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Condition If the coefficient of $x^2$ is 1, think of two numbers which: 1. Add to make the $b$ 2. Multiply to make $c$ Otherwise, utilise the cross method. Cross Method 1. Left sid...
Condition If the coefficient of $x^2$ is 1, think of two numbers which: 1. Add to make the $b$ 2. Multiply to make $c$ Otherwise, utilise the cross method. Cross Method 1. Left sid...
$y = 2$ is a constant, it is the same as $y = 2x^0$. $y = x + 2$ is linear. $y = x^2 + 2x 4$ is quadratic. The gradient of a linear line can be calculated by doing $\frac{\Delta y}...
$y = 2$ is a constant, it is the same as $y = 2x^0$. $y = x + 2$ is linear. $y = x^2 + 2x 4$ is quadratic. The gradient of a linear line can be calculated by doing $\frac{\Delta y}...
Using the mean $\mu = n \times p$ $\text{sample size} \times \text{probability}$ , compare it to the observed value. If it is more, than test upper, otherwise lower. Note: The crit...
Decision: If p value is less than the significance value, reject $H {0}$, otherwise do not reject. Template Sentence: "There is sufficient/insufficient evidence of the x % signific...
See Binomial Expansion. $$ a + b ^2 = a^2 + 2ab + b^2 $$ Example 1. Square the first term: $ 3x ^2 = 9x^2$ 2. Double the product of both: Multiply $3x$ and $10$ together to get $30...
Squared Parentheses Example $$ 7x ^2 = 7^2 \times x^2 = 49 \times x^2 = 49x^2 $$ Squared in squared Parentheses Example WARNING This example was AI generated.
WARNING This document was AI generated. Surds are numbers left in square root or cube root form because their exact value cannot be written as a whole number or a fraction. They ar...
WARNING This document was AI generated. Surds are numbers left in square root or cube root form because their exact value cannot be written as a whole number or a fraction. They ar...
WARNING This document was AI generated. 1. Core Definitions Population: The entire collection of individuals or items that you want to study. Census: An investigation that observes...
A distribution is a way to describe how probabilities are spread across different outcomes. Chapter 6 focuses on the Binomial Distribution . The Four Conditions BINS You can only u...
WARNING This document was AI generated. A hypothesis test is a statistical method used to decide whether there is enough evidence in a sample of data to support a particular belief...
Implicit differentiation is a different way to find the derivative when $y$ is "stuck" inside an equation and you can't or don't want to isolate it as $y = \dots$. Always treat $y$...
You can factorise an equation, such as $9 x^2$, if both values are roots. Difference means subtraction only. Equation: $$ r^1 r^2 r^1 + r^2 $$ Example $$ \begin{array}{0} 9+x^2 \\...
$h x = \frac{1}{f x }$ $$ \begin{array}{0} \text{cosec} \space x = \frac{1}{\sin x} \\ \text{cosec}^2 \space x = \frac{1}{\sin^2 x} \\ \sec x = \frac{1}{\cos x} \\ \cot x = \frac{1...
$$ \begin{array}{0} \frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt} \\ \frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx} \\ \frac{dy}{dx} = \frac{dy}{dx} \end{array} $$
$$ \begin{array}{0} y = \frac{u x }{v x } \\ \frac{dy}{dx} = \frac{v x u' x u x v' x }{v x ^2} \end{array} $$
$$ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} $$ Chain rule is used to decompose working with complex equations within differentiation much easier. It allows you to substit...
Exponential Rules $$ \begin{array}{0} \frac{d}{dx} e^x = e^x \\ \frac{d}{dx} e^{kx} = ke^{kx} \\ \frac{d}{dx} a^x = a^x \ln a \\ \frac{d}{dx} a^{kx} = a^{kx} k \ln a \end{array} $$...
$$ \begin{array}{0} y = uv \\ \frac{dy}{dx} = v \frac{du}{dx} + u \frac{dv}{dx} \end{array} $$ Examples $y = x \sin x$ $$ \begin{array}{0} y = x \sin x \\ u = x \\ v = \sin x \\ \f...
$$ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} $$ Just like you can get $x$ from $y$ by getting the reciprocal of $y$, you can do the same for $\frac{dx}{dy}$ , allowing you to easily...
Using Pascal's triangle or the binomial equation , you can quickly expand an equation such as $ x+h ^x$. To do this you: 1. Look at the $n$ th row of the triangle counting the very...
Using Pascal's triangle or the binomial equation , you can quickly expand an equation such as $ x+h ^x$. To do this you: 1. Look at the $n$ th row of the triangle counting the very...
When you are asked to compare two sets of data, you must compare: A measure of location mean/median\ /mode/quantile/percentile A measure of spread inter quartile range\ , inter per...
Factor theorem is the process of validating whether a factor is a factor of an equation. A factor is a number where the a % b = 0 . For example, $ x + 3 $ is a factor of $f x = x^2...
