A level maths: pure

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Domain

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144 Terms

1

Domain

The set of possible inputs for a function.

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Range

The set of possible outputs of a function.

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Discriminant

b² - 4ac > 0 then two distinct real roots.

b² - 4ac = 0 then one repeated real root.

b² - 4ac < 0 then a quadratic function has no real roots.

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4

Types of Lines for Regions

If y < f(x) or y > f(x) then the curve y = f(x) is not included in the region, and is represented by a dotted line.

If y ≤ f(x) or y ≥ f(x) then the curve y = f(x) is included in the region, and is represented by a solid line.

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Graph Translations

y = f(x) + a is a translation of the graph y = f(x) by a upwards.

y = f(x + a) is a translation of the graph y = f(x) by a to the left.

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Graph Stretches

y = af(x) is a stretch of the graph y = f(x) by a scale factor of a in the vertical direction.

y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1/a in the horizontal direction.

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Graph Reflections

y = -f(x) is a reflection of the graph of y = f(x) in the x-axis.

y = f(-x) is a reflection of the graph of y = f(x) in the y-axis.

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8

Gradient of Equation

m = (y₂ - y₁) ÷ (x₂ - x₁)

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9

Equation of a Line

y - y₁ = m(x - x₁)

with coords (x₁, y₁)

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10

Distance Formula

√((x₂ - x₁)² + (y₂ - y₁)²)

from (x₁, y₁) to (x₂, y₂)

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Perpendicular Bisector

-1/m

where m is original gradient

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12

Standard Equation of a Circle

(x - a)² + (y - b)² = r²

with centre (a, b) and radius r

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13

Equation of a Circle (fg)

x² + y² + 2fx + 2gy + c = 0

with centre (-f, -g) and radius √(f² + g² - c)

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14

Circle Theorems

• Tangent to a circle is perpendicular to the radius of the circle at the point of intersection. • Perpendicular bisector of a chord will go through the circle centre. • If triangle forms across the circle, its diameter is the hypotenuse of the right-angled triangle. • Equations of the perpendicular bisectors of two different chords will intersect at the circle centre.

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Factor Theorem

If f(p) = 0, (x - p) is a factor of f(x)

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16

Mathematical Proofs

• State any info/assumptions • Show every step clearly • Make sure every step follows logically from the previous step • Cover all possible cases • Write a statement of proof at the end of your working

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Truth by Exhaustion

Break the statement into smaller cases and prove each case separately.

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18

Truth by Counter-Example

Find one example that does not work for the statement.

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19

Pascal's Triangle

The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a + b)ⁿ

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20

Factorial Formula

n! = n x (n - 1) x (n - 2) x ... x 2 x 1

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Factorials in Pascal's Triangle

The number of ways of choosing r from a group of n items is: ⁿCᵣ = n! ÷ (r! x (n - r)!)

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Binomial Expansion

(a + b)ⁿ = (ⁿCᵣ)(aⁿ⁻¹bʳ)

(a + b)ⁿ = aⁿ + (ⁿC₁)(aⁿ⁻¹b) + (ⁿC₂)(aⁿ⁻²b²) + ... + (ⁿCᵣ)(aⁿ⁻ʳbʳ) + ... + bⁿ

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23

Cosine Rule (a²)

a^2=c^2+b^2-2bcCosA

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Cosine Rule (cos(A))

CosA=a^2+b^2-c^2/ 2ab

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Sine Rule

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Sine Rule Solutions

Sometimes produces two possible solutions for a missing angle:

sin(θ) = sin(180 - θ)

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Principal Value

When you use the inverse trig function on your calculator, the angle you get is the principal value.

