The student must calculate the number of digits a given decimal number would have when written in a different base. Alternative answers catch some common mal-rules and give appropriate feedback.

Based on table 2 from "diagnosing student errors in e-assessment questions" by Philip Walker, D. Rhys Gwynllyw and Karen L. Henderson.

Numbas representation of the Metacentric Height fluid mechanics laboratory for first year Engineering students. 

Students should answer this question after completing the laboratory and the associated Excel template, including the relevant plots. 

The first question tests their capacity to correctly calculate metacentric heights, the remaining questions evaluate their graph interpretation skills. 

A function of the form (ax+b)/(x+c) is plotted. Student is asked to calculate the shaded area. Area is both above and below the x-axis.

Given  $P(A)$, $P(A\cup B)$, $P(B^c)$ find $P(A \cap B)$, $P(A^c \cap B^c)$, $P(A^c \cup B^c)$ etc..

Question is to calculate the area bounded by a cubic and the $x$-axis. Requires finding the roots by solving a cubic equation. Calculator question

Apply formulas to calculate the areas of various shapes.

Graphs are given with areas underneath them shaded. The student is asked to select the correct integral which calculates its area.

Use the specific heat capacity of an object and the heat released to calculate the final temperature.

Custom marking used to allow for partial credit for common errors (mal rules)

Thermochemistry Revision Sheet Q7

Q1. True/false questions about basic facts.

Q2 and Q3. Velocity-time graphs are given with areas underneath them shaded. The area of the shaded regions are given. From this, changes in position, distances are to be calculated.

Use the specific heat capacity of an object and the change in temperature to calculate the heat added to that object.

Uses Custom marking to allow partial credit for common student errors (mal rules).

Thermochemistry Revision Sheet Q 5

Calculate the local extrema of a function ${f(x) = e^{x/C1}(C2sin(x)-C3cos(x))}$

The graph of f(x) has to be identified.

The first derivative of f(x) has to be calculated. 

The min max points have to be identified using the graph and/or calculated using the first derivative method.  Requires solving trigonometric equation

Given mass and volume of an object in SI units, calculate its density. The units are given at different orders of magnitude.

Just showing how to use the stdev function from the stats extension to calculate the standard deviation of a list of numbers.

Calculate a repeated integral of the form $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$

The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.

Divergence of vector fields.

Find $\displaystyle \int_{\Gamma} \left(x+y \right)\;dx+\left(y-x\right)\;dy,\;\Gamma$ is the line from $(0,0)$ to $(a,b)$.

No description given

Given mean and sd of 1000 sample returns on a scale of 1 to 7 together with a given score, find the z-score.

Also find the 95% confidence interval for the population mean.

Given data on population mean and population standard deviation and three sampling sizes, calculate the probabilities that the sample means are within a specified distance from the population mean.

Given sum of sample from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$. Find MLE of $\mu$ and one of four functions of $\mu$.

Arrivals given by exponential distribution, parameter $\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\theta$.