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• The Binomial Expansion

Pascal's Triangle is used to expand binomial expressions like (a+b)ⁿ. It is created by summing adjacent pairs to find the number beneath this pair, starting from 1. Here are the first five rows: The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a+b)ⁿ Factorial Notation Parts of Pascal's triangle can be calculated quickly using factorial notation, ⁿCr (spoken "n choose r"): Expanding (a+b)ⁿ When n ∈ ℕ (when n is a positive integer) the binomial expansion is in its simplest form: The general term in an expansion of (a+b)ⁿ is given as: Expanding (1+x)ⁿ If n is a fraction or a negative number, you need to use this form of the binomial expansion: It is valid when |x| < 1 and when n ∈ ℝ The general term in this expansion is given as: x-term Coefficient When the x term has a coefficient, so the binomial is in the form (1+bx)ⁿ, treat (bx) as x, and follow the standard expansion for (1+x)ⁿ The expansion for (1+bx)ⁿ is valid for |bx| < 1, or |x| < 1/|b| Double Coefficients If the binomial is in the form (a+bx)ⁿ, you have to take a factor of aⁿ out of each term: The expansion for (a+bx)ⁿ, where n is negative or a fraction, is valid when |bx/a| < 1, or |x| < |a/b| Often, complex expressions can be simplified first by splitting them into partial fractions (see notes sheet on algebraic methods), then by using a binomial expansion.

• Circles

Often when trying to find the equation of a circle, you will be given a line that intersects with the circle twice (it may be the diameter). To work out the circle from this you need to know the midpoint: The perpendicular bisector of a cord will always pass through the centre of a circle. To find this, find the midpoint of the cord and find the equation of the line perpendicular to it (the negative reciprocal of the gradient) Equation of a Circle The points on a circle are all related through the equation of a circle. For a circle with centre (0,0) and radius r, the equation is: x² + y² = r² For a circle with centre (a, b) and radius r, the equation is: (x-a)² + (y-b)² = r² Expanded Form Sometimes, the equation is given or needed in expanded form: x² + y² + 2fx + 2gy + c = 0 (-f, -g) is the centre of the circle √(f² + g² -c) is the radius To go back from expanded form to the standard factorised form for an equation of a circle, it is quickest to complete the square: Intersecting Lines & Circles A straight line can intersect a circle once, twice, or not at all: To find out how many intersections there are between a circle and a straight line without sketching it, you can solve simultaneously: Circle Theorems

• Parametric Equations

An alternative to defining curves with Cartesian equations (one equation per curve with only x and y variables), we can express curves as separate equations of x and y, each in terms of the third variable t. A curve y = f(x) can also be described using x = p(t) and y = q(t). The coordinates of each point on the curve are (p(t), q(t)) It is possible to convert between Cartesian and parametric forms with substitution to eliminate one variable: To convert x = 4t, y = 2t ² + 4 into Cartesian: x = 4t t = ¼x Rearrange to make t the subject y = 2t ² + 4 y = 2(¼x) ² + 4 Substitute into y = q(t) y = ⅛x² + 4 Simplify Domain & Range For parametric equations x = p(t) and y = q(t) where the Cartesian equation is y = f(x): The domain of f(x) is the range of p(t) The range of f(x) is the range of q(t) Trigonometry Often, parametric equations will be given as trigonometric functions. When this is the case, it is often possible to use trig identities (see notes sheet on trigonometry) to convert between parametric and Cartesian. Angles should always be taken in radians. To convert x = sin t + 2, y = cos t - 3 into Cartesian: x = sin t + 2 sin t = x - 2 Rearrange to make sin t the subject y = cos t - 3 cos t = y + 3 Rearrange to make cos t the subject (x - 2)² + (y + 3)² = 1 Use identity sin²t + cos²t ≡ 1 Note that this is the equation of a circle. Sketching Parametric Curves To sketch a curve from parametric equations, either convert into Cartesian form - for example the equation above turned out to be a simple circle, radius 1 and centre (2, -3) - or tabulate the coordinates for different values of t and then plot these.

• Trigonometry

• Integration

• Differentiation

• Vectors

A vector is a quantity with both magnitude and direction, and is typically represented visually by a line segment between two points. There are many ways of representing vectors in notation: The two points that the vector connects with an arrow above them As a bold typeface lower case letter As an underlined lower case letter As a column vector, showing displacement in the x-direction above that in the y-direction As a multiple of unit vectors, i (one unit in the positive x-direction) and j (one unit in the positive y-direction) These notes will predominantly use bold typeface of lower case letters, underlined letters and the unit vectors i and j. Vectors can be multiplied by a scalar, and added and subtracted: Magnitude & Direction The magnitude of a vector is given by Pythagoras' Theorem. Magnitude is noted using straight lines on either side of the letter, like modulus. For the vector a = xi + yj, |a| = √(x²+y²) A unit direction vector, â, can be found as a / |a| A vector can also be defined by giving its magnitude and the angle it makes with one of the coordinate axis. This is called magnitude-direction form. Position Vectors Position vectors are used to give the location of a point relative to a fixed origin. A point (p, q) has a position vector pi + qj Vector Geometry Position vectors can be used to solve geometric problems: Vectors in 3D In three-dimensional geometry, there are x, y, and z-axes. This means each coordinate has three values, (x, y, z). The position vector of this is xi + yj + zk Pythagoras' Theorem still applies, but adjusted for three points: Distance from point (x, y, z) to origin is given as √(x²+y²+z²) The distance between two points, (x₁, y₁, z₁) and (x₂, y₂, z₂) is given as: √( (x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)² ) Vector Geometry in 3D

