Solve for $x$: $\displaystyle ax+b = cx+d$

Find $\displaystyle \frac{a} {b + \frac{c}{d}}$ as a single fraction in the form $\displaystyle \frac{p}{q}$ for integers $p$ and $q$.

Solve for $x$: $\displaystyle \frac{s}{ax+b} = \frac{t}{cx+d}$

Solve for $x$: $\displaystyle \frac{a} {bx+c} + d= s$

What is the value of the expression given a choice of n?

Multiplication of  matrices of different sizes.

Cofactors Determinant and inverse of a 3x3 matrix.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Find the determinant of a $3 \times 3$ matrix.

aij  notation and definition of the order of a matrix.

Exercises in multiplying matrices. 

Dealing with expressions in online tests and in Matlab.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Multiplication of $2 \times 2$ matrices.

Addition, subtraction and multiplication by a scalar for 2 x 2 matrices.

Multiplication of  matrices of different sizes.

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Shows how to use the "openSteps" and "hideSteps" JavaScript events for parts.

Create a truth table for a logical expression of the form $a \operatorname{op} b$ where $a, \;b$ can be the Boolean variables $p,\;q,\;\neg p,\;\neg q$ and $\operatorname{op}$ one of $\lor,\;\land,\;\to$.

Match the equivalence with the rule

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

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

Inverse and division of complex numbers.  Four parts.

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".

Differentiate $f(x) = (a x + b)/ \sqrt{c x + d}$ and find $g(x)$ such that $ f^{\prime}(x) = g(x)/ (2(c x + d)^{3/2})$.

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Differential equations.

rebelmaths

First- and second order recurrence equations, homogenous and nonhomogenous