Content created by Newcastle University

Poles, residues, and contour integral of a complex-valued function.  Single, simple pole.

Contour integral of $z^2$ along any path.

Contour integral of $\mathrm{e}^{-z}$ along any path.

Contour integral of a complex-valued function $f(z)$ with the poles of $f(z)$ either inside or outside the path $C$.

Given a payoff table with two states and five actions, identify which actions are admissible, then the maximax, maximin, and minimax regret actions.

Given a payoff table with two states and five actions, identify which actions are admissible, then the maximax, maximin, and minimax regret actions.

Should state that input is by integers or fractions. Also  the maximax, maximin and minimax regret actions are not asked for.

Given a payoff table with two states and five actions, identify which actions are admissible, then the maximax, maximin, and minimax regret actions.

Given a payoff table with two states and five actions, identify which actions dominate others, and identify admissible actions.

Express $f(z)$ in real-imaginary form, given that $z=x+iy$.

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 $h$.

Express $f(z)$ in real-imaginary form, given that $z=x+iy$, where $f(z)$ involves hyperbolic functions.

Modulus and argument of a single complex number, where $\mathrm{Re}(z)=\mathrm{Im}(z)$.

Polar form of a complex number.

Calculate the principal value of a complex number.

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.

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.