2012 Year 9 &10 Mathematics Marathon (2 Viewers)

RealiseNothing

what is that?It is Cowpea
Joined
Jul 10, 2011
Messages
4,591
Location
Sydney
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

Demento, solution for the triangle question:

The area of a triangle is half times base times height. The longest height will have the shortest base, and the shortest height will have the longest base.

So let the longest height be 'H' and the shortest height be 'h'.

The area of the triangle using the longest height is:



The area of the triangle using the shortest height is:



Since these are just the areas of the triangle, they are equal, that is:



Simplifying gives:



Hence the longest height is twice that of the shortest height, and so k = 1/2.
 
Joined
Apr 1, 2011
Messages
1,012
Location
District 12
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

Just curious, what maths topics have you guys done so far this year? And whether you are 5.1/5.2/5.3 or accelerated. Just want to make sure.
 
Last edited:

kazemagic

Member
Joined
Feb 8, 2012
Messages
626
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

Demento, solution for the triangle question:

The area of a triangle is half times base times height. The longest height will have the shortest base, and the shortest height will have the longest base.

So let the longest height be 'H' and the shortest height be 'h'.

The area of the triangle using the longest height is:



The area of the triangle using the shortest height is:



Since these are just the areas of the triangle, they are equal, that is:



Simplifying gives:



Hence the longest height is twice that of the shortest height, and so k = 1/2.
Wow... you make it sound so simple
 

russ3l

Member
Joined
Dec 12, 2011
Messages
258
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

Just curious, what maths topics have you guys done so far this year? And whether you are 5.1/5.2/5.3 or accelerated. Just want to make sure.
To be honest, I have no fucking clue. I'm one of the top math students in my grade (for 5.3) and I still don't know how to do these questions. Ymceac is accelerated so he doesn't count :p
I feel like some of these questions are based more off 'general knowledge' other than "oh! ive done this topic at school, I know exactly what to do with a question like this!"...you know what I mean? :/
 

kazemagic

Member
Joined
Feb 8, 2012
Messages
626
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

To be honest, I have no fucking clue. I'm one of the top math students in my grade (for 5.3) and I still don't know how to do these questions. Ymceac is accelerated so he doesn't count :p
I feel like some of these questions are based more off 'general knowledge' other than "oh! ive done this topic at school, I know exactly what to do with a question like this!"...you know what I mean? :/
So tru

Will anyone type up a worked solution for this question?


BTW heres a question.
A father in his will left all his money to his children in the following manner: $1k to first born and 1/10 of what then remains, then$2k to the second born and 1/10 of what then remains, then $3k to the third born and 1/10 of what then remains, and so on. When this was done each child had the same amount. How many children were there?

I forgot the answer to this lol
 
Last edited:

Demento1

Philosopher.
Joined
Dec 23, 2011
Messages
866
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

So tru

Will anyone type up a worked solution for this question?
You're coming to meat tomorrow so perhaps I can explain it then. I will be arriving at 10am and you'll just have to find me.

Just curious, what maths topics have you guys done so far this year? And whether you are 5.1/5.2/5.3 or accelerated. Just want to make sure.
Advanced mathematics course in NSW (5.3 i'm guessing?). I don't actually accelerate, not really anyway as our school doesn't allow it.

Demento, solution for the triangle question:

The area of a triangle is half times base times height. The longest height will have the shortest base, and the shortest height will have the longest base.

So let the longest height be 'H' and the shortest height be 'h'.

The area of the triangle using the longest height is:



The area of the triangle using the shortest height is:



Since these are just the areas of the triangle, they are equal, that is:



Simplifying gives:



Hence the longest height is twice that of the shortest height, and so k = 1/2.
Appreciate the explanation, as it's probably a lot more orthodox then my working.
 

russ3l

Member
Joined
Dec 12, 2011
Messages
258
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

So tru

Will anyone type up a worked solution for this question?
^Like this question for example. Is it even in the year 10 syllabus? I've never been taught how to find an independent of something. Maybe it's just me...
 

