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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...

Cubic, Tangent, Circle

The graph of a cubic function with three real roots is drawn. A circle is then drawn so that the circumference of the circle passes through two adjacent roots. A tangent to the cubic is drawn at the point of the centre of the circle. When drawn accurately, the tangent passes...

Integration Stretch

The initial problem looks at an area given between two curves. The first curve is a parabola with the x and y axis intercepts given. The second curve is a stretch parallel to the x-axis, with the challenge being to determine the stretch factor that fits with the area stated....

A Geometric Progression Problem

There is an initial investigation of geometric progressions where the interval between the first and second, and the third and fourth terms are given, with the challenge of discovering two possible values for the first term and associated common ratio. There are ten examples to work on to enable collaboration...

Brackets Out, Brackets In

The problem starts with choosing three distinct, non-zero integers between -5 and 5 that do not sum to zero. These are then used in a six factorisations that make use of different permutations of the chosen numbers. Each factorisation is then expanded and the result summed. The...

Annulus Volume of Revolution

A segment of a circle is created using a chord parallel to the x-axis. The resulting segment of the circle is then revolved 360° about the x-axis. The problem is to calculate the volume of the resulting annulus.

To find a general solution to the problem involves using an elemental disc ring and integrating...

A Circle Journey

The initial task involves using factors of 12 and -12 to set up two points. Compasses are used to draw a circle that has the two points as its diameter. An interesting feature of the resulting circle is that the circumference always goes through the origin. A Geogebra file is...

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