HSC 2014 MX2 Marathon (archive) (2 Viewers)

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FrankXie

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Re: HSC 2014 4U Marathon

for part (ii):



Because the modulus of common ratio is less than 1, the limiting sum exists and







Finally equate the real part of both sides.
 
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Axio

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Re: HSC 2014 4U Marathon

In how many ways can the word:



be arranged in a line so that the two Rs (which are indistinguishable) are separated by at least one other letter?
 

integral95

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Re: HSC 2014 4U Marathon

In how many ways can the word:



be arranged in a line so that the two Rs (which are indistinguishable) are separated by at least one other letter?



i'm not sure hehehe
 

dunjaaa

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Re: HSC 2014 4U Marathon

Insertion method; 11C2 x 10!/(2!2!2!)
 

Axio

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Re: HSC 2014 4U Marathon

I got (assuming same letters are always undistinguishable; I could be wrong lol):

Number of total arrangements - number of arrangements with 2 Rs together
= 12!/(2!2!2!) - (2!11!)/(2!2!2!)
 

FrankXie

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Re: HSC 2014 4U Marathon

I got (assuming same letters are always undistinguishable; I could be wrong lol):

Number of total arrangements - number of arrangements with 2 Rs together
= 12!/(2!2!2!) - (2!11!)/(2!2!2!)
I would also do complementary. But do you mean two R's are different while 2C's, 2S's, 2T's are each pair identical? lol
 

Sy123

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Re: HSC 2014 4U Marathon

Similar to what I have in mind, I just translate the problem into showing:



So it avoids mod-arg form but its essentially the same thing
 

Chlee1998

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Re: HSC 2014 4U Marathon

integral from (6^(1/2) + 2^(1/2))/2 to 1: (1+x^2)/(1+x^4) using the substitution u= x-(1/x)

you cannot use any 4u specific integration formulas such as IBP, partial fractions, inverse tan integral, etc. Only 3u concepts such as u substitution.
 

FrankXie

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Re: HSC 2014 4U Marathon

integral from (6^(1/2) + 2^(1/2))/2 to 1: (1+x^2)/(1+x^4) using the substitution u= x-(1/x)

you cannot use any 4u specific integration formulas such as IBP, partial fractions, inverse tan integral, etc. Only 3u concepts such as u substitution.
Is it allowed to use the standard table of integrals to evaluate ?
 

Chlee1998

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Re: HSC 2014 4U Marathon

Here is my solution using trigonometry. Let AD, DC and CB touch the semicircle at points E, F and G respectively. Let the radius of the circle be r.












can u find another way without using so much trig
 
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FrankXie

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Re: HSC 2014 4U Marathon

oops, how can i forget similar triangles :(



the reason why these pairs of angles are equal is already in my first solution
 

Chlee1998

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Re: HSC 2014 4U Marathon

oops, how can i forget similar triangles :(



the reason why these pairs of angles are equal is already in my first solution



yep good job this was what I did lol. Hence why I told u to find an alternatuve solution which hinted that a very simple did exist
 
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