Demento1

Philosopher.
Joined
Dec 23, 2011
Messages
866
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

^Like this question for example. Is it even in the year 10 syllabus? I've never been taught how to find an independent of something. Maybe it's just me...
The question is just worded a little differently to throw off some students. Really, the question can be interpreted as:

If find an equation which shows the relationship between a and b, without x terms.
 

kazemagic

Member
Joined
Feb 8, 2012
Messages
626
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

We're trying to get rid of x.
O I c. Thanks Ymcaec
Got a question for da genius lol. Have you seen these type of questions b4? Or you did it on the spot
 

RealiseNothing

what is that?It is Cowpea
Joined
Jul 10, 2011
Messages
4,591
Location
Sydney
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

Will anyone type up a worked solution for this question?
Here's my way:

Double the first expression to get:



Now if we expand the brackets we will get:



Add the two expressions together, and also subtract the two expressions, we will get:



Now we just multiply them together:



Now notice the LHS is a difference of two squares and on the RHS the x's will cancel out:



Re-arranging gives the answer of:

 
Last edited:

kazemagic

Member
Joined
Feb 8, 2012
Messages
626
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

Here's my way:

Double the first expression to get:



Now if we expand the brackets we will get:



Add the two expressions together, and also subtract the two expressions, we will get:



Now we just multiply them together:



Now notice the LHS is a difference of two squares and on the RHS the x's will cancel out:



Re-arranging gives the answer of:

Wow

You're coming to meat tomorrow so perhaps I can explain it then. I will be arriving at 10am and you'll just have to find me.
Looking forward to meet you and Ealdoon :)
 

ymcaec

Member
Joined
Apr 25, 2012
Messages
77
Gender
Undisclosed
HSC
N/A
Re: 2012 Year 9 &10 Mathematics Marathon

O I c. Thanks Ymcaec
Got a question for da genius lol. Have you seen these type of questions b4? Or you did it on the spot
i have done some "connecting" questions before
 

ymcaec

Member
Joined
Apr 25, 2012
Messages
77
Gender
Undisclosed
HSC
N/A
Re: 2012 Year 9 &10 Mathematics Marathon

Here's my way:

Double the first expression to get:



Now if we expand the brackets we will get:



Add the two expressions together, and also subtract the two expressions, we will get:



Now we just multiply them together:



Now notice the LHS is a difference of two squares and on the RHS the x's will cancel out:



Re-arranging gives the answer of:

i think this way is better ! :)
 
Joined
Apr 1, 2011
Messages
1,012
Location
District 12
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

^Like this question for example. Is it even in the year 10 syllabus? I've never been taught how to find an independent of something. Maybe it's just me...
Yes, that was in one of my school's past year 10 yearly papers, although admittedly that was one of the last questions.
 
Last edited:

Demento1

Philosopher.
Joined
Dec 23, 2011
Messages
866
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

Wow



Looking forward to meet you and Ealdoon :)
We're the only year 10s, but meh, who cares. To be honest, I reckon russ3l and ymcaec should come tomorrow. It would be interesting. :)
 
Joined
Apr 1, 2011
Messages
1,012
Location
District 12
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

How strong is everyone at probability? I want to post another one hehe.

It'll have more algebra and less getting around the wording of the question :p
 

Demento1

Philosopher.
Joined
Dec 23, 2011
Messages
866
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

How strong is everyone at probability? I want to post another one hehe.

It'll have more algebra and less getting around the wording of the question :p
Do we let something be x? If yes, I will absolutely pour my heart to answer it. :p
 
Joined
Apr 1, 2011
Messages
1,012
Location
District 12
Gender
Male
HSC
2013
Re: 2012 Year 9 &10 Mathematics Marathon

Do we let something be x? If yes, I will absolutely pour my heart to answer it. :p
Yep! Here it is, have fun



Again, tree diagrams can help (although some pro people here don't seem to need them haha), just let p be the probably of winning and 1-p be losing. For part iii it's just basically a similar (but easier) version of the inequality question I posted before by putting everything on one side and proving it for be more than 0
 

Demento1

Philosopher.
Joined
Dec 23, 2011
Messages
866
Location
Sydney
Gender
Male
HSC
2014
Re: 2012 Year 9 &10 Mathematics Marathon

Well for i) Probability of winning first dart = p

Probability of winning with second dart given first dart did not hit

If you expand that you should get:


For ii) To win, she needs to hit the target twice, there are 4 possibilities:

There are 3 possibilities of hitting 2 darts with one missing (i.e. 1st and 2nd, 2nd and 3rd, 1st and 3rd).

The probability of this is:

1 permutation of hitting with all 3 darts:

So the total probability


For iii) I hope I didn't type anything wrong...




If Darcy is to be more likely to win game 1 than game 2, then from above:





Since







Therefore for 0 < p < 1, this statement is true and the induction holds.


For iv), algebra can really finally be used to some extent.



Divide by p, get it into the format of and advance:





Thus, you have a quadratic equation to solve and the quadratic formula is used which I cannot bother with typing unfortunately...

Eventually you get p = 1.39, 0.36. Only one of these values is between our limit 0 < p < 1, so the answer would be when p = 0.36 (2 dp).
 
Last edited:

Users Who Are Viewing This Thread (Users: 0, Guests: 2)

Top