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March 28, 2024, 09:00:52 pm

Author Topic: Mathematics Extension 2 Challenge Marathon  (Read 31635 times)  Share 

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Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #30 on: July 02, 2017, 03:10:41 pm »
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A classic complex numbers question I just did a few minutes ago.

I believe I have a solution that differs from that of Rui's and is as follows.


« Last Edit: July 22, 2017, 12:48:51 am by Ali_Abbas »

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #31 on: July 02, 2017, 03:35:52 pm »
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Tbh all I did was this.

« Last Edit: July 02, 2017, 03:37:57 pm by RuiAce »

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #32 on: July 02, 2017, 03:44:07 pm »
+1

Hint
Can be made much easier using one of the tricks used in the previous problem
Spoiler
Find a contradiction!

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #33 on: July 02, 2017, 08:23:21 pm »
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Tbh all I did was this.
(Image removed from quote.)


You need to be careful when you apply the triangle inequality on a difference of terms as opposed to a sum of terms (occurring within the modulus signs). Although it will still give you a correct upper bound, it doesn't always give the true maximum of the expression. To see this, we can apply the triangle inequality on |alpha-1| + |alpha+1| giving:



But clearly 4 is not the true maximum so I'm not sure if your application of the triangle inequality qualifies as a concrete proof or if it merely gave the true maximum of the sum by shear coincidence.

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #34 on: July 02, 2017, 08:24:35 pm »
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You need to be careful when you apply the triangle inequality on a difference of terms as opposed to a sum of terms (occurring within the modulus signs). Although it will still give you a correct upper bound, it doesn't always give the true maximum of the expression. To see this, we can apply the triangle inequality on |alpha-1| + |alpha+1| giving:



But clearly 4 is not the true maximum so I'm not sure if your application of the triangle inequality qualifies as a concrete proof or if it merely gave the true maximum of the sum by shear coincidence.
The question did not require the least upper bound. I had explicitly stated that you need only prove this for 2sqrt(2).

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #35 on: July 02, 2017, 08:30:31 pm »
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The question did not require the least upper bound. I had explicitly stated that you need only prove this for 2sqrt(2).

But I thought 2sqrt(2) is the least upper bound...isn't it?

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #36 on: July 02, 2017, 08:32:05 pm »
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But I thought 2sqrt(2) is the least upper bound...isn't it?
Yes, but the identification of this was not necessary for this question. Therefore my proof still holds validity.

I did not see the point in forcing the requirement of the least upper bound when giving this question to Ext 2 students, nor did the original question specify this either.

Ali_Abbas

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #37 on: July 02, 2017, 08:40:37 pm »
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Yes, but the identification of this was not necessary for this question. Therefore my proof still holds validity.

I did not see the point in forcing the requirement of the least upper bound when giving this question to Ext 2 students, nor did the original question specify this either.

Alright I see what you're saying but it is my opinion that had the question asked for some other upper bound (e.g. 4), then although one can easily show it, it would be a silly question and a bit misleading to overstate the range of values the sum can take. It seems common sense to assume it to be the least upper bound so that we can be certain that the equation |alpha-1| + |alpha+1| = k will always have a solution when |alpha| <= 1 and k <= 2sqrt(2).

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #38 on: July 02, 2017, 08:44:36 pm »
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Regardless. Whilst it's wasteful to not give the least upper bound, there was still nothing that forces you to ensure you had found it.

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #39 on: July 21, 2017, 12:17:55 am »
+1


RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #40 on: July 21, 2017, 07:27:14 pm »
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« Last Edit: July 21, 2017, 07:54:23 pm by RuiAce »

RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #41 on: July 21, 2017, 07:37:34 pm »
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Ali_Abbas

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RuiAce

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Re: Mathematics Extension 2 Challenge Marathon
« Reply #44 on: July 22, 2017, 01:54:18 pm »
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