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Two questions testing the application of the Sine Rule when given two angles and a side. In this question, the triangle is always acute.

A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).

Dealing with expressions in online tests and in Matlab.

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

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

Match the equivalence with the rule

Shear and bending moment diagram for a beam loaded with an arbitrary parabolic distributed load described with a loading function.  Integrate to find reactions and equations for shear and bending moment.

Simple trig equations with radians

This question demonstrates a few ways of interacting with a Venn diagram drawn using JSXGraph.

Two questions testing the application of the Cosine Rule when given two sides and an angle. In these questions, the triangle is always acute and both of the given side lengths are adjacent to the given angle.

Calculate the centroidal moment of intertia of a rectangle with a excentric rectangular hole.

Compute the experimental probability of a particular score on a die given a sample of throws, and compare it with the theoretical probability.

The last part asks what you expect to happen to the experimental probability as the sample size increases.

Simply supported Beam with three concentrated loads.

Calculate the distance between two points along the surface of a sphere using the cosine rule of spherical trigonometry. Context is two places on the surface of the Earth, using latitude and longitude.

The question is randomised so that the numerical values for Latitude for A and B will be positive and different (10-25 and 40-70 degrees). As will the values for Longitude (5-25 and 50-75). The question statement specifies both points are North in latitude, but one East and one West longitude, This means that students need to deal with angles across the prime meridian, but not the equator.

Students first calculate the side of the spherical triangle in degrees, then in part b they convert the degrees to kilometers. Part a will be marked as correct if in the range true answer +-1degree, as long as the answer is given to 4 decimal places. This allows for students to make the mistake of rounding too much during the calculation steps.

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

The constants in the exponential are randomised.

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".

Consider a binary tree with $2^n$ nodes.

We give generators for the isomorphisms of the tree: at each non-leaf vertex, swap the branches descending from that node.

• What is the action of a given word?
• Write a word which produces the required isomorphism
• Which generators commute?
• What is the order of each generator?
• Write one of the (non-root) generators in terms of the others.
• Which permutations of the leaves are possible?
• What is the order of the group of isomorphisms?
• What is the order of the quotient group obtained by identifying all the leaves of one branch?

This test will assess a students ability to work with piecewise and composite functions.

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Complementary function/particular integral for forced simple harmonic motion (non-resonant)

A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).

Simple trig equations with radians