Factor theorem is the process of validating whether a factor is a factor of an equation. A factor is a number where the a % b = 0 . For example, $ x + 3 $ is a factor of $f x = x^2...
$$ \text{frequency daensity} = \frac{\text{frequency}}{\text{class width}} $$ The area of a bar is proportional to the frequency it represents Area of a bar is proportional to freq...
Mode modal is the most frequent value Median is the middle value Mean is the 'average' value $$ \bar{x} = \frac{\Sigma x}{n} $$
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The First Principles technique focuses on decomposing a question to its fundamentals. This is essential for calculus and vectors . Example A spherical balloon is being inflated. In...
census information from the entire population population the entire data set subset sampling frame a 'slice' of the population random sampling simple random number sampling done by...
Population is the whole dataset. The population contains every possible element that satisfies a specific predicate condition . Sample is a subset/slice of the population, it is of...
To get the local turning point of a quadratic, you can either: 1. Complete the Square Completing The Square 2. Use differentiation anx Equation Remember that the $gradient = 0$ and...
To get the local turning point of a quadratic, you can either: 1. Complete the Square Completing The Square 2. Use differentiation anx Equation Remember that the $gradient = 0$ and...
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NOTE Remember: axes is called a 'plane'. Notes Resultant force = net force in a plane x/y Friction is only effecting if the surface is "smooth" or "rough" Friction applies in the o...
Equations $$ \begin{array}{0} F = ma \\ W = mg \\ force = mass \times acceleration \\ weight = mass \times gravity \end{array} $$ Resolving Axes The resolving axes is represented b...
The formulae for constant motion $v=u+at$ can be used to model an object moving vertically under gravity. This is because all objects accelerate towards the earth at a constant rat...
S = displacement distance U = initial velocity V = final velocity A = acceleration T = time
| Quantity | Unit | Symbol | Derived Quantity | Unit | Symbol | | | | | | | | | Mass | Kilogram | kg | Weight/force | Newton | N = kg m s 2 | | Length/displacement | Meters | m | S...
About Mechanics deals with motion and the action of forces on objects . Mathematical models can be constructed to simulate real life situations but are often more abstract.
NOTE These were also used for Grade 9 questions in GCSE. The Sine Rule and Cosine Rule are used to find unknown sides or angles in non right angled triangles. Sine Rule The Sine Ru...
NOTE These were also used for Grade 9 questions in GCSE. The Sine Rule and Cosine Rule are used to find unknown sides or angles in non right angled triangles. Sine Rule The Sine Ru...
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Vector vs. Scaler A vector has both a magnitude and direction , whereas a scaler has only a magnitude .
A unit vector is any vector with a magnitude of one. Conversion To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide...
A unit vector is any vector with a magnitude of one. Conversion To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide...
TODO Write and explain techniques. Choose Example $$ \begin{array}{0} 1+ax ^10 = 1^10 + 1^9 {10\choose {1}} ax ^1 + 1^8 + {10\choose {2}} ax ^2 \dots \end{array} $$ Year 13 Techniq...
TODO Write and explain techniques. Choose Example $$ \begin{array}{0} 1+ax ^10 = 1^10 + 1^9 {10\choose {1}} ax ^1 + 1^8 + {10\choose {2}} ax ^2 \dots \end{array} $$ Year 13 Techniq...
Limits can be represented as either $ x ^a {b}$ or $\int^a {b}x$. The limits represent both the upper and lower bounds of a definite integral the interval over which to calculate t...
Limits can be represented as either $ x ^a {b}$ or $\int^a {b}x$. The limits represent both the upper and lower bounds of a definite integral the interval over which to calculate t...
WARNING This document was partially AI generated, however adapted. $e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself...
WARNING This document was partially AI generated, however adapted. $e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself...
Integration is the reverse of Differentiation. It uses the formula $y = \frac{k}{n+1}x^{n+1} + c$ as long as $n \not= 1$. It can be summarised in three steps: 1. Add one to the pow...
Integration is the reverse of Differentiation. It uses the formula $y = \frac{k}{n+1}x^{n+1} + c$ as long as $n \not= 1$. It can be summarised in three steps: 1. Add one to the pow...
A logarithm is the exponent to which a base must be raised to produce a given number. Equation $$ \begin{array}{0} a^x = b \\ x = \log {a}b \\ \end{array} $$ Usage Rules You cannot...
A logarithm is the exponent to which a base must be raised to produce a given number. Equation $$ \begin{array}{0} a^x = b \\ x = \log {a}b \\ \end{array} $$ Usage Rules You cannot...
A stationary point on a curve is where the gradient is $0$. This occurs when $f' x = 0$. For example, the curve $y = x^2$ has a stationary point at $ 0, 0 $. Local Maximum and Mini...
A stationary point on a curve is where the gradient is $0$. This occurs when $f' x = 0$. For example, the curve $y = x^2$ has a stationary point at $ 0, 0 $. Local Maximum and Mini...
WARNING This document was AI generated. The equation $ax^n$ is the standard form used for differentiating terms in a polynomial. The Power Rule To differentiate $ax^n$, you multipl...
WARNING This document was AI generated. The equation $ax^n$ is the standard form used for differentiating terms in a polynomial. The Power Rule To differentiate $ax^n$, you multipl...
