Asking users to input coefficients of a system of diff equations so that the phase space is a saddle. All systems input by the user are graphed together with immediate feedback. Also included in the Steps are the graphs of the solutions for $x(t),\; y(t);\; x(0)=-5,\;y(0)=5.$

Calculate the Pearson correlation coefficient on paired data and comment on the significance.

rebelmaths

This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.

This question is the one described in method 1 of the example "Apply a standard integral" in the Numbas documentation.

The student is shown a randomly chosen function to integrate. The function is one of $e^{kx}$, $x^k$, $\cos(kx)$, $\sin(kx)$, with $k$ a randomly chosen integer.

Given the gradient of a slope and the coefficient of friction for a mass resting on it, use the equations of motion to calculate how it moves.

Includes a GeoGebra rendering of the model.

Students are given a line equation (where the coefficient of y is 1) and asked to rearrange it into gradient-intercept form.

Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:
$\displaystyle \frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$

Calculate the Pearson correlation coefficient on paired data and comment on the significance.

Calculate the Pearson correlation coefficient on paired data and comment on the significance.

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

Express $\displaystyle \frac{ax+b}{cx + d} \pm  \frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator. 

First part: Express  $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d},\;a=-c$. Numerator is an integer.

Second part: $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d}+ \frac{r}{sx+t}$ as single fraction 

First part: express as a single fraction: $\displaystyle \frac{a}{x + b} +  \frac{c}{x + d},\; a \neq -c$.

Second part: Find $\displaystyle \frac{a}{x + b} + \frac{c}{x + d}+\frac{r}{x+t}$ as a single fraction.

Standard derivatives asked for (e.g. $x^n$, $1/x^n$, $\sqrt(x)$, $\ln(x)$, $\sin(x)$, etc.) .  

Expression of the (a+bx)^n is given and a couple of coefficients are asked for.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by=0$. Complex roots.

Utilizza un grafico di una retta $y=mx+c$ incluso da Geogebra con coefficienti casuali impostati da NUMBAS. Il grafico contiene anche un punto $P$ di coordinate casuali impostate da NUMBAS. L'esercizio chiede di determinare l'equazione della retta e quelle delle rette parallela e perpendicolare per $P$.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Factorise three quadratic equations of the form $x^2+bx+c$.

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

One method of randomly choosing names for variables. For each variable, we have 4 options. Create a list of 4 numbers, which is 1 for the name we want to use, and 0 otherwise.

Then, whenever we use that variable, multiply each of the possible names by the corresponding number in the list. When the expression is simplified, the unwanted names will cancel to 0, leaving only the name we want.

This is quite clunky!

(This question also uses a custom marking script to check that the student has simplified the expression)

Find a regression equation given 12 months data on temperature and sales of a drink. 

Asking users to interpret a minitab output to give the coefficients of a multiple regression together with a prediction based on the subsequent equation.

Multiple correlation question. Given the correlation coefficent of $Y$ with $X_1$ is $r_{01}$, the correlation coefficent of $Y$ with $X_2$ is $r_{02}$ and the correlation coefficent of $X_1$ with $X_2$ is $r_{12}$ then explain the proportion of variablity of $Y$. Also find the partial corr coeff between $Y$ and $X_2$ after fitting $X_1$ and find how much of the remaining variability in $Y$ is explained by $X_2$ after fitting $X_1$.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''-by=0$. Distinct roots.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by'+cy=0$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

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Calculate the Pearson correlation coefficient on paired data and comment on the significance.