Students are given 2 equations of the form y=mx+b and asked to solve them using either the substitution or the elimination method. The lines are randomised but the solution coordinates are always integers.

Students are shown to intersecting lines in the context of a gym visiting program. They are asked to identify the y-intercept, the gradients, the point of intersection and are asked to interpret the graphs in the context of the word problem.

Studnents are asked to write down equations for cost and income for a business.

They are then asked to graph the two lines.

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:
$\displaystyle \frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$: $\log(ax+b)-\log(cx+d)=s$

Solve for $x$ each of the following equations: $n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.

Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).

Find $\displaystyle \int\frac{ax+b}{(1-x^2)^{1/2}} \;dx$. Solution involves inverse trigonometric functions.

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$. Solution involves the inverse trigonometric function $\arcsin$.

Solve for $x$: $a\cosh(x)+b\sinh(x)=c$. There are two solutions for this example.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Given vectors  $\boldsymbol{A,\;B}$, find the angle between them.

Express $\displaystyle b+  \frac{dx+p}{x + q}$ as an algebraic single fraction. 

Express $\displaystyle \frac{a}{x + b} + \frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator. 

Express $\displaystyle \frac{a}{(x+r)(px + b)} + \frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$.

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Calculate the concentration of a dilute acid given the amount and concentration of dilute alkali to neutralise it. Cribbed from this Bitesize problem.

There are steps to break down the calculation, and the student must give units at each step.

Preparing solutions to given concentrations/dilutions.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by=0$. Complex roots.

Full worked solution using the Extended Euclidian Algorithm

Linear program described in words. Student must write out constraints as equations in standard form, then identify the optimal solution in a finished simplex tableau.

A quadratic equation (equivalent to $(x+a)^2-b$) is given and sketched. Three equations are given that can be solved using the graph. There is a chance there will only be one solution.

A few quadratic equations are given, to be solved by completing the square. The number of solutions is randomised.

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

Introduction to first order recurrence relations with a simple example, including homogenous and non-homogenous solutions.

Question steps student through the process of finding the molar heat of dissolution.

Adaptive marking and partial credit used

Thermochemistry Revision Sheet Q3