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Re: HSC 2014 4U Marathon The function f(x)= \frac{|x-2|}{x^2-5x+6} is not defined for f(2) . Is there a value that could be given for f(2) that would make the function continuous at 2? Give reasons for your answer.

Re: HSC 2014 4U Marathon - Advanced Level

OK Thanks

Yes, that is what they simply want us to do. I just wanted to know how you would do this question using the theorem (without any rigorous proof) hence the reason why I posted the question here..

Re: HSC 2014 4U Marathon - Advanced Level In how many ways can you choose four positive integers a, b, c and d with a<b<c<d , so that \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} is an integer?

Re: HSC 2014 4U Marathon - Advanced Level How many functions f(x)=ax^2+bx+c are there with the property that, for all x , f(x)=f(-x)=f(x^2)

Re: HSC 2014 4U Marathon - Advanced Level How many pairs (a, b) of positive integers are there such that a and b are factors of 6^6 and a is a factor of b ?

Re: HSC 2014 4U Marathon Nice work :). Did you also manage to work out the answer for the other question posted above (finding m+n )?

Re: HSC 2014 4U Marathon A+B=\frac{\pi}{3}

Re: HSC 2014 4U Marathon If x, y and z are real numbers such that x^2+y^2+z^2-xy-yz-zx=8 find the maximum possible difference between any two of x, y and z

Re: HSC 2014 4U Marathon If m and n are positive integers such that \frac{m+n}{m^2+mn+n^2}=\frac{4}{49} find the value of m+n .

OK Thanks

Never mind I got it, as P(3) and P(5) yield opposite signs. Is there another way of doing this question, using IVT or Extreme Value Theorem or some other method? For example the polynomial P(x)=x^3-x+2 has only one real root, how would you be certain in finding the number of real zeroes...

UNSW

OK Thanks :) Another question I was sort of stuck on: How many real zeroes does p(x)=x^3-12x^2+45x-51 have?

Yep, sorry I've fixed it up for you now, what you said is right :) Would you mind explaining the question to me so I can see where to start and learn how to approach this type of question? By the way, the actual question has given a hint by letting g(x)=f(x+\frac{1}{2})-f(x) . I just wanted to...

Can someone please answer the following Thanks

Re: HSC 2014 4U Marathon - Advanced Level Can someone please answer the following : Thanks.

Re: HSC 2014 4U Marathon - Advanced Level Can someone please explain the following question? It would preferable if someone did this question using Gaussian elimination (matrix row reduction).. Thanks