For a sample of size n from a normal distribution, given mean of the sample mean and the standard deviation , find the t-statistic corresponding to a null hypothesis $\mu=m$ and a given confidence level. Check if the result is significant at this level.

A few simple functions are provided of the form ax, x+b and cx+d. Values of the functions, inverses and compositions are asked for. Most are numerical but the last few questions are algebraic.

Laplace from tables: e^(at), cos(bt), sin(bt).

rebelmaths

No description given

No description given

No description given

No description given

Laplace of constants and powers of t

rebelmaths

No description given

No description given

No description given

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

Warning: may take up to 60 seconds to load question!

Students are given six graphs, corresponding to curves $\gamma(t)$. They must match each with its signed curvature function, $\kappa(t)$.

The graphs are generated by calculating $\theta(t)=\int \kappa(t) \mathrm{d}t$ (by hand: these are given to the question as functions of a variable '#', in string form), and solving $x^{\prime}=\cos(\theta(t)-\theta(0))$ and $y^{\prime}(t)=\sin(\theta(t)-\theta(0))$ numerically (using the RKF method) with a JavaScript extension.

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

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

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

Student is asked to sketch $f(x)=\log_2(x)$, by plotting several points and selecting the correct graph.

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

15 questions based on module so far. 

No description given

No description given

Determine the mechanical advantage of a pair of compound lopping shears.

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.

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.

Find the solution of $\displaystyle \frac{dy}{dx}=\frac{1+y^2}{a+bx}$ which satisfies $y(1)=c$

rebelmaths

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

Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.

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

No description given

Converting Radians to Degrees

rebelmaths