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Sine and Cosine Formulae

sin²(θ) + cos²(θ) = 1

tan(θ) = sin(θ) ÷ cos(θ)

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Triangle Law for Vector Addition

A→B + B→C = A→C

If A→B = a, B→C = b and A→C = c, then a + b = c

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Vector Rules

• P→Q = R→S, then line segments PQ and RS are equal in length and are parallel. • A→B = -(B→A) • Any vector parallel to the vector a may be written as λa

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Vector Magnitude

a = xi + yj → |a| = √(x² + y²)

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Unit Vector

In the direction of a, unit vector is a ÷ |a|

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Position Vector

O→P = pi + qj

A→B = O→B - O→A

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34

First Derivative Formula

  • x=(x+h)

  • Any workings on x must be applied to (x+h)

<ul><li><p>x=(x+h)</p></li><li><p>Any workings on x must be applied to (x+h)</p></li></ul>
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Differentiation Formula

dy/dx = anxⁿ⁻¹

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Function's Gradient

Increasing on the interval [a, b] if f'(x) ≥ 0 for all values of a < x < b.

Decreasing on the interval [a, b] if f'(x) ≤ 0 for all values of a < x < b.

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Stationary Point

Any point on the curve y = f(x) where f'(x) = 0. • if f''(a) > 0, the point is a local minimum. • if f''(a) < 0, the point is a local maximum.

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Integration Formula

y = k/(n+1) x xⁿ⁺¹ + c

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39

Area of a Triangle

½ x ab x sin(C)

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40

Natural Log (y = ln x)

The graph of y = ln(x) is a reflection of the graph y = eˣ in the line y = x.

eˡⁿ⁽ˣ⁾ = ln(eˣ) = x

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41

Differentiating eˣ

y = eᵏˣ → dy/dx = keᵏˣ

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Logarithm Laws

  1. logb(xy) = logb(x) + logb(y)

  2. logb(x/y) = logb(x) - logb(y)

  3. logb(x^y) = y logb(x)

  4. logb(x) = logc(x) / logc(b)

<p></p><ol><li><p>logb(xy) = logb(x) + logb(y)</p></li><li><p>logb(x/y) = logb(x) - logb(y)</p></li><li><p>logb(x^y) = y logb(x)</p></li><li><p>logb(x) = logc(x) / logc(b)</p></li></ol>
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y = axⁿ

If y = axⁿ then the graph of log(y) against log(x) will be a straight line with gradient n and vertical intercept log(a).

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y = abˣ

If y = abˣ then the graph of log(y) against x will be a straight line with gradient log(b) and vertical intercept log(a).

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45

Multiplying - rules of indices

a³ x a⁵ = a⁸ add the powers

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Dividing - rules of indices

a⁶ ÷ a² = a⁴ minus the powers

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Brackets - rules of indices

(a⁴)² = a⁸ multiply the powers

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48

Negative and Fractional indices

a^1/m = m√a

a^n/m = m√aⁿ

a^-n = 1/aⁿ

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Multiplying surds

√a x √b = √ab

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50

Dividing surds

√a ÷ √b = √a/b

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51

Rationalising denominators with surds

1/√a x √a/√a

1/a+√b x a-√b/a-√b

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52

What is the form of a quadratic equation?

ax² + bx + c = 0 where a, b and c are real constants and a≠0

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53

What is the quadratic formula?

x = (-b±√b²-4ac)/2a

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54

completing the square

x² + bx = (x+b/2)² - (b/2)²

ax² + bx + c = a(x + b/2a)² + (c - b²/4a²)

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The discriminant

b²−4ac > 0 two real, distinct roots.

b²−4ac = 0 one repeated root.

b²−4ac < 0 no real roots.

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56

Gradient of a straight line

m = y₁ - y / x₁ - x

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57

Equation of a straight line

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

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58

Equation of a line using two known points.

y - y₁ = m(x - x₁) where m = gradient and (x₁,y₁) are the coordinates used.

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Parallel lines

have the SAME gradient.

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60

Perpendicular lines

They meet at a right angle. If line 1 has a gradient of m, the gradient of line 2 is -1/m

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61

Distance between two points

√(x₁-x₂)² + (y₁-y₂)²

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62

equation for midpoint

(x₁+x₂/2 , y₁+y₂/2)

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63

Equation of a circle

(x-a)² + (y-b)² = r² with centre (a,b) and radius r.

