Newcastle University Mathematics and Statistics

Three parts, where the student has to collect together polynomials in $\mathbb{Z}_3$, $\mathbb{Z}_5$ and $\mathbb{Z}_7$, respectively.

The answer to part a has no $X$ term, because they cancel out.

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

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.

Differentiate $\displaystyle \cos(e^{ax}+bx^2+c)$

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

A weighted coin with given $P(H),\;P(T)$ is tossed 3 times. Let $X$ be the random variable which denotes the longest string of consecutive heads that occur during these tosses. Find the Probability Mass Function (PMF), expectation and variance of $X$.

Application of the Poisson distribution given expected number of events per interval.

Finding probabilities using the Poisson distribution.

Moving averages, regression and seasonal adjustments.

Centred moving averages.

Given a probability mass function $P(X=i)$ with outcomes $i \in \{0,1,2,\ldots 8\}$, find the expectation $E$ and $P(X \gt E)$.

Application of the binomial distribution given probabilities of success of an event.

Finding probabilities using the binomial distribution.

Differentiate $\displaystyle (ax^m+b)^{n}$.

Nature of fixed points of a 2D dynamical system.

Given two numbers, find the gcd, then use Bézout's algorithm to find $s$ and $t$ such that $as+bt=\operatorname{gcd}(a,b)$. 

Elementary Exercises in multiplying matrices. 

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

Find $\displaystyle \frac{d}{dx}\left(\frac{m\sin(ax)+n\cos(ax)}{b\sin(ax)+c\cos(ax)}\right)$. Three part question.

Student is given a set of constraints for a linear program. Asked to enter the constraints as inequalities, and then to identify the optimal solution.

Problem with solving the simultaneous equations gven by the constraints - too unwieldy and not given enough marks for doing so. Best if the point of intersection is given graphically by putting the mouse over the intersection.

Student is shown a linear programming problem, with graphical solution. 

Asked to identify binding constraints, and decide if the optimal solution is changed by a named constraint either doubling or halving.

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

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

The derivative of $\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.

Multiple response question (3 correct out of 6) re properties of convergent and divergent sequences. Selection of questions from a pool.

Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.

Differentiate

\[ \sqrt{a x^m+b})\]

Differentiate $\displaystyle (ax^m+bx^2+c)^{n}$.

Differentiate $\displaystyle \ln((ax+b)^{m})$

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

The questions deal with permutations written in cycle or two-line notation.

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