# Thread: HSC 2012 Marathon :)

1. ## Re: HSC 2012 Marathon :)

yeh i typed it up and noticed. But i dont see how that is right.

2. ## Re: HSC 2012 Marathon :)

Consider the straight line y=x. You can multiply the RHS by x/x without changing its value. Thus the straight line becomes y=x^2/x which suddenly can't have a value of 0, even though the line hasn't changed and its still defined at x=0

3. ## Re: HSC 2012 Marathon :)

Originally Posted by deswa1
Consider the straight line y=x. You can multiply the RHS by x/x without changing its value. Thus the straight line becomes y=x^2/x which suddenly can't have a value of 0, even though the line hasn't changed and its still defined at x=0
No no no. You cannot just divide by x and assume that it takes on all real values. As soon as you divide by x you're assuming that it can't equal zero.

4. ## Re: HSC 2012 Marathon :)

Originally Posted by AAEldar
No no no. You cannot just divide by x and assume that it takes on all real values. As soon as you divide by x you're assuming that it can't equal zero.
Yep because x can be any value including zero so multiplying by x/x is erroneous

(i think)

6. ## Re: HSC 2012 Marathon :)

Originally Posted by mirakon
Yep because x can be any value including zero so multiplying by x/x is erroneous
Yep, whoops didn't think of that. Why then does wolfram show the curve continuous at 0?

7. ## Re: HSC 2012 Marathon :)

Originally Posted by deswa1
Yep, whoops didn't think of that. Why then does wolfram show the curve continuous at 0?
Because x=0 is an infinitely small point so barely noticeable.

8. ## Re: HSC 2012 Marathon :)

sorry for turning this thread into a pedantic fest, next question!

solve for x.

e^(2x) - 7e^(x) + 9 > 0 (difficulty: moderately intriguing)

highlight below for the answer:

9. ## Re: HSC 2012 Marathon :)

Originally Posted by barbernator
sorry for turning this thread into a pedantic fest, next question!

solve for x.

e^(2x) - 7e^(x) + 9 > 0 (difficulty: moderately intriguing)

highlight below for the answer:

Is it
x < Ln(3.5 - sqrt(13)/2) and x > Ln(3.5 + sqrt(13)/2)

lol

10. ## Re: HSC 2012 Marathon :)

Originally Posted by Timske
Is it
x < Ln(3.5 - sqrt(13)/2) and x > Ln(3.5 + sqrt(13)/2)

lol
correct

11. ## Re: HSC 2012 Marathon :)

Given f(x) = arccos(1/x)

State it's domain

12. ## Re: HSC 2012 Marathon :)

Originally Posted by Timske
Given f(x) = arccos(1/x)

State it's domain
What would you say the difficulty is?

13. ## Re: HSC 2012 Marathon :)

Originally Posted by Timske
Given f(x) = arccos(1/x)

State it's domain

x>1, x<-1 (greater or equal)

Medium

15. ## Re: HSC 2012 Marathon :)

Originally Posted by barbernator
x>1, x<-1 (greater or equal)
Nice

16. ## Re: HSC 2012 Marathon :)

bit of algebra practice:

rearrange to make the subject y, also, for the first one, put in the form (x-k)(y-h)=c

$2xy-3x-y-2=0\\ and\\ y^2 - 8y + 5x - x^2 - 15 = 0$

(rating: green curry)

17. ## Re: HSC 2012 Marathon :)

1. 2xy - 3x - y - 2 = 0
y(2x - 1) = 3x + 2
y = (3x+2)/(2x-1)
2xy - 3x - y - 2 = 0 -> (x - 1/2)(y - 3/2) = 1.75

2. y^2 - 8y + 5x - x^2 - 15 = 0
y^2 - 8y - 15 = x^2 - 5x
(y-4)^2 - 31 = x^2 - 5x
(y-4)^2 = x^2 - 5x + 31
y - 4 = sqrt(x^2 - 5x + 31)
:. y = sqrt(x^2 - 5x + 31) + 4

18. ## Re: HSC 2012 Marathon :)

The foot D of the altitude BD of a triangle ABC lies between A and C. Prove that:
$CD-AD=\frac{BC^2-AB^2}{AC}$

19. ## Re: HSC 2012 Marathon :)

$TBP: ~CD-AD=\frac{(BC)^2-(AB^2)}{AC}\\ \\ let~AD=x,~DC=y,~DB=h\\ \\ by~pythagoras~theorum\\ \\ RHS=\frac{(BC)^2-(AB^2)}{AC}\\ \\ =\frac{(h^2+y^2)-(h^2+x^2)}{x+y}\\ \\ =\frac{y^2-x^2}{x+y}\\ \\ =y-x\\ \\ =CD-AD\\ \\ =LHS~as~required$
NEW QUESTION:
The letters of the word PERSEVERE are arranged in a row. How many different arrangements are possible.(easy)

for those who want to post solutions but dont know how to properly, http://www.codecogs.com/latex/eqneditor.php

lol whoops need to edit my solution

20. ## Re: HSC 2012 Marathon :)

^ I've used that before but i can't copy/paste it

$x\\ + \\ y$

21. ## Re: HSC 2012 Marathon :)

Originally Posted by Timske
^ I've used that before but i can't copy/paste it
copy the code box from down the bottom and paste that

22. ## Re: HSC 2012 Marathon :)

how come when i press space bar it doesnt actually show a space on the latex image?

23. ## Re: HSC 2012 Marathon :)

Originally Posted by bleakarcher
how come when i press space bar it doesnt actually show a space on the latex image?
To do a space in latex you can either do "\;" and it'll put a space (I think "\," does the same thing). Or you can use the \text{ } and put a space in the braces, OR you can use the  tags and put a space there.

EDIT: Didn't know you could do what barbernator said!

24. ## Re: HSC 2012 Marathon :)

Originally Posted by bleakarcher
how come when i press space bar it doesnt actually show a space on the latex image?
~ between words

25. ## Re: HSC 2012 Marathon :)

my keyboard is fucked. it doesnt have that backward slash but it has the forward slash. FUUUUUUUUUUUUUUUUUUUUUUUUUUUUU

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