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Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force acts in the positive $x$ and positive $y$ direction.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force is applied in the negative $x$ and negative $y$ direction.

Find the eigenfunctions of an irregular Sturm–Liouville problem and hence solve an inhomogeneous boundary value problem by writing the solution as an eigenfunction expansion.

Part (a): Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, give an expression for the function $f(x)$, where $f(x)=g(x)-2h(x)$. 

Part (b): Solve $f(x)=0$.

Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, solve the equation $g(x)=2h(x)$, giving all possible solutions for $x$. 

Using Gaussian Elimination to solve 3X3 systems of equations

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. 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.

Solve simple two step linear equations with feedback.

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

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

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

Solve for $x$: $\displaystyle ax+b = cx+d$

Questions to test if the student knows the inverse of an even power (and how to solve equations that contain a single power that is even). 

Questions to test if the student knows the inverse of an odd power (and how to solve equations that contain a single power that is odd). 

Solve for an angle which will result in equilibrium for a triangle subjected to three couples.   A trial and error solution is recommended.

Some quadratics are to be solved by factorising

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

Matrix multiplication. Has automatically generated "unresolved" matrix product to write in the solution, which is the interesting part of this implementation.

A variety of trigonometric equations which can be solved using inverse operations.

This question tests the student's ability to solve simple linear equations by elimination. Part a) involves only having to manipulate one equation in order to solve, and part b) involves having to manipulate both equations in order to solve. 

Solve quadratic equations (non-simple case, two real, discrete integer roots) using the formula.

Solve quadratic equations (non-simple case, two real, discrete roots) using the formula. Some, random, non integer roots.

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

These first need to be changed to standard form.

Then use factors to solve.

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

Then using factors to find roots.

Some quadratics are to be solved by factorising

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

solve trig equation that requires use of s^s+c^2=1. with worked solutions.

solve trig equation involving sin2x=2sinxcosx in a given interval