Cambridge Maths Academy

Further statistics 1 Table of contents Probability density functions Moment generating functions Moments Edexcel FS1 Ch Exercise 1. Probability density functions The probability dentity function for a normal distribution, $ X \sim \textrm N \left( \mu, \sigma^2 \right) $, is $$ \begin{align} \textrm f(x) = \frac{1}{ \sqrt{2\pi} \sigma} \textrm e^{ - \frac12 \left( \frac{ x - \mu }{ \sigma } \rig..

Further statistics 1 Table of contents Probability mass function Probability generating function Moment generating function Moments Using moment generating function Central moments Standardised moments Edexcel FS1 Ch Exercise (under construction) 1. Probability mass function The probability mass function for a geometric distribution, $ X \sim \textrm{Geo}(p)$, is $$ \begin{align} \mathbb P(x) = ..

Further mechanics 1 Table of contents Hooke's law Units Modulus of elasticity ($\lambda$) Examples Exercise 3A 1. Hooke's law When an elastic string or spring is stretched, the tension, $T$, produced is proportional to the extension, $x$. $$ \begin{align} T \propto x \\ \\ \Rightarrow \qquad T = kx \end{align} $$ where $k$ is a constant. The constant $k$ depends on the unstretched length of the ..

Further mechanics 1 Table of contents Principle of conservation of momentum Examples Edexcel FM1 Exercise 1B 1. Principle of conservation of momentum We can solve problems involving collisions using the principle of conservation of momentum. When two bodies collide, by Newton's 3rd law - the principle of action and reaction - each one exerts an equal and opposite force on the other. They are in ..

Further mechanics 1 Table of contents Momentum = Linear momentum Units First example Impulse The impulse-momentum principle More examples Exercise 1A 1. Momentum = Linear momentum The momentum $p$ of a body of mass $m$ which is moving with velocity $v$ is: $$ \boxed{ \begin{align} \textrm{momentum} &= \textrm{mass} \times \textrm{velocity} \\ p &= mv \end{align} } $$ Mass is a scalar and velocit..

Core pure mathematics 2 Table of contents Introduction Examples Exercise 7D 1. Introduction We can use given boundary conditions to find a particular solution to a second-order differential equation. Since there are two arbitrary constants, we will need two boundary conditions to determine the complete particular solution. 2. Examples Example 1. Find $y$ in terms of $x$, given that $$ \begin{ali..

Core pure mathematics 2 Table of contents Complementary function + Particular integral An example Another example Yet another example An investigation: more general cases Exercise 7C 1. Complementary function + Particular integral Second-order differentail equations of the form $$ \begin{align} a \frac{ \textrm{d}^2y }{ \textrm{d}x^2 } + b \frac{ \textrm{d}y }{ \textrm{d}x } + c y = \textrm f(x)..

Core pure mathematics 2 Table of contents First example Arbitrary constants = constants of integration Linear and non-linear differential equations Finding the general solution with the auxiliary equation More examples Exercise 7B 1. First example A second-order differential equation contains second derivatives. Example 1. Find the general solution to the differential equation $$ \begin{align} \..