In the  ring $\mathbb{Z}[\sqrt{-2}]$ , find the remainder $r=r_1+r_2\sqrt{-2}$, where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{-2}$ by $c+d\sqrt{-2}$ .

Solve $\displaystyle ax + b =\frac{f}{g}( cx + d)$ for $x$.

A video is included in Show steps which goes through a similar example.

Expand $(ax+b)(cx+d)$.

Expanding products of 3 linear  polynomials over $\mathbb{Z}_3,\;\mathbb{Z}_5,\;\mathbb{Z}_7$

$f(X)$ and $g(X)$ are polynomials over $\mathbb{Z}_n$.

Find their greatest common divisor (GCD) and enter it as a monic polynomial.

Hence factorize $f(X)$ into irreducible factors.

Factorise 4 polynomials over $\mathbb{Z}_5$.

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Given polynomial $f(X)$, $g(X)$ over $\mathbb{Q}$, find polynomials $q(X)$ and $r(X)$ over $\mathbb{Q}$ such that $f(X)=q(X)g(X)+r(X)$ and $\operatorname{deg}r(X) \lt \operatorname{deg}g(X)$.

$f(X)$ and $g(X)$ as polynomials over the rational numbers $\mathbb{Q}$.

Find their greatest common divisor (GCD) and enter as a normalized polynomial.

Modular arithmetic. Find the following numbers modulo the given number $n$. Three examples to do.

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

This question tests the student's ability to identify equivalent fractions through spotting a fraction which is not equivalent amongst a list of otherwise equivalent fractions. It also tests the students ability to convert mixed numbers into their equivalent improper fractions. It then does the reverse and tests their ability to convert an improper fraction into an equivalent mixed number. 

Identify well-known fractional equivalents of decimals. Convert obscure decimals and recurring decimals into fractions.

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.

This question aims to assess the student's understanding of the difference between biased and unbiased events and also to assess the student's understanding of the fact that the experimental probability tends towards the theoretical probability as the number of trials increases.

Given two distributions, calculate the measures of average and spread and make some decisions based on the results.

Work with measurements of weight, mass and density.

Calculate outcomes for different configurations of rolling two dice. 

Questions involving the calculation of the volumes of shapes.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.

Use the rule $\log_a(n^b) = b\log_a(n)$ to rearrange some expressions.

Questions on manipulating logarithms.

Calculate a speed in m/s given distance and time taken, then convert that to km/hour

Use the BODMAS rule to determine the order in which to evaluate some arithmetic expressions. 

A graph shows both the speed and acceleration of a car. Identify which line corresponds to which measurement, and calculate the acceleration during a portion of time.

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