# A Level Maths Equations

## A Level Maths Equations Revision

Pure

Arithmetic Series

$S_{n} = \dfrac{1}{2}n(a + l)\\$

$S_{n} = \dfrac{1}{2}n(2a + (n - l)d)\\$

Geometric Series

$S_{n} = \dfrac{a(1 - r^n)}{1 - r}\\$

$S_{\infty} = \dfrac{a}{1 - r}$ for $|r\ < 1\\$

Binomial Expansion

$(a + b)^n = a^n + ^nC_1 a^{n-1} b + ^nC_2 a^{n-2} b^2 + ... + ^nC_r a^{n-r} b^r + ... + b^n$

where $(n \in \N)$ and $^nC_r = \begin{pmatrix}n\\r\end{pmatrix} = \dfrac{n!}{r (n-r)!}$

$(1 + x)^n = 1 + nx + \dfrac{n(n - 1)}{1\times2} x^2 + ... + \dfrac{n(n - 1) ... (n - r + 1)}{1\times2\times...\times r} + ...$

for $|x| < 1$, $n \in \R$

Curved Surface Area

Surface area of sphere $= 4\pi r^2$

Area of curved surface of cone $= \pi r \times$slant height

Exponentials and Logarithms

$\log_a (x) \dfrac{\log_b (x)}{\log_b (a)}\\$

$e^{x\ln{a}} = a^x$

Exponentials and Logarithms

$\sin(A\pm B)\equiv\sin(A)\cos(B)\pm\cos(A)\sin(B)\\$

$\cos(A\pm B)\equiv\cos(A)\cos(B)\mp\sin(A)\sin(B)\\$

$\tan(A\pm B)\equiv\dfrac{\tan(A)\pm\tan(B)}{1\mp\tan(A)\tan(B)} (A\pm B\neq(K+\dfrac{1}{2})\pi)\\$

$\sin(A)\pm\sin(B)=2\sin(\dfrac{A\pm B}{2})\cos(\dfrac{A\mp B}{2})\\$

$\cos(A)+\cos(B)=2\cos(\dfrac{A+B}{2})\cos(\dfrac{A-B}{2})\\$

$\cos(A)-\cos(B)=-2\sin(\dfrac{A+B}{2})\sin(\dfrac{A-B}{2})\\$

Small Angle Approximations

$\sin(\theta)\approx\theta\\$

$\cos(\theta)\approx 1-\dfrac{1}{2}\theta ^2\\$

$\tan(\theta)\approx\theta\\$

Differentiation

First principles: $f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}\\$

Quotient rule: $\dfrac{d}{dx}\left(\dfrac{u(x)}{v(x)}\right)=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}$

 $\textbf f(x)$ $\textbf f'(x)$ $\tan(kx)$ $k\sec^2(kx)$ $\sec(kx)$ $k\sec(kx)\tan(kx)$ $\cot(kx)$ $-k\cosec^2(kx)$ $\cosec(kx)$ $-k\cosec(kx)\cot(kx)$

Integration

$\int u\dfrac{dv}{dx}dx = uv - \int v\dfrac{du}{dx}dx\\$

 $\textbf f(x)$ $\int \textbf f(x) dx$ $\sec^2(kx)$ $\dfrac{1}{k}\tan(kx) + c$ $\tan(kx)$ $\dfrac{1}{k}\ln |\sec(kx)|+ c$ $\cot(kx)$ $\dfrac{1}{k}\ln |\sin(kx)|+ c$ $\cosec(kx)$ $-\dfrac{1}{k}\ln |\cosec(kx)|+\cot(kx)| + c$ $\dfrac{1}{k}\ln |\tan(\dfrac{1}{2}kx)| + c$ $\sec(kx)$ $-\dfrac{1}{k}\ln |\sec(kx)|+\tan(kx)| + c$ $\dfrac{1}{k}\ln |\tan(\dfrac{1}{2}kx + \dfrac{\pi}{4})| + c$

Numerical Methods

Trapezium rule: $\int_{b}{a}y dx = \dfrac{1}{2} h (y_{0} + y_{n} + 2(y_{1} + y_{2} +...+ y_{n-1}))$ where $h=\dfrac{b-a}{h}$

Newton-Raphson iteration for solving $f(x)=0$ is $x_{n+1}=x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}$

Statistics

Measures of Variation

Interquartile Range $=$ IQR $= Q_{3} - Q_{1}\\$

$S_{xx} = \sum (x_{i} - \bar{x})^2 = \left(\sum x_{i}^2\right) - \dfrac{(\sum x_{i})^2}{n}\\$

Standard deviation $=\sqrt{\text{variance}}\\$

$\sigma = \sqrt{\dfrac{S_{xx}}{n}}\\$

$\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}\\$

$\sigma = \sqrt{\dfrac{\sum (x-\bar{x})^2}{n}}\\$

$\sigma = \sqrt{\dfrac{\sum f(x-\bar{x})^2}{\sum f}}\\$

$\sigma = \sqrt{\dfrac{\sum fx^2}{\sum f} - \bar{x}^2}\\$

Probability

$P(A\cup B)=P(A)+P(B)-P(A\cap B)\\$

$P(A\cap B)=P(A)P(B|A)\\$

$P(B|A)=\dfrac{P(A\cap B)}{P(A)}\\$

$P(A')=1-P(A)\\$

$P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}\\$

Independent Events

$P(B|A)=P(B)\\$

$P(A|B)+P(A)\\$

$P(A\cap B)=P(A)P(B)\\$

Binomial Distribution

If $X\sim B(n,p)$, then:

$P(X=x) = \begin{pmatrix}n\\x\end{pmatrix} p^x(q-p)^{n-x}\\$

$\bar{X} = np\\$

$Var(X)=np(1-p)\\$

Normal Distribution

If $X\sim N(\mu ,\sigma ^2)$, then:

$\bar{X}\sim N\left(\mu , \dfrac{\sigma ^2}{n}\right)$

$\dfrac{\bar{X} - \mu }{\dfrac{\sigma }{\sqrt{n}}} \sim N(0,1)$

Mechanics

Motion in a Straight Line

$v=u+at\\$

$s=\dfrac{1}{2}(u+v)t\\$

$s=ut+\dfrac{1}{2}at^2\\$

$s=vt-\dfrac{1}{2}at^2\\$

$v^2=u^2+2as\\$

Motion in Two Dimensions

$\textbf v=\textbf u+\textbf at\\$

$\textbf s=\dfrac{1}{2}(\textbf u+\textbf v)t\\$

$\textbf s=\textbf ut+\dfrac{1}{2}\textbf at^2\\$

$\textbf s=\textbf vt-\dfrac{1}{2}\textbf at^2\\$

Newton’s Laws of Motion

$F = ma$

Friction

$F \leq \mu R$ where $\mu$ is the coefficient of friction

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