Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.

Integration by parts.

An application of quadratic functions based on the Golden Gate Bridge in San Francisco, USA. Student is given an equation representing the suspension cable of the bridge and asked to find the width between the towers and the minimum height of the cable above the roadway. Requires and understanding of the quadratic function and where and how to apply correct formulae.

Student needs to solve a quadratic equation to calculate time taken for a diver to hit the water after diving from a diving board. Height of the board and initial upward velocity of the diver are randomly generated values. student needs to know that surface of the water is height 0, and only positive root of quadratic has physical meaning. Question is set to always give one positieve and one negative root.

The student has to enter three different letters of the alphabet in the three gaps. Their answer is marked as a set: repeated answers only count as one answer.

Each gap has the same custom marking algorithm which marks that gap as correct if the student's answer is in the set of acceptable answers.

Graphing $y=ab^{\pm x+d}+c$

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$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

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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.

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.

Solve a problem using a linear equation.

Finding the current in a simple series RL circuit.

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