Solving a separable differential equation that describes the rate of decay of radioactive isotopes over time with a known initial condition to calculate the mass of the isotope after a given time and the time taken for the mass to reach $m$ grams. 

Decay Constant - Radioactivity - Nuclear Power (nuclear-power.com)

A combination of tasks: checking which matrix products exist, calculating some of these products, calculating transpose matrices. Comparing product of transpose with transpose of product. Experiencing associativity of matrix multiplication. Not much randomisation, only in which matrix product is computed as second option.

Comprehensive solution written out in Advice.

Differentiate the following functions: $\displaystyle x ^ n \sinh(ax + b),\;\tanh(cx+d),\;\ln(\cosh(px+q))$

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The student is shown a diagram containing a line between two points. They're given a gradient and $y$-intercept.

They must manipulate the line or the points so that the line has the given gradient and $y$-intercept.

The student is shown a diagram containing a single point at the origin. They must move the point to the given integer coordinates.

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A demo of the number entry part and its options.

Calculate reactions and shear and bending moment at a point for an overhanging beam with a constant or uniformly varying distributed load.

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 $(x=1)$.

Calculate the dot products of (a) two 2D vectors and (b) two 3D vectors.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

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 $p,\;q$.

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.

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.

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

Find the determinant of  a random (3x3) matrix.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Using indices rules to write powers as negative powers

Writing negative powers as fractions

Writing fractins as negative powers

Using indices rules to write expressions as negative powers.

Using indices rules to write fractions as negative powers.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Using rules for roots and powers to write expressions with only powers