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March 28, 2024, 07:58:36 pm

Author Topic: Experimental Questions  (Read 1698 times)  Share 

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Ali_Abbas

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Experimental Questions
« on: September 07, 2016, 09:21:04 pm »
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Hi all,

Just thought I'd create a thread where I (as well as anyone else) can post their own self-made questions for any level of HSC Mathematics they like.

Solving questions is one thing, but composing your own is an entirely different ball game. One must uncover new patterns/properties/scenarios and results and encode them such that they can only be unravelled by a cunning manipulation, a theorem application, an insightful observation, or a geometric interpretation (no idea how I made that all rhyme :P). Formulating a mathematically-coherent, non-trivial question takes a lot of experimenting and planning. As such, you can expect that questions appearing on here will be rare occurrences (at least to begin with). Nevertheless, it's a place to start.

Here's my first question for people to attempt (I would classify it as a 3-Unit question):

Suppose a circle is inscribed entirely within the interior of the parabola x^2 = 4ay, i.e. the region x^2 <= 4ay, such that there are exactly two (tangential) points of intersection between them.

a) Show that this occurs precisely when the distance from the Origin to the centre of the circle is equal to twice the focal length of the parabola plus the y-value at either point of intersection. You may assume the centre of the circle is situated somewhere along the positive y-axis.

b) Without using the result from part a), prove (geometrically) that the distance from the centre of the circle to the focus of the parabola is equal to the distance from either point of intersection to the directrix.

c) Challenge exercise:
 Using part a), show that the average concavity of the lower arc of the circle bounded by the two intersection points is equal to the concavity of the parabola.

P.S. feel free to give any sort of feedback/criticisms regarding any aspects of my questions.

Cheers
« Last Edit: September 07, 2016, 09:36:02 pm by Ali_Abbas »

Ali_Abbas

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Re: Experimental Questions
« Reply #1 on: September 07, 2016, 10:30:18 pm »
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Here's a diagram to give you an idea of the set-up.

Ali_Abbas

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Re: Experimental Questions
« Reply #2 on: September 10, 2016, 10:06:18 pm »
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Some nice 2-Unit limit questions for y'all: