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

These practice questions cover:

• resolving axial forces in a truss structure
• stress & strain in a closed thin-walled pressure vessel (an example of 2D plane stress)
• use of Mohr's circle for determining principal stresses for a 2D plane stress case
• calculation of invariants for general 3D stress case, leading to von Mises stress and principal stresses

Two questions on solving systems of simultaneous equations.

Solving pair of simultaneous (linear) equations

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

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

Some quadratics are to be solved by factorising

Solving quadratic equations using a formula,

Solving cosx in given interval. With random variation and worked solutions.

Solving sinx in given interval. With random variation and worked solutions.

Solving tanx in given interval. With random variation and worked solutions.

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

Solving 2nd order differential equation for pendulum, with and without damping.

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Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Introductory question where the numbers come out quite nice with not much dividing. Set-up is meant for formative assessment. Adapated from a question copied from Newcastle.

Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Set up so that sometimes it has infinitely many solutions (one free variable), sometimes unique solution. Scaffolded so meant for formative. The variable d determines the cases (d=1: unique solution, d-0: infinitely many solutions). The other variables are set up so that no entries become zero for some randomisations but not others.

Adding and subtracting vectors of random size, including resolving brackets. Advice (i.e. solution) has conditional visibility to show only the correct size.

Calculating with vectors of random size, including resolving brackets. Advice (i.e. solution) has conditional visibility to show only the correct size.

Calculating with vectors in \(\mathbb{R}^4\), including resolving brackets. The fixed vector size is so that a test is fair to all students.

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Quiz covering basic arithmetic with complex numbers, solving a quadratic with complex solutions and converting to/from polar and rectangular forms.

Basic differentiation techniques, including evaluations, brackets/fractions, solving equation, and an application

Straightforward solving linear equations question.

Adapted from 'Simultaneous equations by elimination 1 with parts' by Joshua Boddy.

Straightforward solving linear equations question

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.