More work on differentiation with fractional coefficients.

Identify whether or not an expression is a polynomial. From HELM Book 2.7.1.

Inputting algebraic expressions into Numbas.

Knowing the half-life of Carbon-14 and the initial mass of Carbon-14 when a tree was cut (a) write an expression that describes the relationship between the remaining mass and time, (b) calculate the remaining mass after $t$ years, and (c) given the remaining mass calculate how many years ago the tree was cut down. 

The relationship between the frequency of an allele A, $x$, at a genetic locus in a diploid population and the fitness of a population with this frequency of allele A, $w$, is described by the function $w=ax^2+x(b-x)+c(b-x)^2$ . The aims are (a) ti simplify the algebraic expression, (b) calculate the fitness of a population with a given allele A frequency, and (c) calculate the allele A frequency when the fitness of the population is given.

A mathematical expression part with a pattern restriction to ensure that the student has extracted the highest common factor of two terms.

The answer must be of the form $a(b+cx)$, where $b$ and $c$ are coprime.

evaluate a function with randomised alphanumeric expressions as inputs.

A custom marking algorithm picks out the names of the constants of integration that the student has used in their answer, and tries mapping them to every permutation of the constants used in the expected answer. The version that agrees the most with the expected answer is used for testing equivalence.

If the student uses fewer constants of integration, it still works (but they must be wrong), and if they use too many, it's still marked correct if the other variables have no impact on the result. For example, adding $+0t$ to an expression which otherwise doesn't use $t$ would have no impact.

Examples of the following custom part types: Yes/no, List of numbers, Give a numerical input for an expression, Number entry modulo.

This question shows a JSXGraph diagram above a mathematical expression input, containing a plot of the expression the student enters.

Simplify, if possible,
(a) $\frac{2}{3}x^2 + \frac{x^2}{3}$, (b) $0.5x^2 + \frac{3}{4}x^2 - \frac{11}{2}x$, (c) $3x^3 - 11x + 3yx + 11$, and (d) $-4\alpha x^2 + \beta x^2$ where $\alpha$ and $\beta$ are constants.

The numbers and letters are all randomised.

Part of HELM Book 1.3

Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.

Apply the factor and remainder theorems to manipulate polynomial expressions

Explain why one variable is a factor but the other is not. Factorise the expression, whose factor is alphanumeric. Part of HELM Book 1.3

Factorise a 2-term expression where the gcd can be alphanumeric. Part of HELM Book 1.3

Factorise an expression of 2 or 3 terms where the gcd is a letter times a number. Part of HELM Book 1.3.4

Factorise a 2-term expression by pulling out a numeric gcd. Part of HELM Book 1.3

Expand something like a(b+c) or (b+c)a or a(bc), where a, b and c can be +ve or -ve expressions. Part of HELM Book 1.3

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

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

Show one of several blocks of text depending on the value of a question variable.

As well as a simple check for the value of a variable, the condition to display a block of text can be a complex expression in any of the question variables - in this example, depending on the discriminant of the generated quadratic.

Removing brackets from expressions of the form (a+b)(c+d). Part of HELM Book 1.3

Rewriting fractions involving surds by rationalising the denominator.

Rewriting fractions involving surds by rationalising the denominator.

Rewriting expressions involving surds into the form $a+b \sqrt{c}$, where $a$, $b$ and $c$ are integers.

Rewiting expressions involving sums or differences of surds into the form $a \sqrt{b}$.

Rewiting expressions involving sums or differences of surds into the form $a \sqrt{b}$.

Simplifying expressions of the form $(a_1+b_1 \sqrt{c}) + (a_2 + b_2 \sqrt{c})$.

Rewriting expressions involving surds into the form $a+b \sqrt{c}$, where $a$, $b$ and $c$ are integers.

Rewriting expressions involving surds into the form $a+b \sqrt{c}$, where $a$, $b$ and $c$ are integers.