Content created by Newcastle University

Linear program described in words. Student must write out constraints as equations in standard form, then identify the optimal solution in a finished simplex tableau.

Express $\displaystyle a \pm  \frac{c}{x + d}$ as an algebraic single fraction.

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..

Compute the information rate and minimum distance of some binary codes.

Curl and divergence of a vector field.  Determine whether the vector field is irrotational or solenoidal.

Given random set of data (between 13 and 23 numbers all less than 100), find their stem-and-leaf plot.
This version of the question asks for 10 fields to be filled rather than the full 25, although the question statement asks for 25. I am sure that the first version asked for all 25.

Correct an error in a received message which was encoded using Hamming's square code

Decode a message of two codewords encoded using Hamming's [7,4] code, with at most one error per codeword.

Find upper and lower bounds on the number of codewords in three maximal codes given their codeword lengths and minimum distances.

Uses Hamming, Singleton and Gilbert-Varshamov bounds.

List all vectors in a spanning set.

(repeated 3 times)

Write down a lexicographic parity check matrix for a Hamming code and correct two received codewords.

Write down the lexicographic parity check matrix for a Hamming code, and correct two received codewords.

Compute the word length, minimum distance and dimension of some given Hamming codes.

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

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.

Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\boldsymbol{r=r_0+\lambda a+\mu b}$.

The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.

When are vectors $\boldsymbol{v,\;w}$ orthogonal?

Part b) is not answered in Advice, the given solution is for a different question.

Determine if various combinations of vectors are defined or not.

Find angle between plane $\Pi_1$, given by three points, and the plane $\Pi_2$ given in Cartesian form.

The calculation of $cos(\alpha)$ at the end of Advice has fractionNumbers switched on and so the result is presented as a fraction, which can be misleading. Best if calculation is followed through without using fractionNumbers.

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.

The correct solution is given, however the accuracy of 0.001  is not enough as in some cases answers near to the correct solution are also marked as correct.

Calculation of the length and alternative form of the parameteric representation of a curve.

Parametric form of a curve, cartesian points, tangent vector, and speed.

Parametric form of a curve, cartesian points, tangent vector, and speed.

Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.

Three 3 dim vectors, one with a parameter $\lambda$ in the third coordinate. Find value of $\lambda$ ensuring vectors coplanar. Scalar triple product.

(Green’s theorem). $\Gamma$ a rectangle, find: $\displaystyle \oint_{\Gamma} \left(ax^2-by \right)\;dx+\left(cy^2+px\right)\;dy$.

Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$

Given a pair of 3D position vectors, find the vector equation of the line through both.  Find two such lines and their point of intersection.