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March 29, 2024, 06:50:55 am

Author Topic: Mathematics Extension 1 Challenge Marathon  (Read 26608 times)  Share 

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birdwing341

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #15 on: April 30, 2016, 02:17:36 pm »
+2


Required knowledge
Preliminary 2U Trigonometric Ratios, Preliminary 2U Introduction to Calculus, HSC 3U Inverse Functions and the Inverse Trigonometric Functions
Are you sure the last part of the root is x^3? My answer is with an x^2.

Is this right?

jamonwindeyer

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #16 on: May 01, 2016, 12:12:57 am »
0
Are you sure the last part of the root is x^3? My answer is with an x^2.

Is this right?

I think (pending the thumbs up from Rui, I hadn't tried this question before you mentioned the possible mistake) that you are indeed correct!! Great solution, well set out, great job!  ;D

RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #17 on: May 01, 2016, 03:55:46 pm »
+1
I don't know where I got x3 from either, sorry

Edit: Original post fixed!  :)
« Last Edit: May 05, 2016, 01:52:01 pm by jamonwindeyer »

Empathy

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #18 on: May 05, 2016, 12:26:03 pm »
0
'Infinitely' times more easier than what meets the eye


Answer/method in spoiler
Spoiler
Is the answer 8?
arctan(infinity)=pi/2.
e^(-infinity)=0
cos(infinity) oscillates between 1, -1 but this is actually irrelevant.

After removing/subbing these in (aside from the cos one), divide the top and bottom by x^8 as usual with lim questions. The reason the cos was irrelevant is that its basically (1/x^7)*((cosx)/x) which is 0/x^7. After cancelling out all the non whole numbers, I got 8/1=8
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jamonwindeyer

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #19 on: May 05, 2016, 01:47:58 pm »
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Answer/method in spoiler
Spoiler
Is the answer 8?
arctan(infinity)=pi/2.
e^(-infinity)=0
cos(infinity) oscillates between 1, -1 but this is actually irrelevant.

After removing/subbing these in (aside from the cos one), divide the top and bottom by x^8 as usual with lim questions. The reason the cos was irrelevant is that its basically (1/x^7)*((cosx)/x) which is 0/x^7. After cancelling out all the non whole numbers, I got 8/1=8

Good thinking putting the solution in a spoiler, I'm going to go back and put other solutions in a spoiler as well  ;D but yes, pending the Rui tick of approval, that is correct!! Nicely done!  ;D

Empathy

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #20 on: May 05, 2016, 02:57:06 pm »
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Good thinking putting the solution in a spoiler, I'm going to go back and put other solutions in a spoiler as well  ;D but yes, pending the Rui tick of approval, that is correct!! Nicely done!  ;D

One of my friends just pointed out to me that i wasted alot of time on that, I could have just skipped everything up to the last step and solved it in 1-2 lines... lol
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RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #21 on: June 22, 2016, 01:51:27 pm »
+1
Didn't even realise that someone actually attempted one of these questions.




For anyone's interest, the formal way to evaluate them is:

1. By considering -1≤cos(x)≤1, apply the squeeze theorem by choosing an appropriate function to multiply every side with
2. Use one application of L'Hopital's rule
3. (DOABLE) Split the limit into the limit of arctan(x) and the limit of x-n which was included in the given answer
4. (DOABLE) This is just 0 * 1/inf
« Last Edit: June 22, 2016, 02:02:54 pm by RuiAce »

conic curve

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #22 on: July 10, 2016, 10:27:43 am »
+1
Just in case anyone's interested feel free  to try out this question

1. a. SKetch f(x)=(e^x)-4, showing clearly the points of intersection with the axes and the equations of any asymptotes
b. On the same diagram, sketch the graph of the inverse function f^-1(x), showing clearly any important features
c. Explain why the x-coordinate of any points of the intersection y=f(x) and y=f^-1(x) satisfies e^x - x - 4=0
d. Show the equation e^x - x - 4 =0 between x=0 and x=2 and use the method of 'halving by intervals' to find this root correct to the nearest whole number

Also anyone here wishing to try a few challenge circle geometry questions feel free to


relativity1

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #23 on: July 12, 2016, 08:23:42 pm »
+1
for c) Would the answer be becuase the inverse is a reflection about the line y=x so the point of intersection is where x=y ie e^x-4=x therefore e^x-x-4=0?

RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #24 on: July 12, 2016, 09:00:15 pm »
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for c) Would the answer be becuase the inverse is a reflection about the line y=x so the point of intersection is where x=y ie e^x-4=x therefore e^x-x-4=0?

Yes.

Paradoxica

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #25 on: October 05, 2016, 06:32:45 pm »
+1
The acute triangle ABC has sides a>b>c, where A is opposite a and likewise for B and C.

A square is inscribed with vertices on the edges of the triangle.

It is given that only three such squares exist, and that one side of each square is concurrent with the side of a triangle.

Suppose the side length of the squares are p,q,r (with any correspondence you want), and the altitudes of the triangle are a',b',c', which connect the vertices A,B,C with the sides a,b,c respectively.

i) Find expressions for each of the three squares in terms of all the variables described above.

ii) Determine which of the three squares has the largest area.

iii) Hence, or otherwise, find the smallest possible area of any such triangle ABC which encloses a square of unit area.

RuiAce

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Mathematics Extension 1 Challenge Marathon
« Reply #26 on: June 08, 2017, 10:16:21 am »
+1



« Last Edit: June 08, 2017, 10:56:24 am by RuiAce »

RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #27 on: July 21, 2017, 07:34:53 pm »
+1

RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #28 on: July 22, 2017, 01:57:56 pm »
+1



RuiAce

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Re: Mathematics Extension 1 Challenge Marathon
« Reply #29 on: August 14, 2017, 06:10:43 pm »
+2

« Last Edit: October 29, 2017, 02:26:13 pm by RuiAce »