HSC 2012 MX1 Marathon #2 (archive) (2 Viewers)

Carrotsticks · Jun 17, 2012

Re: HSC 2012 Marathon

nightweaver066 said:
The question that you got from one of your students; that i indirectly sent you

Oh right yes yes I remember now.

Here's an interesting question:

A and B are two positive integers such that A > B and A =/= B.

Now consider A^B and B^A.

Which one is larger? Do any conditions need to be imposed to satisfy each inequality?

barbernator · Jun 17, 2012

Re: HSC 2012 Marathon

nightweaver066 said:
Same, then i realised his question is troll.

nah it wasn't a troll, i have just had my definition of integration wrong the whole time for exponents lol. its only e^f(x)/f'(x) for linear factors.... my bad

deswa1 · Jun 17, 2012

Re: HSC 2012 Marathon

Carrotsticks said:
Oh right yes yes I remember now.

Here's an interesting question:

A and B are two positive integers such that A > B and A =/= B.

Now consider A^B and B^A.

Which one is larger? Do any conditions need to be imposed to satisfy each inequality?

Consider A=5, B=2, B^A>A^B. Now consider A=3, B=2, A^B><b^a. now="" consider="" a="3," b="2," a^b="">B^A. Therefore conditions must be placed to satisfy the inequalities. Unfortunately I don't have time now to work it out- I'll try it tomorrow though because it looks cool</b^a.>

Kingportable · Jun 17, 2012

Re: HSC 2012 Marathon

Lol plox someone help me solve this question, its driving me crazy!

John Fitzpatrick 3U Mathematics 25(c) Question 3

A particle movies in a straight line. At time t seconds, its displacement x cm from a fixed poin O in the line is given by x=5sin((pi/2)T + pi/6). Express the acceleration in terms of x only and hence show that the motion is simple harmonic. Find:

iii) The speed when x=-2 1/2 or x = - 5/2
iv) the acceleration when x = -2 1/2 or x = - 5/2

nightweaver066 · Jun 17, 2012

Re: HSC 2012 Marathon

barbernator said:
nah it wasn't a troll, i have just had my definition of integration wrong the whole time for exponents lol. its only e^f(x)/f'(x) for linear factors.... my bad

Oh.. Sorry about that then.

I wouldn't use that formula, stick to the simple $\int f'(x)e^{f(x)} dx = e^{f(x)} + c$ and stick to 'balancing' to integrate.

RealiseNothing · Jun 17, 2012

Re: HSC 2012 Marathon

Carrotsticks said:
Oh right yes yes I remember now.

Here's an interesting question:

A and B are two positive integers such that A > B and A =/= B.

Now consider A^B and B^A.

Which one is larger? Do any conditions need to be imposed to satisfy each inequality?

The bolded bit seems a bit redundant given A > B lol.

But I'll give it a go now.

Carrotsticks · Jun 17, 2012

Re: HSC 2012 Marathon

Oh woops haha! Not thinking straight.

Sanjeet · Jun 24, 2012

Re: HSC 2012 Marathon

Kingportable said:
Lol plox someone help me solve this question, its driving me crazy!

John Fitzpatrick 3U Mathematics 25(c) Question 3

A particle movies in a straight line. At time t seconds, its displacement x cm from a fixed poin O in the line is given by x=5sin((pi/2)T + pi/6). Express the acceleration in terms of x only and hence show that the motion is simple harmonic. Find:

iii) The speed when x=-2 1/2 or x = - 5/2
iv) the acceleration when x = -2 1/2 or x = - 5/2

$x = 5sin \left ( \frac{\pi}{2}t + \frac{\pi}{6} \right ) \\ \dot{x} = \frac{5\pi}{2}cos\left ( \frac{\pi}{2}t + \frac{\pi}{6} \right ) \\ \ddot{x} = \frac{-5\pi^2}{4}sin \left ( \frac{\pi}{2}t + \frac{\pi}{6} \right ) \\ \ddot{x} = a = -\frac{\pi^2}{4}x \therefore $ It is in SHM as acceleration is proportional to displacement$ \\ \\ (iii)$ When $x = \frac{-5}{2},\\ \frac{-5}{2} = 5sin \left ( \frac{\pi}{2}t + \frac{\pi}{6} \right ) \\ \\ \frac{-1}{2} = sin \left ( \frac{\pi}{2}t + \frac{\pi}{6} \right ) \\ \\ \therefore \frac{\pi}{2}t + \frac{\pi}{6} = \frac{\pi}{6} + \pi = \frac{7\pi}{6}$ (ASTC)$ \\ \dot{x} = \frac{5\pi}{2}cos \left \frac{7\pi}{6} \right \Rightarrow -\frac{5\sqrt{3}\pi}{4} \\ \therefore speed = \frac{5\sqrt{3}\pi}{4}cms^{-1} \\$

(iv) Not even gonna bother to do this.

