Three equilateral triangles are divided equally into 3, 4 and 5 parts respectively. Calculate the distance between two marked points.

Based on a puzzle by Catriona Shearer, shared on Twitter.

$f(x)= ae^{-bt}+c$ is given and plotted. A few points are plotted on the curve. $x$-coordinates are provided for two of them and $y$-coordinate provided for third. Student is required to determine other coordinates.

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Calculate the local extrema of a function ${f(x) = e^{x/C1}(C2sin(x)-C3cos(x))}$

The graph of f(x) has to be identified.

The first derivative of f(x) has to be calculated. 

The min max points have to be identified using the graph and/or calculated using the first derivative method.  Requires solving trigonometric equation

Give the student three points lying on a quadratic, and ask them to find the roots.

Then ask them to find the equation of the quadratic, using their roots. Error in calculating the roots is carried forward.

Finally, ask them to find the midpoint of the roots (just for fun). Error is carried forward again.

Construct a line in a GeoGebra worksheet by writing its definition string by hand.

This isn't a very neat way of doing this. It's easier to define two points in GeoGebra, then make a line through those points. You can set the positions of the points from Numbas using vectors.

Construct a line through two points in a GeoGebra worksheet. Change the line by setting the positions of the two points when the worksheet is embedded into the question.

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.

Find angle between plane $\Pi_1$, given by three points, and the plane $\Pi_2$ given in Cartesian form.

The calculation of $cos(\alpha)$ at the end of Advice has fractionNumbers switched on and so the result is presented as a fraction, which can be misleading. Best if calculation is followed through without using fractionNumbers.

Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.

The correct solution is given, however the accuracy of 0.001  is not enough as in some cases answers near to the correct solution are also marked as correct.

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

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

Given a pair of 3D position vectors, find the vector equation of the line through both.  Find two such lines and their point of intersection.

Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

Intersection points, tangent vectors, angles between pairs of curves, given in parametric form.

Find all points for which the gradient of a scalar field is orthogonal to the $z$-axis.

Should warn that multiplied terms need * to denote multiplication.

$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 < t<1$. Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \leq m^k t^m$ for all $m \geq N$. 

A graphical approach to aiding students in writing down a formal proof of discontinuity of a function at a given point.

Uses JSXgraph to sketch the graphs and involves some interaction/experimentation by students in finding appropriate intervals.

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Find a regression equation.

Statistics and probability. 2 questions. Both simple regression. First with 8 data points, second with 10. Find $a$ and $b$ such that $Y=a+bX$. Then find the residual value for one of the data points.

Trying out something: get the student to enter a set for each of "regular singular points" and "essential singular points".

Find and classify singular points of a second-order ordinary differential equation.  One equation is chosen from a selection of 10.

Given 32 datapoints in a table find their minimum, lower quartile, median, upper quartile, and maximum.

Converting odds to probabilities.

Given 32 datapoints in a table find their minimum, lower quartile, median, upper quartile, and maximum.

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$. Input constant of integration as $C$.

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.