• Best of luck to the class of 2024 for their HSC exams. You got this!
    Let us know your thoughts on the HSC exams here
  • YOU can help the next generation of students in the community!
    Share your trial papers and notes on our Notes & Resources page
MedVision ad

HSC 2012 MX2 Marathon (archive) (3 Viewers)

deswa1

Well-Known Member
Joined
Jul 12, 2011
Messages
2,256
Gender
Male
HSC
2012
Re: 2012 HSC MX2 Marathon

Thanks guys. The way I did it was Carrotsticks way but I don't get how that can only be a one mark question...
 

seanieg89

Well-Known Member
Joined
Aug 8, 2006
Messages
2,662
Gender
Male
HSC
2007
Re: 2012 HSC MX2 Marathon

It's good to see new methods pop up. I never thought of this. Nice work!

However, It is a wee bit slow. Another problem I can see coming is that you will need to actually show that it is in fact max/min, which can get quite tedious.
It can be made pretty fast...observe that arctan is an increasing function of the positive reals and that with equality attained at x=2.

So

with equality attained for some z.
 
Last edited:

Carrotsticks

Retired
Joined
Jun 29, 2009
Messages
9,494
Gender
Undisclosed
HSC
N/A
Re: 2012 HSC MX2 Marathon

Thanks guys. The way I did it was Carrotsticks way but I don't get how that can only be a one mark question...
If it's only a 1 mark question, then a solution by simple inspection will suffice (seanieg89's way).

If it is more, then perhaps a more 'definite' proof may be required.
 

maths lover

Member
Joined
Feb 22, 2011
Messages
292
Gender
Male
HSC
2012
Re: 2012 HSC MX2 Marathon

The method I'm using below can be handy sometimes, because it gives me max AND min argument immediately. Of course there are other ways, but it's really just preference.:











EDIT:

How did you use calculus for this problem?
i am going to rep you for this very intuitive method(me thinks). avoids allot of the tedious trigonometry i usually use for these types of questions.
 

rolpsy

Member
Joined
Apr 9, 2011
Messages
94
Gender
Male
HSC
2012
Re: 2012 HSC MX2 Marathon



 
Last edited:

SpiralFlex

Well-Known Member
Joined
Dec 18, 2010
Messages
6,960
Gender
Female
HSC
N/A
Re: 2012 HSC MX2 Marathon

By differentiation.








b)

Let

Solve from there. By using RHS then it will be in the form of LHS.

Laptop stuffing up so I can't type much.

For the last parts.

Use



Substitute and solve.
 
Last edited:

SpiralFlex

Well-Known Member
Joined
Dec 18, 2010
Messages
6,960
Gender
Female
HSC
N/A
Re: 2012 HSC MX2 Marathon

Rolpsy, you are really tempting me to not sleep and do questions. (I'll miss Scischool tomorrow. :()





By using the identity of

Taking


But substitute instead,

 
Last edited:

rolpsy

Member
Joined
Apr 9, 2011
Messages
94
Gender
Male
HSC
2012
Re: 2012 HSC MX2 Marathon

Rolpsy, you are really tempting me to not sleep and do questions. (I'll miss Scischool tomorrow. :()
ok I'll stop lol – until tomorrow


 
Last edited:

IamBread

Member
Joined
Oct 24, 2011
Messages
757
Location
UNSW
Gender
Male
HSC
2011
Re: 2012 HSC MX2 Marathon

First time using latex, that took awhile...
 

IamBread

Member
Joined
Oct 24, 2011
Messages
757
Location
UNSW
Gender
Male
HSC
2011
Re: 2012 HSC MX2 Marathon

cool (although you should probably include limit swap step: )
Yeah I did on paper, but didn't want to write more then was absolutely necessary with latex:cool:
 

IamBread

Member
Joined
Oct 24, 2011
Messages
757
Location
UNSW
Gender
Male
HSC
2011
Re: 2012 HSC MX2 Marathon

Where do you guys get these questions from?
 

Carrotsticks

Retired
Joined
Jun 29, 2009
Messages
9,494
Gender
Undisclosed
HSC
N/A
Re: 2012 HSC MX2 Marathon

Many 'difficult' or 'interesting' Extension 2 questions are proofs for cool identities.

These proofs can be found everywhere on the Internet, or any higher-level Maths textbooks.

ie: The question rolpsy just posted was actually a 'proof' (though technically flawed, the idea is correct) of the Taylor Series expansion for the function ln(x+1). It also leads to the fact that the Alternating Harmonic Series converges to ln(2).
 

rolpsy

Member
Joined
Apr 9, 2011
Messages
94
Gender
Male
HSC
2012
Re: 2012 HSC MX2 Marathon

oh.. well anyway:





new question:
 

Users Who Are Viewing This Thread (Users: 0, Guests: 3)

Top