Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

$g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min? 

$I$ compact interval. $\displaystyle g: I \rightarrow I, g(x)=\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min? 

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force acts in the positive $x$ and positive $y$ direction.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force is applied in the negative $x$ and negative $y$ direction.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force is applied in the negative $x$ direction but the positive $y$. 

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

Using the chain rule within product rule problems.

Using e to solve equations involving the natural log

A question to practise the following skills in the context of Earth Sciences:

• graph-reading skills
• using a graph to find the equation of a line
• used adpative marking

Unit circle definition of sin, cos, tan using degrees

Using $\cos^2\theta+\sin^2\theta=1$ to evaluate expressions.

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

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

Use a piecewise linear graph of speed against time to find the distance travelled by a car.

Finally, use the total distance travelled to find the average speed.

Given a factor of a cubic polynomial, factorise it fully by first dividing by the given factor, then factorising the remaining quadratic.

This question tests the student's ability to find remainders using the remainder theorem. 

Unit circle definition of sin, cos, tan using degrees

Determine a finite sum and a limiting sum of a geometric series written using sigma notation. I'm not sure how to eliminate rounding errors...

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Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Simplify logarithmic expressions using rules on logarithms.

Differentiate the function $f(x)=(a + b x)^m  e ^ {n x}$ using the product and chain rule. Find $g(x)$ such that $f^{\prime}(x)= (a + b x)^{m-1}  e ^ {n x}g(x)$. Non-calculator. Advice is given.

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

An example of using the GeoGebra extension to ask the student to create a geometric construction, with marking and steps.

Three graphs are given with areas underneath them shaded. The student is asked to calculate their areas, using integration.  Q1 has a polynomial. Q2 has exponentials and fractional functions. Q3 requires solving a trig equation and integration by parts.

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

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

This creates a 4x4 times table grid using numbers from 2 to 9. The step gives some advice on how to work out times tables (products, multiplications) if you can't remember them.