Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

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

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Simple probability question. Counting number of occurences of an event in a sample space with given size and finding the probability of the event.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

Intorduces strong induction and uses it to verify the solutions of a second order linear recurrence.

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

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

A quadratic is given and sketched. Based on the sketch, task is to determine the number of solutions to the equation $f(x)=0$.

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

Find the solution of $\displaystyle x\frac{dy}{dx}+ay=bx^n,\;\;y(1)=c$

rebelmaths

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

Find the solution of a first order separable differential equation of the form $(a+x)y'=b+y$.

Find the solution of a first order separable differential equation of the form $axyy'=b+y^2$.

Equations which can be written in the form

\[\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(y),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)f(y)\]

can all be solved by integration.

In each case it is possible to separate the $x$'s to one side of the equation and the $y$'s to the other

Solving such equations is therefore known as solution by separation of variables

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx $ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

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

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

Quiz covering basic arithmetic with complex numbers and solving roots for a quadratic with complex solutions

Find the solution of a first order separable differential equation of the form $axyy'=b+y^2$.

Using trig identities to find solutions to equations

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

Independent events in probability. Choose whether given three given pairs of events are independent or not.

Short Video Lectures, followed by questions and video solutions

Some impossible-looking questions about quadratic equations which can be solved with a bit of thinking.

Evaluate $\int_0^{\,m}e^{ax}\;dx$, $\int_0^{p}\frac{1}{bx+d}\;dx,\;\int_0^{\pi/2} \sin(qx) \;dx$. 

No solutions given in Advice to parts a and c.

Tolerance of 0.001 in answers to parts a and b. Perhaps should indicate to the student that a tolerance is set. The feedback on submitting an incorrect answer within the tolerance says that the answer is correct - perhaps there should be a different feedback in this case if possible for all such questions with tolerances.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.