A set of MCQ designed to help Level 2 Engineering students prepare/practice for the on-line GOLA test that is used to assess the C&G 2850, Level 2 Engineering, Unit 202: Engineering Principles.

Tags: algebra, equations (linear), equations (quadratic)

Last updated Sep 2019

In class test contributes about 30% of your assesment work. Make sure you write your REG number.

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Solve linear equations with unkowns on both sides. Including brackets and fractions.

In this question the students have to solve a linear recurrence of order 1. The sequence is asked in recurrence form and the goal is to find its closed form.

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

Linear combinations of $2 \times 2$ matrices. Three examples.

The CSS preamble adds a vertical line down the input for part b, to separate the two parts of the matrix.

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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".

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.

This question uses a Geogebra applet to solve a linear program with two variables using the graphical method. It contains three steps:

1. Construct the feasible area (polygon) by adding the constraints one by one. The students can see what happens when the constraints are added.
2. Add the objective function, and the level set of the objective value is shown, as well as its (normalised) gradient.
3. Compute the optimal solution by moving the level set of the objective around.

Find the equation of the line that passes through given points

Extra practice on some basic algebra skills, including solving linear equations. You can try as many times as you like and also generate new versions of the questions for extra practice.

The student is given the equations of a line and a circle, and has to find the coordinates of the points of intersection. They're always at integer coordinates.

The student is shown a set of axes with three lines. They must move the lines so they match the given inequalities, then move a point inside the region satisfied by the inequalities.

Solve a problem using a linear equation.

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

A random dataset given by a linear function with noise (gradient and y-intercept of the linear function are randomised as is distribution of x values).

Identifying gradient and $y$-intercept from $y=mx+b$.

GIve a random linear program and ask the students to convert it to canonical form.

Basic solving of linear equations

Solving linear equations