$A,\;B$ $2 \times 2$ 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.

Friedman test, 5 subjects, 4 treatments.

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.

Choosing whether given random variables are qualitative or quantitative.

Elementary examples of multiplication and addition of complex numbers. Four parts.

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

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 $5$ 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.

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.

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

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

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

rebelmaths

Directional derivative of a scalar field.

Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.

Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

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

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.

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

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