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HSC 2015 MX2 Marathon (archive) (3 Viewers)

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Ekman

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Re: HSC 2015 4U Marathon

Is it ok to prove it like this?







 

VBN2470

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Re: HSC 2015 4U Marathon

Yes, that should be fine too, as long as you can prove those other expressions are imaginary, the proof is valid.
 

FrankXie

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Re: HSC 2015 4U Marathon

the line from the center of the semi circle to the point where one of the small circles are tangent to the cyrved bit of the semi circle also goes rhrough the center of the small circle?
yes of course, because the semicircle and the smaller circle are also tangent(internally), the centres and the point of contact are colinear. isn't this a theorem in circle geometry?
 

Ekman

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Re: HSC 2015 4U Marathon

Simple question:

 

InteGrand

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Re: HSC 2015 4U Marathon

Suppose , where a and b are coprime positive integers.

Then .

This is a contradiction because the LHS ends in 0 in base 10, whereas the RHS cannot end in 0 (the last digits of powers of 8 go in the repeating cycle 8,4,2,6,8,4,2,6,8,....).
 

braintic

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Re: HSC 2015 4U Marathon

Suppose , where a and b are coprime positive integers.

Then .

This is a contradiction because the LHS ends in 0 in base 10, whereas the RHS cannot end in 0 (the last digits of powers of 8 go in the repeating cycle 8,4,2,6,8,4,2,6,8,....).
Why did you need the 'coprime' ?
 

InteGrand

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Re: HSC 2015 4U Marathon

Why did you need the 'coprime' ?
I didn't for this proof, but I'm used to writing a rational number as a fraction in its lowest terms, so I wrote coprime without thinking too much about it.
 

Fizzy_Cyst

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Re: HSC 2015 4U Marathon

A car takes a banked curve of a racing track at p m/s, the lateral gradient angle being designed to reduce the tendency to side-slip to zero for a lower speed q.

Show that the coefficient of friction necessary to prevent side-slip for the greater speed p must be at least:

 

FrankXie

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Re: HSC 2015 4U Marathon

A car takes a banked curve of a racing track at p m/s, the lateral gradient angle being designed to reduce the tendency to side-slip to zero for a lower speed q.

Show that the coefficient of friction necessary to prevent side-slip for the greater speed p must be at least:

when the speed is q, no side force, we have which yields ....(1)

when the speed is p>q, the side force F is downwards, we have which (by dividing them) result in sub in the equation (1) for gr, and let which is the coefficient of friction, it follows finally solving the last equation for completes the proof
 
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Ekman

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Re: HSC 2015 4U Marathon

next question

Loved it! :)



























However, every time I try finding the radius of x^2 + y^2 = a^2 in terms of c, I keep on getting the same result of c being equal to 1...
 
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