Calculating the integral of a function of the form $ax^2 e^{bx}$ using integration by parts.

Calculating the integral of a function of the form $ax^2 \cos(bx)$ using integration by parts.

Calculating the integral of a function of the form $x^n \ln(ax)$ using integration by parts.

Calculating the integral of a function of the form $(a+bx)\cos(x)$ using integration by parts.

Calculating the integral of a function of the form $ax e^{bx}$ using integration by parts.

Calculating the integral of a function of the form $ax \cos(bx)$ using integration by parts.

Calculating the integral of a function of the form $ax \sin(bx)$ using integration by parts.

Calculating the area enclosed between a cosine function and a quadratic function by integration. The limits (points of intersection) are given in the question.

Calculating the definite integral $\int_{n_1}^{n_2}a \sin(bx) dx$, where $n_1$ and $n_2$ are multiples of $\frac{\pi}{12}$.

Calculating the definite integral $\int_{n_1}^{n_2}a \sin(bx) dx$, where $n_1$ and $n_2$ are multiples of $\frac{\pi}{12}$.

Calculating the definite integral $\int_{n_1}^{n_2}a_1x^{b_1}+a_2x^{b_2}+a_3x^{b_3} dx$.

Solving a differential equation of the form $\frac{dy}{dx}=a \cos(x) e^{-y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=ax^n e^{-y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=\frac{a \cos(x)}{y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=x(y-a)$ using separation of variables.

Rewriting fractions involving surds by rationalising the denominator.

Rewriting fractions involving surds by rationalising the denominator.

Example of an explore mode question. Student is given a 2x2 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

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Indefinite integration of basic functions.

Demonstration of randomisation: many elements in this question are randomised. The names of the products and clients are randomly chosen, as are the prices and order amounts.

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No time limit - unlimited regeneration of questions allowed from these groups:

• Notation and Algebra
• Calculus - Differentiation
• Calculus - Integration
• Trigonometry and Matrices

Educational calculation tool rather than "question".

This allows the student to input a linear system in augmented matrix form (max rows 5, but any number of variables). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the system. This question has no marks and no feedback as it's just meant as a "calculator".

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.

Gives a plot of a velocity-time graph using JSXgraph, and a description of the motion shown in the graph. The student is asked to calculate the acceleration at different stages of the motion, and the displacement at the end of the motion.

Educational calculation tool rather than "question".

This allows the student to input a square matrix (max rows 5). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the matrix and the identity matrix (or what it has got to). This question has no marks and no feedback as it's just meant as a "calculator". It has some checks in so students know when they are not entering a square matrix or a valid row number etc.

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.