Example of an explore mode question. Student is given a 2x2 matrix with eigenvalues and eigenvectors, and is asked to decide if the matrix is invertible. If yes, second and third parts are offered where the student should give the eigenvalues and eigenvectors of the inverse matrix.

Assessed: remembering link between 0 eigenvalue and invertibility. Remembering link between eigenvalues and eigenvectors of matrix and its inverse.

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students are given a ratio, and the total number of dancers. They must calculate the number of girls (the larger share). The ratio, and total number are randomly generated.

Used for LANTITE preparation (Australia). MG = Measurement & Geometry strand. NC = Non Calculator. The start time for Period 1 (randomised) and the end time for Period 2 (randomised) are given, and students are asked to find the middle of those two times. 

Dividing Cubic expressions to find the quotient

Given the original formula the student enters the transformed formula

Find a regression equation.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

Based on Chapter 8, quite loosley.Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.

Given reach and frequency, calculate TARPS. Multiple choice question.

Find a percentage of a given population. Percentage and population are both randomised.

Find a percentage of a number (number of TARPs spent on breakfast programs), then divide by the number of programs to find the average number of TARPs per program.

Identify co-efficients, then use quadratic formula to find roots.

All set to be distinct and real. Some can be non-integer (0.5 steps)

Find a percentage of a given population. Percentage and population are both randomised.

Questions asking for the factoring of simple case (a=1)

Then using factors to find roots.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

This is a copy of a question by Ben Brawn. It replaces the JavaScript construction of the diagram with Eukleides.

This is a copy of a question by Ben Brawn. It replaces the JavaScript construction of the diagram with JSXGraph+JME.

Add three vectors by determining their scalar components, summing them and then resolving the rectangular components to find the magnitude and direction of the resultant

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

Draws a right angled triangle based on a length and an angle.

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Using trig identities to find solutions to equations

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

rebelmaths

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

Add three vectors by determining their scalar components, summing them and then resolving the rectangular components to find the magnitude and direction of the resultant

Using trig identities to find solutions to equations

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

rebelmaths

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.