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A question to test understanding of set cardinality and intersections when limited information is known about the size of certain sets.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Example of an explore mode question. Student is given a 3x3 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 eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

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

Unit 1: Sequences and Functions

This question tests the students ability to factorise simple quadratic equations (where the coefficient of the x^2 term is 1) and use the factorised equation to solve the equation when it is equal to 0.

This exam test students understanding of determinants and inverses of 3x3 matrices.

This question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.

Exam to test students understanding of matrices multiplied by a scalar.

This exam gives tests for an overview of the skills students need to be able to perfom matrix multiplications.

This question tests if a students understands when matrices are conformable

This is a collection of questions to test different aspect to consider when representing a matrix.

This question tests understanding of subscript notation for matrices

Statistics and probability. 2 questions, 1 on two sample t-test and 1 on paired t-test.

Factorising polynomials using the highest common factor

Basic solving of linear equations

Solving linear equations

Using the chain rule with polynomials

Split $\displaystyle \frac{ax+b}{(cx + d)(px+q)}$ into partial fractions.

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Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

A basic introduction to differentiation

Find the gradient of  $ \displaystyle ax^b+\frac{c}{x^{d}}+f$ at $x=n$

Quotient and remainder, polynomial division.

$I$ compact interval, $g:I\rightarrow I,\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$

More work on differentiation with trigonometric functions

Differentiating further exponentials