

Graph of y=In(k)

Passes through (1,0) and y-axis is an asymptote

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68

Convert y=ax^n

log(y) = log(a) + nlog(x)

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69

log(y) = log(a) + nlog(x)

y-intercept is log(a) and gradient is n

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70

Binomial theorem

(a+b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + … + ⁿCₙa⁰bⁿ

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71

ⁿCr

n! / r!(n-r)!

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72

n!

n × (n-1) × … × 3 × 2 × 1

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73

0!

1

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74

sin(x)

sin(180-x) = sin(x+360)

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75

sin(180+x)

sin(-x) = -sin(x)

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76

cos(x)

cos(-x) = cos (x+360)

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77

cos(180-x)

cos(180+x) = -cos(x)

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78

tan(x)

tan(x+180) = tan(x+360) = …

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79

Tan graph asymptotes

x = 90° , x = 270° etc

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80

Period of sine and cosine graph

360°

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81

Period of tan graph

180°

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82

cos(90-x)

sin(x)

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83

sin(90-x)

cos(x)

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84

tan(x) identity

tan(x) ≡ sin(x) / cos(x)

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85

sin(x) and cos(x) identity

sin²(x) + cos²(x) ≡ 1

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86

Sine rule

a / sin(A) = b / sin(B) = c / sin(C)

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87

Two answers by sine rule

A and 180-A

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88

cosine rule

c² = a² + b² - 2abcos(C)

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89

Area of a triangle

Area = 1/2 absin(C)

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90

Magnitude of vector a = pi + qj

√p² + q²

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91

Unit vector

Magnitude is one

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92

Parallelogram

AB = DC

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93

Rhombus

|AB| = |BC|

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94

If vectors a and b are parallel

b = ta

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95

Displacement from A to B

AB = b-a

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96

Midpoint of line AB

1/2 (a+b)

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97

If graph is increasing

Positive gradient

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98

If graph is decreasing

Negative gradient

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99

When gradient = 0

Stationary point

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100

Differentiation from first principles

f’(x) = lim h→0 (f(x+h) - f(x)) / h

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