Newcastle University Mathematics and Statistics

Expressing $\log(f(i))$ in the form $u+iv$.  Principal values of log.

Find the roots of $\sin(z)=a$.

Multiple response question (2 correct out of 4) covering properties of continuity and differentiability. Selection of questions from a pool.

Can choose true and false for each option. Also in one test run the second choice was incorrectly entered, rest correct,  but the feedback indicates that the third was wrong.

Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.

Questions on Pearson and Spearman correlation coefficients.

For given optimisation problems, determine maximin, maximax, and minimax regret actions, expected value criteria, expected value of perfect information.

Questions on differentiation from first principles, and continuity and differentiability.

Use the chain rule to differentiate various functions.

Use the product rule to differentiate various functions.

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

Composite multiplication and division of complex numbers. Two parts.

Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.

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

Determine the long-term behaviour of 1D dynamical systems.

Fixed points of 2D dynamical systems.

Eight questions on finding least upper bounds and greatest lower bounds of various sets.

Fixed points of a 1D dynamical system.

Approximate $f(x)=(a+h)^{m/n}$ by $f(a)+hf^{\prime}(a)$ to 5 decimal places and compare with true value.

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An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed. 

Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots. 

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$. 

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$.  (Equal roots example).

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 $x^2$.

Finding the distance between two complex numbers using the modulus of their difference. Three parts.

Finding the modulus of four complex numbers; includes finding the modulus of a product, a power and a quotient.

Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.

Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.