Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

Questions on powers, the laws of indices, and exponential growth.

This question tests to see if students can recognise an alternating series and based on the alternating series test and the divergence test determine if it is convergent or divergent.

At the moment there is no example included with $b_n\rightarrow 0$ which isn't eventually monotonically convergent. 

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Predicting the probability of an unbiased coin landing on heads based on the results of previous throws.

Uses the $\chi^2$ test to see if there is any significant difference in preferences.

This demonstrates how to:

• Generate a fixed number of observations of data in two categories.
• Display the data in a table, with row and column totals.
• Ask the student to enter the data in a spreadsheet which is automatically marked.

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".

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 $a,\;b,\;c,\;d$.

$I$ compact interval, $g:I\rightarrow I,\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$

$I$ compact interval, $g:I\rightarrow I,\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$

Question asks students to find the time taken for an object thrown vertically upward from a platform to reach the ground. Set up randomly chooses environment to be on Earth, Mars or the Moon and uses appropriate acceleration due to gravity. The initial velocity of the body and height of the platform above the ground are randomly selected. In this version students are expected to write up their working and submit it seperately to the Numbas question. Students are expcted to recognise that only the positive solution has physical significance.

Calculating the gradient of a quadratic equation at a specific point and finding the stationary point (maximum) in a contextualised problem. 

The subtraction algortihm using the borrow and pay back method with integers.

Subtracting a decimal with 3 decimal places from a decimal with 2 or 3 decimal places. borrowing is necessary. This was modified from a subtraction question using integers with each number divided by 1000 so the variables have names referring to ones, tens, hundreds etc.

There's one question, which you have to get right 5 times in a row. If you get it wrong, you have to start again.

This makes more sense if the question is randomised!

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

Recovering original function given some information such as derivative and value at some point.

The student is shown a table with rows for each class in a school, and two columns of numbers: books, and children. They have to work out the column totals, and the number of books per child, and enter them in the table.

Dane są średnia i odchylenie standardowe z próby. Należy przeprowadzić test istotności dla średniej i zweryfikować postawioną hipotezę.

Subtracting a decimal with 3 decimal places from a decimal with 2 or 3 decimal places. borrowing is necessary. This was modified from a subtraction question using integers with each number divided by 1000 so the variables have names referring to ones, tens, hundreds etc.

The subtraction algortihm using the borrow and pay back method with integers.

The student is shown an exponential graph and asked to evaluate the function at some given value.

They are also asked whether or not the model is valid for all real inputs, but they only give a yes/no response. The reasoning is explained in the advice but is not required from students.

Students are given an exponential equation and asked to evaluate it at two points.

The constants in the exponential are randomised.

Solving a separable differential equation that describes the population growth over time with a known initial condition to calculate the population after $n$ years. 

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.

Determinant of n x m matrix by Laplace Expansion across top row.