Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.

Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.

Questions on the least upper bounds and greatest lower bounds of sets of the form $\{ f(x) : x \in \mathbb{Z} \text{ or } \mathbb{R} \}$.

Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$.  Find both roots of the equation $q(x)=0$.

Given two sets of data, sample mean and sample standard deviation, on performance on the same task, make a decision as to whether or not the mean times differ. Population variance not given, so the t test has to be used in conjunction with the pooled sample standard deviation.

Link to use of t tables and p-values in Show steps.

Provided with information on a sample with sample mean and standard deviation, but no information on the population variance, use the t test to either accept or reject a given null hypothesis.

Provided with information on a sample with sample mean and known population variance, use the z test to either accept or reject a given null hypothesis.

Exercise using a given uniform distribution $X$, calculating the expectation and variance. Also finding $P(X \le a)$ for a given value $a$.

Find $\displaystyle \frac{a} {b + \frac{c}{d}}$ as a single fraction in the form $\displaystyle \frac{p}{q}$ for integers $p$ and $q$.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Entering the correct roots in any order is marked as correct. However, entering one correct and the other incorrect gives feedback stating that both are incorrect.

Questions on adding, subtracting, multiplying and dividing numbers in standard form.

Add two numbers in standard form, then subtract two numbers in standard form.

Convert a variety of numbers from decimal to standard index form.

Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.

Factorise a quadratic expression of the form $x^2+akx+bk^2$ for $x$, in terms of $k$. $a$ and $b$ are constants.

Determine the way in which an image A has been reflected onto an image B.

Identify the centre of enlargement and the scale factor in a transformation of an image A to an image B

Substitute given values into formulas.

Solve a linear equation of the form $ax+b = c$, where $a$, $b$ and $c$ are integers.

The answer is always an integer.

Identify the plane transformation performed to create a given diagram.

Given a geometric sequence, find the common ratio (negative in this question), write down the formula for the nth term and use it to calculate a given term.

Given the first four terms of a quadratic sequence, write down the formula for the $n^\text{th}$ term.

Given the first three terms of a sequence, give a formula for the $n^\text{th}$ term.

In the first sequence, $d$ is positive. In the second sequence, $d$ is negative.

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.

Given a formula for a sequence, calculate a given term.

Given the first and last terms of a finite arithmetic sequence, calculate the number of elements and then the sum of the sequence.

Each part is broken into steps, with the formula given.

This question assesses the student's ability to use some given information involving two different units of measurement to rewrite the information as a compound measure.

Rearrange expressions in the form $ax^2+bx+c$ to $a(x+b)^2+c$.

This question tests the students ability to calculate the area of different 2D shapes given the units and measurements required. The formulae for the areas are available if required but students are encouraged to try to remember them themselves.

The shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.

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