Content created by Newcastle University

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.

Solving an equation of the form $ax \equiv b\;\textrm{mod}\;n$ where $a$ and $n$ are coprime.

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

Differentiate the function $f(x)=(a + b x)^m  e ^ {n x}$ using the product rule. Find $g(x)$ such that $f^{\prime}(x)= (a + b x)^{m-1}  e ^ {n x}g(x)$.

Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean of a sample. The population variance is given and so the z values are used. Various scenarios are included.

Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean of a sample. The population variance is given and so the z values are used. Various scenarios are included.

Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\pi$ and $\pi$ and careful with quadrants!

Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find positive powers of the complex numbers.

Find modulus and argument of the complex number $z_1$ and find the $n$th roots of $z_1$ where $n=5,\;6$ or $7$.

Factorise $\displaystyle{ax ^ 2 + bx + c}$ into linear factors. 

Given a generating matrix for a linear code, give a parity check matrix

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Rolling a pair of dice. Find probability that at least one die shows a given number.

English sentences which are propositions are given and the appropriate logical expression chosen for the negation of the sentence.

English sentences which are propositions are given and for each the appropriate proposition  involving quantifiers is to be chosen. 

Write down the lexicographic parity check matrix and generator matrix for a Hamming code, which is the dual of a Simplex code, then determine if a given word is a codeword of the corresponding Simplex code.

Compute tables of Hamming distances in given codes, then determine which codes are equivalent.

Given a generating matrix for a binary linear code, construct a parity check matrix, list all the codewords, list all the words in a given coset, give coset leaders, calculate syndromes for each coset, correct a codeword with one error.

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.