Hard Questions (4 Viewers)

[Nailgun]

Well, it is clear z > 0. By setting z to 1 (or any other positive constant), it's clear that the horizontal cross-section is that of an ellipse, which is stretched with major axis being the y-axis. By considering y or x being constant, it is clear the shape traced is that of a parabola. Hence the shape is an "elliptic paraboloid" (is that the right term?). Basically it's like a 3-d parabola that's stretched on the y-axis.

masterful paint skills View attachment 33073

wow thats actually pree impressive paint skills lol

would rep but

"You must spread some Reputation around before giving it to KingOfActing again."

 

[KingOfActing]

wow thats actually pree impressive paint skills lol

would rep but

"You must spread some Reputation around before giving it to KingOfActing again."

thanks man

feeling the love

 

[wu345]

 

[KINGOM 885]

View attachment 33077

Nice Question

 

[tywebb]

Solution attached

 

[tywebb]

It's the 2005 imo Q3.

My upload is from the solutions of the 2005 imo shortlist http://www.imomath.com/imocomp/sl05_0707.pdf , in particular, the shortlist question 5 from Korea.

 

[tywebb]

There is a second solution but it refers to theorems in a book which you might not have (IMO compendium, 1st edition, Theorems 2.26 and 2.27 on page 9). I have it so have attached them as well as the second solution.

 

[tywebb]

Here's a funny one from Fort Street Leaving Certificate trial 1959 Question 3a:

Differentiate x^xx

 

[Paradoxica]

Here's a funny one from Fort Street Leaving Certificate trial 1959 Question 3a:

Differentiate x^xx

Trivial by inspection.

 

[leehuan]

Here's a funny one from Fort Street Leaving Certificate trial 1959 Question 3a:

Differentiate x^xx

y=x^x^2

ln(y)=x^2.ln(x)

Meh...

1/y dy/dx = x(2ln(x)+1)

 

[tywebb]

Yeah. Well that's hard. But jokes aside, do you know that the Riemann Zeta function WAS actually examined in the HSC in 1975? So here is a not-so-hard question - HSC 1975 4 unit Q9i:



See if you can do it.

Actually this was not the first time this was examined. It was also in the Fort Street 1960 Leaving certificate trial Question 5a.

So that's interesting that the HSC in 1975 pinched a question from a school trial question 15 years prior - and from a Leaving certificate trial too!

 

[tywebb]

y=x^x^2

No. y=x^x^x

Anyway, here is the correct answer:

x^xx(x^x-1+x^xlnx(1+lnx))

 

[InteGrand]

I suppose logarithmic differentiation was explicitly taught before?

 

[Nailgun]

Here's a funny one from Fort Street Leaving Certificate trial 1959 Question 3a:

Differentiate x^xx

this question wouldve been so scary before integrand and drsoccerball taught me logarithmic differentiation ahahah

 

[tywebb]

I suppose logarithmic differentiation was explicitly taught before?

Yeah. It was in the Leaving certificate syllabus.

 

[leehuan]

No. y=x^x^x

Anyway, here is the correct answer:

x^xx(x^x-1+x^xlnx(1+lnx))

That's what happens when I can't see the question clearly enough because it is too small.

ln(y)=x^x.ln(x)

result quoted here out of laziness: d/dx x^x = x^x(1+ln(x))

1/y dy/dx = x^x(1+ln(x)).ln(x) + x^(x-1)

 

[Carrotsticks]

Actually this was not the first time this was examined. It was also in the Fort Street 1960 Leaving certificate trial Question 5a.

So that's interesting that the HSC in 1975 pinched a question from a school trial question 15 years prior - and from a Leaving certificate trial too!

I don't think the HSC would have pinched it from a school trial.

Almost all standard courses dealing with infinite series will at some point have the proof of the p-series converging for p>1, and it most likely would have been the typical integral test proof.

 

[tywebb]

Here are some more papers from 1956-1962 containing some hard questions:

http://4unitmaths.com/lc1956-1962.pdf

By "hard" I mean compared to what is expected of current year 11's. Back in those days there was no year 12. That began in 1967 with the first hsc. They left school 1 year earlier than they do now and went to uni 1 year earlier. Some would say better prepared too.

 

[dan964]

OP is a Year 12 student Extension 1. These are not in the scope of the HSC syllabus.

Also, these aren't particularly difficult problems. I am almost certain that my Year 11 class, who just learned Chain Rule less than a week ago, could do the second problem, given a 30 second introduction on how to compute partials.

yeah I know (they aren't too difficult, and not in the scope of syllabus), this thread is in the extra-curricular section.

 

[KINGOM 885]

Why?
I am an extension 1 student