Timske · Jun 24, 2012

Re: HSC 2012 Marathon

<a href="http://www.codecogs.com/eqnedit.php?latex=x = -2\frac{1}{2} ~ and~\ddot{} x=-\frac{\pi^{2}}{4}x \\ Sub ~x ~in~,~\ddot{} x=-\frac{\pi^{2}}{4} *-2\frac{1}{2} = \frac{5\pi^{2}}{8} ~cm/s^2" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x = -2\frac{1}{2} ~ and~\ddot{} x=-\frac{\pi^{2}}{4}x \\ Sub ~x ~in~,~\ddot{} x=-\frac{\pi^{2}}{4} *-2\frac{1}{2} = \frac{5\pi^{2}}{8} ~cm/s^2" title="x = -2\frac{1}{2} ~ and~\ddot{} x=-\frac{\pi^{2}}{4}x \\ Sub ~x ~in~,~\ddot{} x=-\frac{\pi^{2}}{4} *-2\frac{1}{2} = \frac{5\pi^{2}}{8} ~cm/s^2" /></a>

barbernator · Jun 24, 2012

Re: HSC 2012 Marathon

if sally the problem gambler has 6 dice, all of which are 50% biased to a different number (i.e. for one die, chance of rolling a 1 - 50%, 2,3,4,5,6 - each 10%)

if she chooses 2 die at random and rolls them, what is the chance that she rolls a 1 and a 6?

If she wanted fair odds on the bet, what would the odds be?

RealiseNothing · Jun 24, 2012

Re: HSC 2012 Marathon

barbernator said:
if sally the problem gambler has 6 dice, all of which are 50% biased to a different number (i.e. for one die, chance of rolling a 1 - 50%, 2,3,4,5,6 - each 10%)

if she chooses 2 die at random and rolls them, what is the chance that she rolls a 1 and a 6?

If she wanted fair odds on the bet, what would the odds be?

Just having a quick look at the question, but wouldn't the fact that the even distribution of bias cancels out mean that the question is just the probability of rolling a 1 and 6 from 2 dice, which is 1/18?

barbernator · Jun 25, 2012

Re: HSC 2012 Marathon

RealiseNothing said:
Just having a quick look at the question, but wouldn't the fact that the even distribution of bias cancels out mean that the question is just the probability of rolling a 1 and 6 from 2 dice, which is 1/18?

yep lol

Fus Ro Dah · Jun 25, 2012

Re: HSC 2012 Marathon

More questions please =) I'm bored..

barbernator · Jun 25, 2012

Re: HSC 2012 Marathon

ok hmmm.

a projectile is fired at an angle of 40 degrees to the positive x axis. If the particle just clears a 4m high wall, 5.5m away from the firing spot on its ascent, what initial velocity V is it fired at, and at what distance away would it just clear a 2m high wall on its decline

Fus Ro Dah · Jun 25, 2012

Re: HSC 2012 Marathon

Maybe a 'cooler' one that isn't just calculations and number crunching please? I like proofs more :3

barbernator · Jun 25, 2012

Re: HSC 2012 Marathon

ok ill dig through a past cssa paper=

barbernator · Jun 25, 2012

Re: HSC 2012 Marathon

a lower half of a hemisphere is obtained by rotating x^2+y^2-20y=0 around the x axis, and cutting it through the centre. Water is poured in, Find equations for the volume of the water, and the surface area of the water. lol thats super simple haha ill find a better one

ok, heres one.

find the limiting value of 1 + 1 + 3/4 + 1/2 + 5/16 + 3/16 + .... and i won't give you the general term because you want a challenge

Fus Ro Dah · Jun 25, 2012

Re: HSC 2012 Marathon

We can break the above series into two sub-series and use the limiting sum for each one.

barbernator · Jun 25, 2012

Re: HSC 2012 Marathon

Fus Ro Dah said:
We can break the above series into two sub-series and use the limiting sum for each one.

which are? i will add a few more terms if you want

Trebla · Jun 26, 2012

Re: HSC 2012 Marathon

The second term of a geometric series is 1/4 and its limiting sum is -1/3. Find the value of the fifth term of the series.

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