Application of the Poisson distribution given expected number of events per interval.

Finding probabilities using the Poisson distribution.

Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Manipulate surds and rationalise the denominator of a fraction when it is a surd.

Application of the Poisson distribution given expected number of events per interval.

Finding probabilities using the Poisson distribution.

rebelmaths

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Using Pythagoras' Theorem to find a missing side. Illustrated using simple Eukleides diagram

rebelmaths

Calculating the square root of a complex number using De Moivre.

Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.

Intorduction to using the defnition to prove simple stamtements.

Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

Manipulate surds and rationalise the denominator of a fraction when it is a surd.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Two quadratic graphs are sketched with some area beneath them shaded. Question is to determine the area of shaded regions using integration. The first graph's area is all above the $x$-axis. The second graph has some area above and some below the $x$-axis.

Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$.  Find both roots of the equation $q(x)=0$.

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

A graph shows both the speed and acceleration of a car. Identify which line corresponds to which measurement, and calculate the acceleration during a portion of time.