• Exponentials and Logarithms

An exponential function is one a constant is raised to the power of a variable: The larger the coefficient, the steeper the graph All exponential functions in the form y = a^x pass through (0, 1) The value of the function decreases as x tends to 0 Functions in the form y = a^x where 0 < a < 1 are the other way around: The gradient graphs of an exponential function are always very similar, but when a = 2.72 (e), the gradient graph is exactly the same: For all real values of x: if f(x) = e^x, f'(x) = e^x if f(x) = e^kx, f'(x) = ke^kx Functions in e can be used to model growth or decline where the rate of increase in number is proportional to the number. Logarithms Logarithms are the inverse of exponential functions. Just as indices laws apply to exponentials, there are a series of laws that apply to logarithms - these are known as the log rules or laws of logarithms: When working with difficult equations with exponents, it can be helpful to apply logarithms to both sides: The Natural Logarithm, ln The natural logarithm, ln, is the logarithm in base e. The graph of ln(x) is reflection of e^x in the line y=x: The y=axis is an asymptote in this graph, meaning that ln(x) is only defined when x > 0 Logarithms and Non-Linear Data Often, it is helpful to model non-linear data as linear. This can be achieved by taking logs of one or both sides of the equation: If the equation is in the form y = ax^n, a graph of log y against log x will give a straight line where log a is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log y against x will give a straight line where log a is the y intercept and log b the gradient.

• Numerical Methods

A root of a function is when the function = 0. On a graph, this is represented by an x-axis intersection, where y-value changes from positive to negative or vice versa. Therefore, we can locate roots by showing that the sign has changed. However, this only applies if the function is continuous throughout the given interval, notated using square brackets [a, b]. This means that the graph does not jump from one value to another, but changes smoothly. If there is a vertical asymptote in the interval, it is not continuous. If f(x) is continuous on the interval [a, b], and f(a) and f(b) have opposite signs, then there is a root between the two Iteration It is possible to estimate roots using an iterative method. Generally, the equation needs to be manipulated slightly before it works, as the linear variable (x) has to be the subject. To solve an equation in the form f(x) = 0 using iteration, rearrange it into the form x = g(x) and apply iteration Applying iteration means you use put the value you get out of g(x) back into the function, and repeat this multiple times. There are multiple rearrangements for each function f(x), giving different variations of x = g(x). These different iterations will either converge to a root, or diverge: If the iteration converges closer and closer to the root, always in the same direction, it forms a staircase diagram. If the iteration converges in an alternating pattern above and below the root, a cobweb diagram is formed. If the iteration diverges, it moves away from the root very quickly. The Newton-Raphson Method The Newton-Raphson method is the method used by many calculators to solve equations in the form f(x) = 0. The formula for the Newton-Raphson method is: It works using tangent lines with increasing accuracy to approximate a root: As you can see, it relies on differentiation, and so cannot work when f'(x) equals or is close to equalling zero. If the starting value x₀ is close to a turning point, the method does not work, because the gradient of the tangent is very small, so the tangent crosses the x-axis a long way away from x₀ If any value of x is on a turning point, the method does not work because the gradient of the tangent is 0, and so never crosses the x-axis.

• Work, Energy & Power

Work done is the amount of energy transferred from one form to another when a force causes movement. The units of work are Joules, J W = Fx Work Done = Force × distance moved (in the same direction as the force) When the force is not in the same direction as the movement of the object, you need to work out the correct component force using trigonometry. You must remember to do this! Energy One of the fundamental laws of physics is the idea that energy cannot be created. This is known as the Principle of Conservation of Energy and states: The total energy in a closed system always remains constant - energy can never be created or destroyed, it can only ever be transferred from one form to another. Some of these forms include: Kinetic Gravitational Potential Elastic Potential Electrical Potential Nuclear Internal (heat/thermal) Kinetic Energy All moving objects have kinetic energy: KE = 1/2 m v^2 Kinetic Energy = 1/2 x mass x velocity squared This equation can be derived easily from work done and SUVAT: Gravitational Potential Energy Gravitational Potential Energy is the energy of an object due to its positioning in a gravitational field. The greater the height, the greater the GPE. GPE gain is work done in moving an object: GPE = mgh This can be derived as such: W = Fh → F = ma = mg → GPE = mgh (g is the acceleration due to free fall, 9.81 m/s^2) Because all energy is conserved, falling objects increase in kinetic energy as they decrease in GPE. Power Power is the rate at which work is done: P = W/t power = work done / time taken This can also be written as: P = Fv power = force × velocity A derivation of this is: P = W/t → P = Fx/t → P = Fv (Because v = x/t) The units of power are Watts, W, or Joules per second, J/s Efficiency Efficiency is the proportion of input energy that is transferred into useful output energy - for example in a motor, it is the portion of input energy that comes out as kinetic energy, rather than thermal or sound energy. Efficiency can be calculated as a percentage: 100 x useful output energy / total input energy

