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Find f(2)

The maximum value of a quadratic function is given, together with the value of f(3) and the information that this is equal to f(-1). The challenge is to determine the coefficients of the quadratic. This involves the use of symmetry and solution of a simultaneous equation.

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How Close

The coordinates of two points on a line are given. The task is to find the shortest distance from the line of a third given point. The task involves finding the equation of the line joining the first two points, and then calculating a normal to this line that passes through the...

Parabola in a Parallelogram

A parabola is given with the x-coordinate of the minimum point, and two points on the parallelogram shown. The points are joined and then a parallel line is drawn that is a tangent to the parabola. The task is to determine the ratio of the area bounded by the upper line and the...

Volume of a Tetrahedron

The coordinates of the vertices of a tetrahedron is given. The challenge is to work out the volume of the tetrahedron. This involves calculating the area of the base, working out the height, and then using the formula for the volume of a pyramid. The calculation uses a combination...

Survey Plot

There are a series of 8 sided polygons for which the areas are to be calculated. The area of each one turns out to be the same. This provides a lead in to asking how the polygons themselves were formed.

A method is introduced...

Skew Lines

This problem involves using vectors to model 3D space. Two skew lines are presented and the challenge is to find the shortest distance between the lines. In addition, the problem requires determination of the co-ordinates of the points giving rise to the minimum distance....

Area Between Parabolas

Two parabolas are shown, one with a positive quadratic coefficient, the other with a negative quadratic coefficient. Each has the same linear coefficient and between the two parabolas the quadratic coefficient and constant term are transposed. The problem is to determine the area...

A Fractional Sequence

The initial investigation uses an inductive formula and two initial values to derive a sequence that results in a series of fractions before returning to the initial two values of the sequence. A spreadsheet is provided that performs the calculations given in the inductive formula. This is helpful for both checking...

The Shortest Race

The introduction to this problem involves modelling a race from one tree to another, and the requirement to touch a fence on the way. The challenge is to determine the point on the fence that should be touched to minimize the distance to run.

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Hyper-parabola

A hyperbola is given with a parabola ax2  touching each branch of the hyperbola at single points. The task is to determine the value of a.

A formal proof of the general situation involves substituting the equation of the parabola into the equation of the hyperbola. The...

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