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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".

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.

This question uses a Geogebra applet to solve a linear program with two variables using the graphical method. It contains three steps:

1. Construct the feasible area (polygon) by adding the constraints one by one. The students can see what happens when the constraints are added.
2. Add the objective function, and the level set of the objective value is shown, as well as its (normalised) gradient.
3. Compute the optimal solution by moving the level set of the objective around.

Calculate the marginal and average cost for a given cost function. Find the corresponding startup/shutdown price.
Maximize the profit function at a given price.

Drag points on a graph to the given Cartesian coordinates. There are points in each of the four quadrants and on each axis.

Convert numbers greater than 1 into standard form/scientific notation.

Find the equation of the line that passes through given points

Given a randomised square root function select the possible ways of writing the domain of the function.

Extra practice on some basic algebra skills, including solving linear equations. You can try as many times as you like and also generate new versions of the questions for extra practice.

Decide whether statements about square and cube numbers are always true, sometimes true or never true.

German translation of https://numbas.mathcentre.ac.uk/question/22768/always-sometimes-or-never-square-and-cube-numbers/ von Stanislav Duris.

Schreibe $\displaystyle \frac{a} {b + \frac{c}{d}}$ als einen gekürzten Bruch $\displaystyle \frac{p}{q}$ mit ganzen Zahlen $p$ und $q$.

Angepasste, übersetzte und erweiterte Version von https://numbas.mathcentre.ac.uk/question/11701/simplifying-fractions/ von Newcastle University Mathematics and Statistics.

Some questions for our open day, to give students a first taste of Numbas.

The student is given the equations of a line and a circle, and has to find the coordinates of the points of intersection. They're always at integer coordinates.

Expand $(az^2+bz+c)(pz+q)$.

The student is shown a set of axes with three lines. They must move the lines so they match the given inequalities, then move a point inside the region satisfied by the inequalities.

This is a copy of a question from the Numbas demos project, with references to the editor removed.

The student is shown a plot of a mystery function. They can enter values of $x$ check, within the bounds of the plot.

They're asked to give the formula for the function, and then asked for its value at a very large value of $x$.

A plot of the student's function updates automatically as they type. Adaptive marking is used for the final part to award credit if the student gives the right value for their incorrect function.

Solving a system of simultaneous congruences using the Chinese Remainder Theorem.

An explore mode question which allows you to write a solution straight away, or break it down into steps.

After solving a system of three congruences, you can ask for a new problem with more congruences.

Solve a problem using a linear equation.

Finding the current in a simple series RL circuit.

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A practice test to get used to the NUMBAS environment.

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Given the number of international students enrolled on a course of $n$ students, calculate the percentage of 'home' students.