• Energy, Power & Resistance

Potential Difference Potential Difference (p.d.) is the work done per unit charge – the amount of energy converted from electrical to another form per coulomb of charge passing through a component. W = VQ work done = p.d. × charge Electromotive Force Electromotive Force is the amount of energy converted into electrical energy per coulomb of charge through the source: W= εQ work done = e.m.f x charge Essentially, p.d. is voltage used up in a circuit, whereas e.m.f. is the voltage provided to the circuit by a cell or power supply. Both p.d. and e.m.f. are often referred to as voltage, because they both shave the Volt as their unit. The Volt is 1J of energy per 1C of charge. Accelerating Electrons A cathode-ray tube (also called an electron gun) is used to accelerate electrons. A hot metal filament releases electrons by thermionic emission, and then these are accelerated in a vacuum between the filament and a metal plate with a small hole in it. The two are connected and a very high p.d. is applied, making the filament a cathode and the plate an anode. The electrons released from the filament are attracted to the positively charged anode plate, and pass through the hole. This creates a concentrated beam of high velocity electrons. When an electron is accelerated in this way, the energy transferred equals the wok done on the electron, which equals the kinetic energy: eV = 1/2 m v^2 work done on electron = gain in kinetic energy of electron Resistance All conductors have resistance that obstructs the flow of charge through them. The higher the resistance of a conductor, the more energy is required to push the charge through, as more energy is lost in the process. Ohm's Law Ohm's Law states that the current flowing through a conductor is directly proportional to the p.d. across it, provided that the physical conditions (such as temperature) remain constant. V = IR p.d. = current x resistance The units of resistance are Ohms, Ω, defined as the resistance of a conductor when 1 volt produces a current of 1 ampere through that conductor. I-V Characteristics To investigate the I-V relationships of components, a test circuit as shown can be set up and the current recorded for a variety of potential differences. Taking averages increases reliability, measuring to 3 sig-figs increases accuracy. A graph can be plotted to see its ohmic or non-ohmic properties. The ammeter must be connected in series, and the voltmeter in parallel. A Resistor/Metallic Conductor obeys Ohm's Law - the I-V characteristic is a straight line through the origin, where the inverse of the gradient is the resistance (R = V/I). A Filament Lamp is an example of temperature affecting resistance. As the p.d. increases, so does the brightness and temperature of the bulb and with it, its resistance. This is because at higher temperatures, the positive ions in the filament have greater internal energy, meaning they vibrate more. Therefore, the probability of the charge carriers colliding with them is greater. In this instance, Ohm's law does not apply. A Diode only works with current flowing thorough it in one direction - this is known as the forward bias. Typically, there is a threshold of around 0.6V before they conduct at all. Light Dependent Resistors and NTC Thermistors (Negative Temperature Coefficient) lose resistance as the energy on them increases. This is because they are semiconductors, that release more electrons for conducting when they have more energy. This means the electron density increases, so the current increases and resistance decreases. Resistivity Resistivity is a property of a material, as different structures make for different conducting properties. It is defined as the resistance from one side to another of a 1m cube. R = ρL/A resistance = resistivity × length / cross-sectional area This relationship means that resistance is directly proportional to length, while it is inversely proportional to the cross-sectional area: R ∝ L R ∝ 1/A Resistivity is measured in ohm meters, Ωm Investigating Resistivity To determine resistivity, you need to work out the cross-sectional area using a digital caliper. Then, clamp the wire in place with a voltmeter in parallel and an ammeter in series with a fixed power supply. Vary the length of wire (use a ruler) and measure voltage and current to establish the resistance. Repeat at several different lengths. The resistivity of a metal increases with temperature. This is because charge is carried by moving electrons, but heat makes this motion harder as the ions in the structure vibrate more (see I-V characteristics of a filament lamp above). The resistivity of a semiconductor decreases with temperature, as more electrons are released for conducting (See LDR/Thermistors above). Energy & Power Power is the rate of transfer of energy, where one Watt is one joule of work done per second. P = IV power = current x voltage Using the equation V = IR, we can rewrite this power equation to other, often more useful forms: If power is energy per second, then we can use it to work out the electrical energy (work done): W = Pt work done = power x time Again, this can be rewritten in multiple ways: A joule is a very small unit of measuring energy. In fact, it is so small that it is rarely used commercially, because we typically use thousands or millions of joules at a time. Therefore, the Kilowatt-hour is used domestically: 1 kWh = 3.6 million joules This can quickly be calculated: 1 kW = 1000W 1 h = 60x60s = 3600s 1 kWh = 1000W x 3600s = 3,600,000 Ws = 3,600,00 J Energy Transferred (kWh) = Power (kW) x Time (h)

• Materials