WARNING This does not have any practical example. It is just a demonstration. See Binomial Expansion for technique. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
WARNING This does not have any practical example. It is just a demonstration. See Binomial Expansion for technique. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
$ x a ^2 + y b ^2 = r^2$ $ a, b $ is the centre of the circle. $ x, y $ is the current position. $r$ is the radius of the circle. Example: Finding the equation of a circle Given: c...
$ x a ^2 + y b ^2 = r^2$ $ a, b $ is the centre of the circle. $ x, y $ is the current position. $r$ is the radius of the circle. Example: Finding the equation of a circle Given: c...
$$ \begin{array}{1} a^\frac{1}{m} = \sqrt m { a } \\ a^\frac{n}{m} = \sqrt m { a^n } \\ a^ m = \frac{1}{a^m} \\ a^0 = 1 \end{array} $$
$$ \begin{array}{1} a^\frac{1}{m} = \sqrt m { a } \\ a^\frac{n}{m} = \sqrt m { a^n } \\ a^ m = \frac{1}{a^m} \\ a^0 = 1 \end{array} $$
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \begin{array}{1} \left \frac{x {1} + x {2}}{2}, \frac{y {1} + y {2}}{2} \right \end{array...
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \begin{array}{1} \left \frac{x {1} + x {2}}{2}, \frac{y {1} + y {2}}{2} \right \end{array...
NOTE Revise. $$ a x + h ^2 + k $$ To get the co ordinates of the maximum/minimum turning point. $ x + h = 0$ or $ h$ provides the $x$ co ordinate. $k$ provides the $y$ co ordinate....
NOTE Revise. $$ a x + h ^2 + k $$ To get the co ordinates of the maximum/minimum turning point. $ x + h = 0$ or $ h$ provides the $x$ co ordinate. $k$ provides the $y$ co ordinate....
WARNING This document was AI generated. Equation $$ f' x = \lim {h \to 0} \frac{f x+h f x }{h} $$ Examples Function: $x^2$ Step 1: Find $f x+h $ $$ f x+h = x+h ^2 = x^2 + 2xh + h^2...
WARNING This document was AI generated. Equation $$ f' x = \lim {h \to 0} \frac{f x+h f x }{h} $$ Examples Function: $x^2$ Step 1: Find $f x+h $ $$ f x+h = x+h ^2 = x^2 + 2xh + h^2...
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$$ m = \frac{\Delta y}{\Delta x} $$
$$ m = \frac{\Delta y}{\Delta x} $$
To find the length of a line, you can use Pythagoras' Theorem. $$ A^2 = B^2 + C^2 $$ as: $$ d = \sqrt{ x^2 x^1 ^2 + y^2 y^2 ^2 } $$
To find the length of a line, you can use Pythagoras' Theorem. $$ A^2 = B^2 + C^2 $$ as: $$ d = \sqrt{ x^2 x^1 ^2 + y^2 y^2 ^2 } $$
Parallel lines are two lines which have the same gradient but never touch. Example $$ \begin{array}{1} y {1} = 5x + 2 \\ y {2} = 5x + 4 \end{array} $$ These linear lines are parall...
Parallel lines are two lines which have the same gradient but never touch. Example $$ \begin{array}{1} y {1} = 5x + 2 \\ y {2} = 5x + 4 \end{array} $$ These linear lines are parall...
Perpendicular lines are two lines which meet at 90°. If a line has the gradient of $m$, a line perpendicular will have the gradient of $ \frac{1}{m}$. If two lines are perpendicula...
Perpendicular lines are two lines which meet at 90°. If a line has the gradient of $m$, a line perpendicular will have the gradient of $ \frac{1}{m}$. If two lines are perpendicula...
Components The circle is decomposed into several components: Centre the centre of the circle Radius the distance from the centre of the circle to the circumference diameter is $2 \...
A ratio shows the relationship between two or more quantities . For instance, $1:5$ is for every $1$ part of the first quantity, there is every $5$ parts of the second quantity. Al...
Sin sine , cos cosine and tan tangent are the core three functions used in trigonometry and are based on a right angled triangle. "Opposite" is opposite to angle θ "Adjacent" is ad...
When plotting the coordinates of Sin and Cos, they will create a circle due as long as each point cos θ, sin θ lies on a circle of radius 1 centred at the origin. This circle is ca...
Cartesian coordinates is composed of X horizontal, going left and Y vertical, going up coordinates. As a general consensus: Demonstration
1. Introduction 1. 1.1. Introduction to Sin, Cos and Tan 2. 1.2. Unit Circles and Sin, Cos and Tan 2. unfinished.
Sine and Cosine $\sin x $ and $\cos x $ are periodic functions with a period of 360° or $2\pi$ radians . Both sine and cosine graphs oscillate between 1 and 1. The sine graph start...
Sine and Cosine $\sin x $ and $\cos x $ are periodic functions with a period of 360° or $2\pi$ radians . Both sine and cosine graphs oscillate between 1 and 1. The sine graph start...