Content created by Newcastle University

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

Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.

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

Given normal distribution $\operatorname{N}(m,\sigma^2)$ find $P(a \lt X \lt b),\; a \lt m,\;b \gt m$ and also find the value of $X$ corresponding to a given percentile $p$%. 

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

Questions testing understanding of the index laws.

Questions testing rather basic understanding of the index laws.

Find $\displaystyle \frac{a} {b + \frac{c}{d}}$ as a single fraction in the form $\displaystyle \frac{p}{q}$ for integers $p$ and $q$.

Two questions testing the application of the Sine Rule when given two angles and a side. In this question, the triangle is always acute.

A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).

Add/subtract fractions and reduce to lowest form.

Express $\displaystyle \frac{ax+b}{x + c} \pm  \frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator. 

Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.

Inverse and division of complex numbers.  Four parts.

Differentiate $f(x) = (a x + b)/ \sqrt{c x + d}$ and find $g(x)$ such that $ f^{\prime}(x) = g(x)/ (2(c x + d)^{3/2})$.

Two questions testing the application of the Cosine Rule when given two sides and an angle. In these questions, the triangle is always acute and both of the given side lengths are adjacent to the given angle.

Find the determinant of a $4 \times 4$ matrix.

Find the determinant of a $3 \times 3$ matrix.

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