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64

What is a tangent?

A tangent is a line that only touches one point on the circumference of a circle.

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What is a chord?

A line joining two points on the circumference of a circle, but not passing through the centre (that is the diameter).

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Tangent and chord properties on circles.

A tangent to the circle is perpendicular to the radius at the point of intersection. The perpendicular bisector of a chord will go through the centre of a circle.

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67

Factor Theorem

if f(p)=0 then (x-p) is a factor of f(x).

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What is proof by deduction?

Starting from known facts or definitions, then using logical steps to reach the desired conclusion.

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What is proof by exhaustion?

Breaking the statement into smaller cases and proving each case separately.

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What is disproof by counter-balance?

Giving an example that doesn't work for the statement.

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71

Pascal's Triangle

It is formed by adding adjacent pairs of numbers to find the numbers of the next row. The (n+1)th row of Pascal's Triangle gives the coefficients in the expansion of (a+b)ⁿ.

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72

Trigonometry for right-angled triangles

sinθ = opp/hyp cosθ = adj/hyp tanθ = opp/adj

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73

What is the cosine rule?

Length: a² = b² + c² - 2bc cosA Angle: cosA = (b² + c² - a²)/2bc

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What is the sine rule?

Length: a/sinA = b/sinB = c/sinC Angle: sinA/a = sinB/b = sinC/c

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75

Area of a triangle with two sides and angle in between

area = ½absinC

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76

Graph of y = sin(x)

Repeats every 360°. Crosses x-axis at -180, 0, 180, 360 etc. Has a max value of 1 and a min of -1.

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Graph of y = cos(x)

Repeats every 360°. Crosses x-axis at -90, 90, 270, 450 etc. Has a max value of 1 and a min of -1.

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Graph of y = tan(x)

Repeats every 180°. Crosses x-axis at -180, 0, 180, 360 etc. Has no max or min value Has vertical asymptotes at x=-90, 90, 270 etc.

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79

What are the two trig identities?

sin²θ + cos²θ ≡ 1

tanθ ≡ sinθ/cosθ

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80

Derivative or gradient function equation

Where h represents a small change.

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81

Differentiating x^n

If f(x) = xⁿ then f'(x) = nxⁿ⁻¹

If f(x) = axⁿ then f'(x) = anxⁿ⁻¹

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82

Differentiating a quadratic

If y = ax² + bx + c then dy/dx = 2ax + b

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83

Tangents to curves.

The tangent to the curve y=f(x) at (a, f(a)) has the equation: y-f(a) = f'(a)(x-a).

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84

Normals to curves.

The normal to the curve y=f(x) at (a, f(a)) has the equation: y-f(a) = -1/f'(a) x (x-a)

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85

Increasing and decreasing functions.

Increasing: f'(x) > 0 Decreasing: f'(x) < 0

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86

Second order derivative

f''(x) or d²y/dx² a.k.a differentiate twice.

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Types of stationary points.

Local maximum, local minimum and point of inflection.

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Sketching gradient functions

y=f(x) y=f'(x) max or min cuts x-axis point of inflection touches x-axis +ve grad above x-axis -ve grad below x-axis vertical asymp. vertical asymp. horizontal asymp. horiz asymp at x-axis

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89

the discriminant

b^2-4ac=0

>for two roots =for one repeated root <no real roots

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90

turning point of a graph

complete the square

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91

quadratic formula

x=(-b±√(b^2-4ac))/2a

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y=f(x)+a

moves (0/a)

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y=f(x+a)

moves (-a/0)

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y=af(x)

movement of a in y (vertical) direction

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y=f(ax)

movement by scale factor of 1/a in x (horizontal) direction

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y=-f(x)

reflection in x axis

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y=f(-x)

reflection in y axis

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98

gradient equation

m = (y2-y1) / (x2-x1)

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Distance between two points

d = √(x₁-x₂)² + (y₁-y₂)²

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midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

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