Maximisation or Optimisation Problems

There is only so much to go around, even more so now with the Earth’s resources running low and the price of everything. We have to make the most of what we have, and decide what things should be used for so that the most use can be got out of them.. W will consider […]

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The Remainder Theorem

The Remainder Theorem is the generalized form of The Factor Theorem. The Remainder Theorem When a polynomial expressionis divided by a linear factorthe remainder isIfthenis a factor of Example: Show thatis a factor of thereforeis a factor. Example If the remainder whenis divided byis 3 and the remainder whenis divided byis 4, findand Remainder ofon […]

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Minima and Maxima 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 ofcomplete the square to getthen the minimum is at To find the […]

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Coordinate Geometry – Triangles and Normals

    Diagram not to scale Typically in coordinate geometry we are concerned with lines and normals, distances and circles. We may be of, for the above diagram: a)Show that AC is perpendicular to AB. b)Find the distance AB. a) Gradient of Gradient of The product of the gradients is -1 so AC and AB […]

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Combinatorics and Factorials

If we have to choose a group of 6 people from a selection of 10, there are possible ways of making a choice of six. The order of selection doesn’t matter here. So that although each of the 10 are distinct, once they are picked they can line up in any order, and the order […]

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Simplifying Algebraic Expressions

We have to be able to simplify expressions of the formand We must be clear about the rules for indices and cancelling fractions. The first one is simplified as follows: The second expression factorises on the top and bottom: is a common factor hence we can cancel it to give the answer: We may also […]

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Rearranging Equations

Many equations are quite easy to rearrange. If an equation only has one incidence of x and we have to make x the subject then there is usually not much difficulty. Example: Make x the subject of the equation Make x the subject of Notice above that by was moved as a single term. Some […]

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Intersections of Curves

When two curves intersect they have the same x and y values This means that given the equations of the curves, we can solve them like simultaneous equations to find the points of intersection. Example: Find the points of intersection of the curve(1) and(2). The curve and the line are illustrated below.   We can […]

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Equations of Circles

A circle centrein theplane and radiushas a cartesian equation given by This can be seen from the graph below. We obtain the equationfor the circle by applying Pythagoras theorem to the right – angled triangle shown. In general with can define a a circle by three points which lie on it, a radius and two […]

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Simultaneous Equations With One a Quadratic

When solving ordinary – linear – simultaneous equations we multiply the equations by constant factors to make the coefficient of some variable the same in magnitude, then add or subtract the equations to eliminate that variable. For example, solve (1) (2) (1)*2-(2)*3 eliminatesto give Substitution of this value ofinto (1) to find a gives If […]

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