HSC 2012 MX1 Marathon #1 (archive) (1 Viewer)

[Drongoski]

Re: 2012 HSC MX1 Marathon

View attachment 24385



QUESTION:
Prove by induction that n lines in the plane (no two being parallel and no three concurrent) divide the plan into regions.

By the way:

= no of pairs from n+1 distinct objects.

Edit

= no of possible pairs from (n+1) distinct objects.

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

2Cos(360-48) = 2cos48 which = cos(180-48) right/wrong?

right.

Wrong.

cos(180 - 48) = -cos(48), which is not the same thing as 2cos48.

 

[RivalryofTroll]

Re: 2012 HSC MX1 Marathon

Wrong.

cos(180 - 48) = -cos(48), which is not the same thing as 2cos48.

I think Examine deleted the second bit so yeah..... Nooblet is referring to only cos(360-x) = cosx
I already mentioned cos(180-x) = -cosx anyways :/

 

[Examine]

Re: 2012 HSC MX1 Marathon

Need halp again

Attempted this question though got it wrong numerous times, or maybe the answer in the booklet is wrong, not sure which one.

Let x=cos48, Express 2cos312xcos132/5cos228+3cos132 in terms of x

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

By the way:

An interesting problem would be to prove the question from scratch, and somehow relating it to that particular arithmetic series.

 

[Examine]

Re: 2012 HSC MX1 Marathon

Nvm I know what I got wrong now.

 

[Examine]

Re: 2012 HSC MX1 Marathon

So what is=

sin(360-x)
cos(360-x)
tan(360-x)

AND
sin(360+x)
cos(360+x)
tan(360+x)

Need it for future references

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

So what is=

sin(360-x)
cos(360-x)
tan(360-x)

AND
sin(360+x)
cos(360+x)
tan(360+x)

Need it for future references

Use the unit circle.

Check your pm.

 

[math man]

Re: 2012 HSC MX1 Marathon

An interesting problem would be to prove the question from scratch, and somehow relating it to that particular arithmetic series.

do you mean prove the ap sum formula for that particular series?

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

do you mean prove the ap sum formula for that particular series?

Oh, I meant proving from scratch that the 2D plane is divided into 1/2 x n x (n+1) regions are created by drawing n non-parallel lines.

 

[nightweaver066]

Re: 2012 HSC MX1 Marathon

Oh, I meant proving from scratch that the 2D plane is divided into 1/2 x n x (n+1) regions are created by drawing n non-parallel lines.

The proof has a nice trick to it. Teacher showed it to us in class.

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

The proof has a nice trick to it. Teacher showed it to us in class.

I can think of a combinatoric proof. What is the one that your teacher showed you?

 

[Drongoski]

Re: 2012 HSC MX1 Marathon

So what is=

sin(360-x)
cos(360-x)
tan(360-x)

AND
sin(360+x)
cos(360+x)
tan(360+x)

Need it for future references

In each case you get the same value as with the 360 omitted.

e.g. cos(360 - x) = cos(-x) = cos x

tan(360+x) = tan x

 

[nightweaver066]

Re: 2012 HSC MX1 Marathon

I can think of a combinatoric proof. What is the one that your teacher showed you?

Graphical. Drew it out and explained it. By the way it was a proof by induction.

The question was worded differently but i'm pretty sure they are the same thing.

 

[someth1ng]

Re: 2012 HSC MX1 Marathon

it's a lot more fun.

I would have to disagree with this bit - Extension 2 isn't fun. lol.

 

[Carrotsticks]

Re: 2012 HSC MX1 Marathon

I would have to disagree with this bit - Extension 2 isn't fun. lol.

Must... not.... neg....

 

[AAEldar]

Re: 2012 HSC MX1 Marathon

I would have to disagree with this bit - Extension 2 isn't fun. lol.

Clearly doing something wrong then! It was by far the most interesting and enjoyable subject I did (besides Music, but that was because it was prac 90% of the time).

 

[math man]

Re: 2012 HSC MX1 Marathon

I would have to disagree with this bit - Extension 2 isn't fun. lol.

Someth1ng you are completely and 200% right, extension 2 isn't fun at all
English is just a terrible subject in general, all levels not just ext 2

 

[Examine]

Re: 2012 HSC MX1 Marathon

So how do you do addition/subtraction/multiplication/division of trigonometry without calcs or tables?

For example:

Tan^260/sin45-cos30

OR

Sin^2 15 + sin^2 30 + cos^2 15 - cos^2 45

 

[AAEldar]

Re: 2012 HSC MX1 Marathon

So how do you do addition/subtraction/multiplication/division of trigonometry without calcs or tables?

For example:

Tan^260/sin45-cos30

OR

Sin^2 15 + sin^2 30 + cos^2 15 - cos^2 45

For the first one, use your exact values. You know them for 30, 45 and 60 and so just by subbing them in you'll reach the answer.

For the second one, same thing. But you don't know the exact value of 15 however you do know that so group the terms and use that.