HSC 2013 Maths Marathon (archive) (1 Viewer)

Menomaths · Aug 26, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
$A standard fair die is rolled$ \ n \ $times, the product of the scores is calculated$

$What is the probability that this product is even?$

0.75?

Sy123 · Aug 26, 2013

Re: HSC 2013 2U Marathon

Menomaths said:
0.75?

you're answer should be dependent on 'n' =)

Menomaths · Aug 26, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
you're answer should be dependent on 'n' =)

My bad, 0.75n?

Sy123 · Aug 26, 2013

Re: HSC 2013 2U Marathon

Menomaths said:
My bad, 0.75n?

If n = say 10, then your probability becomes greater than 1

Just imagine n to be any other regular number, do it for 5 numbers first, then try to do it for any number n

Menomaths · Aug 26, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
If n = say 10, then your probability becomes greater than 1
Just imagine n to be any other regular number, do it for 5 numbers first, then try to do it for any number n

Oh my god....0.75ⁿ?

Sy123 · Aug 26, 2013

Re: HSC 2013 2U Marathon

Menomaths said:
Oh my god....0.75ⁿ?

$\frac{2^n - 1}{2^n}$

I'll leave it to you to explain why it is. i.e. I've turned the question into a 'Show that'

Menomaths · Aug 26, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
$\frac{2^n - 1}{2^n}$

I'll leave it to you to explain why it is. i.e. I've turned the question into a 'Show that'

Shit, I'll think about it this time...My name holds true so far

Sy123 · Sep 1, 2013

Re: HSC 2013 2U Marathon

$$By using differentiation from first principles of the function$ \ \ f(x) = \sin x$

$Prove$

$\lim_{t \to 0} \frac{\sin t}{t} = 1$

Sy123 · Sep 1, 2013

Re: HSC 2013 2U Marathon

$\int \frac{1+\cot x}{1-\cot x} \ dx$

integral95 · Sep 1, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
$\int \frac{1+\cot x}{1-\cot x} \ dx$ [/QUOTE

1. Change cotx into cosx/sinx
2. multiply top and bottom of the fraction by sinx
3. Now you should realise that the differentiation of the bottom equals to top.

Hence the answer is ln(sinx - cosx)

omgiloverice · Sep 1, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
$A standard fair die is rolled$ \ n \ $times, the product of the scores is calculated$

$Show that the probability that this product is even is$

$\frac{2^n-1}{2^n}$

$P(Even\ for\ 1 \ throw)=\frac{1}{2}\\P(Even\ for\ 2 \ throws)=1-(\frac{1}{2})^{2}\\P(Even\ for\ 3 \ throws)=1-(\frac{1}{2})^{3}\\\therefore P(Even\ for\ n \ throws)=1-(\frac{1}{2})^{n}$

omgiloverice · Sep 1, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
$$By using differentiation from first principles of the function$ \ \ f(x) = \sin x$

$Prove$

$\lim_{t \to 0} \frac{\sin t}{t} = 1$

I don't think 2 unit learn angle sums...

Sy123 · Sep 1, 2013

Re: HSC 2013 2U Marathon

omgiloverice said:
I don't think 2 unit learn angle sums...

don't be fooled, you will not need to use any 3U knowledge here

You don't need to actually differentiate sin(x) using first principles, I am asking that using the differentiation from first principle, i.e.:

Using the fact that:

$\cos x = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}$

Prove the limit above.

omgiloverice · Sep 1, 2013

Re: HSC 2013 2U Marathon

$$A ladder is resting again an wall makes an $ 60^{\deg}\ $angle with the ground.$ $ When the base off the ladder slides 1 metre further from the wall it makes an $45^{\deg}\ $ angle with the ground. $$ Find the length of the ladder.$

Sy123 · Sep 1, 2013

Re: HSC 2013 2U Marathon

omgiloverice said:
$P(Even\ for\ 1 \ throw)=\frac{1}{2}\\P(Even\ for\ 2 \ throws)=1-(\frac{1}{2})^{2}\\P(Even\ for\ 3 \ throws)=1-(\frac{1}{2})^{3}\\\therefore P(Even\ for\ n \ throws)=1-(\frac{1}{2})^{n}$

This proof would probably be accepted at 2U level, same concept is used in loan repayments/annuities etc
However above that I don't think it would be accepted at 3U and above

Better proof:

$P($even product$) = P($ at least 1 even result$) = 1- P($no even results$) = 1-P($all odd results$) = 1 - \frac{1}{2^n}$

omgiloverice · Sep 1, 2013

Re: HSC 2013 2U Marathon

Ahh... that makes alot more sense, I was wondering how to show the solution in an more formal sense.

Sy123 said:
don't be fooled, you will not need to use any 3U knowledge here

I'm sorry Sy I will never doubt you again.

anthony_wheels · Sep 2, 2013

Re: HSC 2013 2U Marathon

omgiloverice said:
$$A ladder is resting again an wall makes an $ 60^{\deg}\ $angle with the ground.$ $ When the base off the ladder slides 1 metre further from the wall it makes an $45^{\deg}\ $ angle with the ground. $$ Find the length of the ladder.$

Looks correct...I think

HeroicPandas · Sep 2, 2013

Re: HSC 2013 2U Marathon

anthony_wheels said:
View attachment 28580

Looks correct...I think

so the ladder is extendable?

anthony_wheels · Sep 2, 2013

Re: HSC 2013 2U Marathon

No. Sorry, that was my bad drawing! The line corresponding to the 45degree angle should be lower than the first line (I had intended to draw it like that initially, but as you can see I forgot to). The length of the ladder is given by 'L' which is the same on both lines

Sy123 · Sep 3, 2013

Re: HSC 2013 2U Marathon

$$i) Sketch the line$ \ \ y=A-x \ \ $for$ \ x\geq 0 \ , \ y \geq 0 \ , \ A>0$

$ii) The sum of 2 non-negative numbers is$ \ \ A \ \ $what is the maximum$

$and the minimum value of the sum of the squares of these numbers$

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