Integrating a polynomial functions which describe the rate of change of a population over time to find and use an equation that describes the total population according to time.

A demonstration of the random person extension, which picks representative names of people.

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students must do a simple interest calculation (geven principal, rate and months, all of which are randomly generated) then subtract their answer from the supplied compound interest amount to find the difference.

Demonstrating a feedback loop in a Numbas part: which of the required properties does your answer satisfy?

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.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force acts in the positive $x$ and positive $y$ direction.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force is applied in the negative $x$ and negative $y$ direction.

This is out of date. There is a new "getting statrted" exam at https://numbas.mathcentre.ac.uk/exam/21108/getting-started/

6 questions which introduce the user to the Numbas system.

Identifying which graph shows the function $y=\frac{x-a}{x+b}$

No description given

Use the BODMAS rule to determine the order in which to evaluate some arithmetic expressions. 

Divisor is single digit. There is a remainder which we express as a decimal by continuing the long division process. Rounding is required to one decimal place. The working suggests determining the second decimal place so the student knows whether to round up or down.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

No description given

A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).

Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.

English sentences which are propositions are given and for each the appropriate proposition  involving quantifiers is to be chosen. 

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

Determine the single force which is equivalent to a force and a couple.

Solve for an angle which will result in equilibrium for a triangle subjected to three couples.   A trial and error solution is recommended.

Student finds a basis for kernel and image of a matrix transformation. Any basis can be entered; there is a custom marking algorithm which checks if it is a correct basis.

There are options to adjust this question fairly easily, for example to get different variants for practice and for a test, by changing the options in the "pivot columns" in the variables. You should be careful to think about and test your pivot options, as some are easier or harder than others, and some don't work very well.

Matrix multiplication. Has automatically generated "unresolved" matrix product to write in the solution, which is the interesting part of this implementation.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Randomized size of the matrices and formula.

The interesting part about the implementation is the way the output is generated for "Advice".

The student is shown two labelled graphs. They are asked:

• Number of vertices in each
• Number of edges in each
• Degree sequences for each
• Is there an isomorphism between them? If so, write one.

The number of vertices is always equal, so this is a gimme.

If the edges or degree sequences are different, the student is expected to realise that there cannot be an isomorphism.

If these values are the same, then there will be an isomorphism (else the question is a bit too tricky).

Numbas expects a particular isomorphism, but there may be more than one, all of which would be accepted.

Consider a binary tree with $2^n$ nodes.

We give generators for the isomorphisms of the tree: at each non-leaf vertex, swap the branches descending from that node.

• What is the action of a given word?
• Write a word which produces the required isomorphism
• Which generators commute?
• What is the order of each generator?
• Write one of the (non-root) generators in terms of the others.
• Which permutations of the leaves are possible?
• What is the order of the group of isomorphisms?
• What is the order of the quotient group obtained by identifying all the leaves of one branch?

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

Given a total budget and the cost of each sale for three different scenarios, which scenario produces the most sales?

A function which renders the factorisation of a number in LaTeX.

Four sinusoidal graphs are given. Student should select the one which is sine and cosine.

Four sinusoidal graphs are given. Student should select the one which is sine and cosine.