The roots of a polynomial whereare constants are given by the solutions to. Suppose though that we have the roots and want to find the polynomial. We can form the polynomialexpand the brackets and factorise. Example: Find the polynomial with roots 1, 3, 5. Often the roots are surds. Still we can follow the above […]

## Minima and Maxima of Quadratics and Maxima of Reciprocals of Quadratics

To find the minimum or maximum of a quadratic we complete the square expressing the function in the form Ifthe minimum will be wheresoand the minimum is at Ifthe maximum will be wheresoand the maximum is at For example, to find the minimum of complete the square to getthen the minimum is at To find […]

## Proof of Formula for Curved Surface Area of Cone

For the cone of slant heightand base radiusthe curved surface areais give by the equation To prove this equation, draw the net of the cone. The net of the cone is a sector of a circle, with radiusand arc length equal to The area of the sector is(1) whereis in radians. The circumference of the […]

## Venn Diagrams – A Summary

Some of the Subsets of a Venn Diagram are shown below: AB={The set of elements of A that are also in B} A/B (also written A-B or AB‘) ={The set of elements of A that are not in B} B/A (also written B-A or BA‘) ={The set of elements of B that are not in […]

## Plotting Lines From Exponential Equations

Given a straight line plotted on a graph we can estimate the values of the gradient and either using a point on the line or by estimating the y – intercept, we can find the equation of the line in the form The most convenient form in which to analyse the relationship between two variables […]

## Formulae

The equation is solved using You can choose r from n possibilities in possible ways. You can line up r from n possibilities in different ways Binomial Theorem : For any triangle ∆ABC , The straight line passing through has gradient and equation The distance between two points, is A straight linepassing though points A […]

## Combinations and Permutations

This topic deals with the numbers of ways ways we can pick a selection from a number of possible combinations. For instance, suppose we have 10 people lined up and we have to pick a team of 4. The number of ways we can pick 4 from 10 is written or Working from first principles […]

## Solving Logarithmic Equations

A summary of log equations will be useful (1) (2) (3) (4) (5) Ifthen (6) In the examples that follow I will indicate where each rule has been used. Example: Solve Apply (1) to give Apply (6) to give x cannot be -4 becausedoes not exist, so Example: Solve (5) gives Substitute to give Multiply […]

## Solving Trigonometric Equations

The basicandcurves are given on the left below andon the right below: – blueblack We have typically to solve equations such as We start by making cosx the subject: We take the inverse cos: Now is the tricky part. There is more than one solution forWe have found one. The other solutions are given by […]

## Finding The Area Between Two Curves

The task is to find the shaded area above between the two curvesandwhich intersect at the points I labelled the curves top and bottom. Using this sophisticated notation, the area is Sometimes it is not quite so obvious what the equation of either the top or bottom curve is. The graph isWe have to find […]