Equivalent fractions, simplifying, adding, subtracting, multiplying and dividing fractions. Converting between mixed numbers and improper fractions.

Students explore the relationship between length and area of a rectangle.

The perimeter of the rectangle is randomised. Students are given 11 different lengths, and asked to compute rectangle width and area for each. They are then asked to graph the function, identify it as a parabola, and estimate the maximum value.

Students are given the bearings and distances of 2 consecutive straight line walks. They are asked to find the distance from the starting point to the endpoint. They are given a diagram to assist them.

The bearings and distances are randomised (any bearing, distances between 1.1 and 5.). Bearings can be given as either compass bearings or true bearings.

Students are given the bearings and distances of 2 consecutive straight line walks. They are asked to find the distance from the starting point to the endpoint. They are given a diagram to assist them.

The bearings and distances are randomised (any bearing, distances between 1.1 and 5.). Bearings can be given as either compass bearings or true bearings.

This question assesses the students ability to calculate and convert between different types of compound units, including rates of pay, speed and unit pricing.

Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.

Calculating the area enclosed between a linear function and a quadratic function by integration. The limits (points of intersection) are not given in the question and must be calculated.

Paired t-test to see if there is a difference between times taken to complete a task.

Convert figures for car engine sizes between cc (cm^3), litres, and m^3.

Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\pi$ and $\pi$ and careful with quadrants!

Calculate the vector product between two vectors.

Given three 3-dimensional vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$, calculate the scalar product between $\mathbf a$ and $\mathbf b$, the angle between $\mathbf a$ and $\mathbf b$, and $\mathbf a (\mathbf b \cdot \mathbf c)$,

Given three 2-dimensional vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$, calculate the scalar product between $\mathbf a$ and $\mathbf b$, the angle between $\mathbf a$ and $\mathbf b$, and $\mathbf a (\mathbf b \cdot \mathbf c)$,

Given the coordinates of three 2-dimensional points $A$, $B$ and $C$, find the vectors $\vec{AB}$, $\vec{AC}$ and $\vec{CB}$.

Calculating the area enclosed between a cosine function and a quadratic function by integration. The limits (points of intersection) are given in the question.

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

This question will test the ability of test taker to distinguish between solving mean and solving weighted mean problem

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".

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 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.

The student is shown two labelled graphs. They are asked:

• Number of vertices in each
• Number of edges in each
• Degree sequences for each
• Is there an isomorphism between them? If so, write one.

The number of vertices is always equal, so this is a gimme.

If the edges or degree sequences are different, the student is expected to realise that there cannot be an isomorphism.

If these values are the same, then there will be an isomorphism (else the question is a bit too tricky).

Numbas expects a particular isomorphism, but there may be more than one, all of which would be accepted.

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). 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. 

Short question in recognising errors in binary strings transmitted over a binary symmetric channel, as well as calculating probabilities of these errors occuring. Randomised 7-bit strings between 64 and 128, randomised probability between 0.05 and 0.1. Randomised error pattern.

The student is given a number in base 10 and asked to write it in a given base, between 2 and 16. The number has at most 3 digits in the other base.

Until it's possible to derive the expected answer for a part in the marking algorithm (see the issue tracker), this question has "show expected answer" turned off, because it just shows the base 10 number.

Practice of conversion between SI units of mass, volume & length.

5 questions on vectors. Scalar product, angle between vectors, cross product, when are vectors perpendicular, combinations of vectors defined or not.

Used for LANTITE preparation (Australia). MG = Measurement & Geometry strand. Two people walk away from a central flagpole, in different directions. The multiple choice question asks for the angle between them. There are seven potential scenarios.

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given trig function applied to tides. students need to find max depth, time of low tide and times between which a boat of given depth can use the port.