Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

An experiment is performed twice, each with $5$ outcomes

$x_i,\;y_i,\;i=1,\dots 5$ . Find mean and s.d. of their differences $y_i-x_i,\;i=1,\dots 5$.

A quadratic is and a graph of it is given. A tangent is also sketch. The equation of the tangent line is asked for.

For a sample of size n from a normal distribution and given the mean of the sample and the standard deviation of the distribution, find the MLE for the mean. Also the expected information and a confidence interval for the mean.

For a sample of size n from a normal distribution, given mean of the sample mean and the standard deviation , find the t-statistic corresponding to a null hypothesis $\mu=m$ and a given confidence level. Check if the result is significant at this level.

Recognising the equation of a circle.

Recognising that $(x-a)^2+(y-b)^2=r^2$ is a circle of radius $r$ with centre $(a,b)$ 

Recognising that $(x-a)^2+(y-b)^2=r^2$ is a circle of radius $r$ with centre $(a,b)$ 

Draws a triangle based on 3 side lengths.

Draws a triangle based on 2 angles and a side length.

Choosing whether given random variables are qualitative or quantitative.

Deciding whether or not three sampling methods are simple random sampling, stratified sampling, systematic or judgemental sampling. 

Finding probabilities from a survey giving a table of data on the alcohol consumption of males. This can be easily adapted to data from other types of surveys.

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

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A graphical approach to aiding students in writing down a formal proof of discontinuity of a function at a given point.

Uses JSXgraph to sketch the graphs and involves some interaction/experimentation by students in finding appropriate intervals.

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Quadratic factorisation that does not rely upon pattern matching.

A graphical introduction to the concept of even functions a symmery

Using the chain rule with polynomials and negative powers

Two shops each have different numbers of jumper designs and colours. How many choices of jumper are there?

Express $\log_a(x^{c}y^{d})$ in terms of $\log_a(x)$ and $\log_a(y)$. Find $q(x)$ such that $\frac{f}{g}\log_a(x)+\log_a(rx+s)-\log_a(x^{1/t})=\log_a(q(x))$

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Other method. Find $p,\;q$ such that $\displaystyle \frac{ax+b}{cx+d}= p+ \frac{q}{cx+d}$. Find the derivative of $\displaystyle \frac{ax+b}{cx+d}$.

Find $\displaystyle \frac{d}{dx}\left(\frac{m\sin(ax)+n\cos(ax)}{b\sin(ax)+c\cos(ax)}\right)$. Three part question.

Reconociendo que $ (x-a)^2 + (y-b)^2 = r^2 $ es un círculo de radio $r$ con el centro $(a, b) $

Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.