Maths - 1997 Paper 1 Q7 (c) - Help

alan4cult

Sorry I posted incorrect question!

I can do part (i) but cannot do part (ii)

Any ideas?

ZorbaTehZ

I have a result but I'm not bothered posting the whole thing up cause it's so long.
Can you post up how far you got and I can say where you went wrong, or help?

ninebeanrows

This is the type of question i don't bother even attempting! Liklyhood is it will not come up in the Post-Modern Leaving Cert and if it does i will not be doing the question.

alan4cult

I posted the incorrect problem before (Sorry)
The new one is here
www.icefusionf1.com/prob.jpg

I cannot do part (ii)

JC 2K3

Corrected below.

alan4cult

Hey man,

You used the wrong equation yet that isn't a problem. Yes your version's the best I've seen, better than the dudes on yahoo answers.

Thanks,
Alan

JC 2K3

lol, sorry bout that, I mixed up a sign with the other equation, which made it work out the same anyway

Correction:
y = (x-1) + 1/(x-1)
xy - y = x^2 - 2x + 1 + 1
x^2 + x(-2-y) + (2+y) = 0

For x to be real:
b^2 - 4ac > 0
ie. (-2-y)^2 - 4(2+y) > 0
(-2-y)^2 > 4(2+y)
4 + 4y + y^2 > 8 + 4y
y^2 > 4
y > 2 or y > -2

ie. -2 < y > 2

alan4cult

Nice work!

seandoiler

JC 2K3 wrote:

y > 2 or y > -2

ie. -2 < y > 2

to be honest i suggest that you do not write your answer as -2< y >2, this is really a bad way to present the answer, you are better off with the first line y>2 or y<-2, hence y not an element of [-2,2]...just my suggestion...

JC 2K3

How so?

I see it written like that all the time....

Irish Wolf

JC 2K3 wrote:

y^2 > 4
y > 2 or y > -2

ie. -2 < y > 2

Not strictly correct it should be y > 2, or y < -2 and hence -2 > y > 2

seandoiler

Irish Wolf wrote:

Not strictly correct it should be y > 2, or y < -2 and hence -2 > y > 2

this is not correct either, you must be very carefully writing things like that, the reson is that "<" and ">" are what are called transitive relations so if you write -2>y>2 you are basically saying that -2 is greater 2, which is obviously rubbish.....now while -2 < y > 2 is just wrong as you pointed out.....simply write y<-2 or y>2

EDIT: I missed the inequality sign being the wrong way around in my first post

Fremen

ok, feeling a bit lazy so I didn't read through the whole thread.
It's quite easy to solve: pretend y is a constant C, then multiply across by (x-1)
You're left with a quadratic in terms of X and that constant.

Use the quadratic formula to show that if C is in a certain range, the square root term is complex.

I haven't done it, but I bet -2< C < 2 for a complex C

Edit: looks like JC2k3 beat me to it

seandoiler

Fremen wrote:

I haven't done it, but I bet -2< C < 2 for a complex C

jaysus i wonder how you managed to guess that simply from it saying it in the actual question :P

Irish Wolf

seandoiler wrote:

if you write -2>y>2 you are basically saying that -2 is greater 2

Nope... in this case you are saying that in order for x to be real y must be a real number which is either less than -2 or greater than 2 - so the above notation is correct.

x is a real number for (-2 > y > 2) y ∈R

seandoiler

Irish Wolf wrote:

Nope... in this case you are saying that in order for x to be real y must be a real number which is either less than -2 or greater than 2 - so the above notation is correct.
x is a real number for (-2 > y > 2) y ∈R

look sorry but that notation is just not correct... > is a transitive relation, he has written on the same line -2 > y > 2, follow it through and it implies -2 > 2, in order to write the answer in a correct, clear format it must be written as y < -2 or y>2, these are distinct cases and will have to be separated by the "or"....

another way to think of it is to realise that > introduces an ordering on the number line, by writing the answer as -2 > y > 2, the ordering of the number line is completely screwed up.....

alright i'm not writing on more on this....just realise that they are two separate cases that need dealing with and can not be treated as simultaneous occurences

Irish Wolf

seandoiler wrote:

follow it through and it implies -2 > 2

I can't understand how you can arrive at this conclusion? The notation is perfectly correct.

End of.

seandoiler

Irish Wolf wrote:

I can't understand how you can arrive at this conclusion? The notation is perfectly correct.

End of.

no not end of, please read my previous comment clearly....(1) as i said previoulsy ">" is a transitive relation, this means that if a > b and b > c, then a > c...is this okay? do you follow that?

now given (1) above and the notation you use you and -2 > y >2, the way this is written actually means -2 > y and y > 2, now since ">" is transitive this means that -2 > 2.....is this okay? do you follow the logic?

now clearly -2 > 2 is rubbish, this is why the answer must be written in the form y<-2 OR y >2, the OR here is vital as it implies that we are dealing with two completely separate and distinct cases, which is not implied in the way that you have written the answer......is this okay? again do you follow the logic?

Irish Wolf

seandoiler wrote:

no not end of, please read my previous comment clearly....(1) as i said previoulsy ">" is a transitive relation, this means that if a > b and b > c, then a > c...is this okay? do you follow that?

now given (1) above and the notation you use you and -2 > y >2, the way this is written actually means -2 > y and y > 2, now since ">" is transitive this means that -2 > 2.....is this okay? do you follow the logic?

now clearly -2 > 2 is rubbish, this is why the answer must be written in the form y<-2 OR y >2, the OR here is vital as it implies that we are dealing with two completely separate and distinct cases, which is not implied in the way that you have written the answer......is this okay? again do you follow the logic?

But we're dealing here with a set of values y. Once y satifies either side of the relationship then x is a real number. So as I have said previously the notation is correct. It's an interval.

That is my final word here - PM me if you'd like to continue the discussion.

seandoiler

i'll continue via pm

Irish Wolf

We're done here... I was making the mistake of using total ordering. What I should have been writing was x is NOT a real number for -2 < y < 2.

The correct solution to the above question is indeed -2 > y or y > 2.

Agreed?

seandoiler

well seeing as i think we're the only two discussing it, i'm absolutely fine with my answer being right : P

Tomlowe

seandoiler is right.

JC 2K3

*Takes notes.