Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

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

$I$ 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 $I$

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

$I$ 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 $I$

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Rationalise the denominator with increasingly difficult examples involving compound denominators. 

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

The subtraction algortihm using the borrow and pay back method with integers.

Manipulate surds and rationalise the denominator of a fraction when it is a surd.

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

The subtraction algortihm using the borrow and pay back method with integers.

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

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.

Find the determinant of a $4 \times 4$ matrix.

Find the determinant of a $3 \times 3$ matrix.

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

Use matrix multiplication to get an equation for \(k\) which is then to be solved.

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Find the inverse of three $2 \times 2$ invertible matrices.

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Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Student is given a set of constraints for a linear program. Asked to enter the constraints as inequalities, and then to identify the optimal solution.

Problem with solving the simultaneous equations gven by the constraints - too unwieldy and not given enough marks for doing so. Best if the point of intersection is given graphically by putting the mouse over the intersection.

Student is shown a linear programming problem, with graphical solution. 

Asked to identify binding constraints, and decide if the optimal solution is changed by a named constraint either doubling or halving.

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

Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions. 

Find the determinant of three $2 \times 2$ invertible matrices.

Express $\displaystyle a \pm  \frac{c}{x + d}$ as an algebraic single fraction.