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. 

The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

The derivative of  $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

The derivative of $\displaystyle \frac{a+be^{cx}}{b+ae^{cx}}$ is $\displaystyle \frac{pe^{cx}} {(b+ae^{cx})^2}$. Find $p$.

The derivative of $\displaystyle \frac{a+be^{cx}}{b+ae^{cx}}$ is $\displaystyle \frac{pe^{cx}} {(b+ae^{cx})^2}$. Find $p$.

English sentences are given and for each the appropriate proposition involving quantifiers is to be chosen. Also choose whether the propositions are true or false.

English sentences are given and for each the appropriate proposition involving quantifiers is to be chosen. Also choose whether the propositions are true or false.

Given two propositions in mathematics using quantifiers, choose the corresponding negation of the proposition. For example, the negation of: $\displaystyle \exists a \in \mathbb{R^+},\;\exists b \in \mathbb{N},\;\exists c \in \mathbb{N}\;\left[(c \lt b+1) \land \left(\frac{1}{2^n} \geq 3a\right)\right]$

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Enumerate the elements in some sets defined using set builder notation.

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. 

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.

Find $\displaystyle \frac{d}{dx}\left(\frac{m\sin(ax)+n\cos(ax)}{b\sin(ax)+c\cos(ax)}\right)$. Three part question.

Other method. Find $p,\;q$ such that $\displaystyle \frac{ax+b}{cx+d}= p+ \frac{q}{cx+d}$. Find the derivative of $\displaystyle \frac{ax+b}{cx+d}$.

Solve for $x$ and $y$:  \[ \begin{eqnarray} a_1x+b_1y&=&c_1\\   a_2x+b_2y&=&c_2 \end{eqnarray} \]

The included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.

Two shops each have different numbers of jumper designs and colours. How many choices of jumper are there?

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Inputting algebraic expressions into Numbas.

Inputting ratios of algebraic expressions.

Inverse and division of complex numbers.  Four parts.

Multiplication and addition of complex numbers. Four parts.

Multiplication and addition of complex numbers. Four parts.

Find the equation of the straight line parallel to the given line that passes through the given point $(a,b)$.

Find the points of intersection of a straight line and a circle.