Newcastle University Mathematics and Statistics

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Modulus and argument of a single complex number $z=z_1/z_2$, where $\mathrm{Re}(z_1)=\mathrm{Im}(z_1)$ and $\mathrm{Re}(z_2)=-\mathrm{Im}(z_2)$.

Calculation of modulus, argument, multiplication by complex conjugate, given two complex numbers.

Express $\displaystyle ax+b+  \frac{dx+p}{x + q}$ as an algebraic single fraction. 

Solving simple linear equations in $\mathbb{Q}$ and $\mathbb{Z}_n$ for $n= 13, \;17$ or $19$.

Determine the correct parametric representation of a given curve.  Curve is randomly chosen from a set of 20.

The graph of the curve was not displayed on my machine.

No description given

Use the simplex method to solve a linear program.

Questions on Kruskal-Wallis and Friedman tests.

No description given

Friedman test, 5 subjects, 4 treatments.

Given linear programming problems in standard form, write down the dual problem.

Questions used in a university course titled "Coding theory".

Ready to use as all 20 questions are stated as ready to use.

Given $f(x)=(x+b)^n$. Find the gradient and equation of the chord between $(a,f(a))$ and $(a+h,f(a+h))$ for randomised values of $a$, $b$ and $h$.

Uses the $\chi^2$ test to see if there is any significant difference in preferences.

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.

Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean and standard deviation of a sample. The population variance is not given and so the t test has to be used. Various scenarios are included.

Multiplication and addition of complex numbers. Four parts.

Directional derivative of a scalar field.