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STEP Prep Thread 2016 (Mark. II)

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STEP II 2015:

Spoiler

(edited 7 years ago)
Original post by Zacken
sqrt(1+a) approx 1 + a/2 by the binomial expansion


Okay, many thanks!
Original post by Vesniep
I've done the paper today.


Well done!

Q 13:

Spoiler

Original post by EnglishMuon
Just a quick check: when showing arcoshx= ln{1+(x^2-1)^1/2} I am able to look at cosh { ln{1+(x^2-1)^1/2}}= x and deduce the result from this?


And mention ln(x + root(x^2 - 1)) is positive so that it is arcosh x and not -arcosh x?
Original post by 16Characters....
And mention ln(x + root(x^2 - 1)) is positive so that it is arcosh x and not -arcosh x?


and yep, did put that in my workings :smile:
Original post by sweeneyrod
Well done!

Q 13:

Spoiler



E(C) = P(X<y) * E(ky) + P(X>y) * E(ky+a(X-y) I Y>y ) =ky+ a e^(-λy) (E(X I X>y) - y)
E( X I X>y)= [ integral of (λe^(-λχ)) from y to infinity ] / P(X>y) = y + 1/λ
Comment : I know it was quiet difficult and it got me into trouble .
part ii was too much , very messy integrals I don't see the point
(edited 7 years ago)
just did 2013 ii - 105
Original post by Vesniep
E(C) = P(X<y) * E(ky) + P(X>y) * E(ky+a(X-y) I Y>y ) =ky+ a e^(-λy) (E(X I X>y) - y)
E( X I X>y)= [ integral of (λe^(-λχ)) from y to infinity ] / P(X>y) = y + 1/λ
Comment : I know it was quiet difficult and it got me into trouble .
part ii was too much , very messy integrals I don't see the point


I see, I forgot the given part. Thanks
When explaining a simple idea such as the first part of q5 III 2001, is it usually acceptable to use a diagram instead of words? i.e. in this case is it enough to differentiate and show f(x) is an increasing function and then give a rough sketch of a strictly increasing cubic (instead of saying as x tends to +- infinity....)? (followed by stating the conclusion)
Original post by 16Characters....
I also did STEP II 2015 today

Spoiler


Spoiler

Original post by Number Nine
just did 2013 ii - 105


which q's
Original post by Vesniep

Spoiler



Spoiler

Original post by EnglishMuon
When explaining a simple idea such as the first part of q5 III 2001, is it usually acceptable to use a diagram instead of words? i.e. in this case is it enough to differentiate and show f(x) is an increasing function and then give a rough sketch of a strictly increasing cubic (instead of saying as x tends to +- infinity....)? (followed by stating the conclusion)


It would be more rigorous especially if you mention Bolzano's theorem for continuous functions (at least one root) and that it's a 1-1 function (at most 1 root) but perhaps they'd accept it
Original post by physicsmaths

Spoiler


Didn't try it yet . I thought geometry => let's move on
Original post by gasfxekl
which q's


1,2,3,5,7,8
Original post by Vesniep
It would be more rigorous especially if you mention Bolzano's theorem for continuous functions (at least one root) and that it's a 1-1 function (at most 1 root) but perhaps they'd accept it


I mean i cant imagine mentioning bolzanos theorem would get u any more credit especially as you are not required to know it and it is just the same as saying there is only 1 root if u dont give proof.
Original post by EnglishMuon
Just a quick check: when showing arcoshx= ln{1+(x^2-1)^1/2} I am able to look at cosh { ln{1+(x^2-1)^1/2}}= x and deduce the result from this?


Well yes.


Posted from TSR Mobile
Original post by EnglishMuon
When explaining a simple idea such as the first part of q5 III 2001, is it usually acceptable to use a diagram instead of words? i.e. in this case is it enough to differentiate and show f(x) is an increasing function and then give a rough sketch of a strictly increasing cubic (instead of saying as x tends to +- infinity....)? (followed by stating the conclusion)


Yep that is perfectly fine. Infact much better.


Posted from TSR Mobile
Original post by EnglishMuon
I mean i cant imagine mentioning bolzanos theorem would get u any more credit especially as you are not required to know it and it is just the same as saying there is only 1 root if u dont give proof.


At least one root not one root.
Also you are not required to write the name of the theorem but I can remember from a-levels that sign change in continuous functions means a root in that interval so I guess you can use it and since it's an established result , wouldn't it be better ?
Original post by Vesniep
At least one root not one root.
Also you are not required to write the name of the theorem but I can remember from a-levels that sign change in continuous functions means a root in that interval so I guess you can use it and since it's an established result , wouldn't it be better ?


Question is 1 root not atleast. English muons idea is the best I feel, I would write it like this,
the curve is strictly increasing hence does not turn and come back for another root.
-infinity to infinity so it crosses atleast once, since it is strictly increasing it crosses exactly once.
For the atleast once root you can just say complex solutions come in pairs hence atleast one real root.

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