MASH Bath: Question Bank

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.

The proportion of the sodium carbonate, $p$, which has dissolved by time $t$ seconds is given by the formula $ p=\frac{bt-at^2}{c}$. The aim is to calculate the proportion of sodium carbonate in a solution at a given time and vice versa.

Estimating the proportion of sodium carbonate in a solutionat a specific timepoint and vice versa, depicted as a quadratic graph.

Interpreting line graphs depicting the decrease of temperature in a mixture over time. Estimating the temperature of the mixture at a given time point and vice versa.

Calculating the angle between a vector and the positive $x$-axis.

Solving $\cos(nx)=a$ for $x\in (0,\pi)$, where $n$ is an integer and $a\in(0,1)$.

Solving a pair of simultaneous equations of the form $a_1x+b_1y=c_1$ and $a_2 x^2+b_2y^2=c_2$ by forming a quadratic equation.

Solving a pair of simultaneous equations of the form $a_1xy=c_1$ and $a_2x+b_2y=c_2$ by forming a quadratic equation.

Calculating the area enclosed between a linear function and a quadratic function by integration. The limits (points of intersection) are given in the question.

Calculating the definite integral $\int_{n_1}^{n_2}ax^b dx$.

Solving a pair of linear simultaneous equations with integer solutions.

Solving a pair of linear simultaneous equations, giving answers as integers or fractions.

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.

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

Simplifying surds from $a\sqrt{b^2c}$ to $ab\sqrt{c}$, where $a$, $b$ and $c$ are positive integers.

Simplifying surds from $\sqrt{a^2b}$ to $a\sqrt{b}$, where $a$ and $b$ are positive integers.

Rewriting fractional expressions involving $\sqrt[n]{x^m}$ using rules to combine and simplify indices. 

Finding the product of a linear function of the form $mx+c$ and a cubic function of the form $ax^3+bx^2+cx+d$.

Finding the product of two linear functions of the form $mx+c$ and a quadratic function of the form $ax^2+bx+c$.

Finding the product of three linear functions of the form $mx+c$.

Finding the product of two linear functions of the form $mx+c$.

Given two quadratic expressions $f(x)$ and $g(x)$, calculate $f(x)(g(x)-f(x))$. 

Calculate the product of two quadratic expressions of the form $ax^2+bx+c$.