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# cambridge interview for maths watch

1. (Original post by fishpaste)
It'a case of saying that a set of triple primes will be in the form

n, n + 2, n + 4

and in a set of three consecutive integers you have

n, n + 1, n + 2

one must be divisible by three

n and n + 2 cannot be divisible by three if these numbers appear in a set of triple primes
so n + 1 must be divisible by three, but if n +1 is divisible by three then n + 4 must be, and so n + 4 is not prime unless n = 3, which is why 3,5,7 is the only set.
that is sort of wot i said isnt it? i have seen much harder qs than that
2. well i now know that e^pi is bigger...but how could u show that again?
was it by drawing the graphs of y=pi^x and y=e^x and then seeing which was bigger when x=e and x=pi respecitively
3. yeurck you get a test before your interview for cambridge? how AWFUL! thats so nasty, i dont think i could cope with it! i mean...interviews are bad enough, hmm..
4. What like medicine?
5. (Original post by Dill)
yeurck you get a test before your interview for cambridge? how AWFUL! thats so nasty, i dont think i could cope with it! i mean...interviews are bad enough, hmm..
Interviews are much worse. They're like tests, with impatient tutors glaring at you.
6. (Original post by lgs98jonee)
that is sort of wot i said isnt it? i have seen much harder qs than that
Well when presented with a sheet of A3 paper, a pen, and told "Prove that the only triple primes are 3, 5 and 7." I was rather distressed. It's a pretty plain, uninteresting proof to look back on, but when you've never seen it before, and don't know where to start, I assure you it's a really terrifying prospect.
7. (Original post by fishpaste)
Well when presented with a sheet of A3 paper, a pen, and told "Prove that the only triple primes are 3, 5 and 7." I was rather distressed. It's a pretty plain, uninteresting proof to look back on, but when you've never seen it before, and don't know where to start, I assure you it's a really terrifying prospect.
i believe u....i am not looking forward to a cambridge interview at all + a hard test...isnt the trinity one, called 'the trinity quiz'.
just hope i get a q that i have seen be4
8. (Original post by lgs98jonee)
i believe u....i am not looking forward to a cambridge interview at all + a hard test...isnt the trinity one, called 'the trinity quiz'.
just hope i get a q that i have seen be4
I had my interview in december at Trinity College, I attempted the following questions (applied for maths with comp sci):

(yes Trinity test is called Trinity mathematics Quiz 2003)

1) Prove that the product of four consecutive numbers is always a multiple of 24.
2)What is the highest power of 2 which will divide 20! as close as possible i.e. near 1 and what will be the highest power of 10 which will divide 20!?
3)Definite Integral [dx/1+sinx] with limits being pi and 0
4)Find all pairs of integers m and n which satisfy m^n = n^m.
5)y^2 + (x-2)^2 = k^2 and y = kx find all possible values of k.

I did extremely bad did not even get one question right!! and guess what got rejected..

I know the answers for all except (4) which is solved by theone anyway.

Anyone wnat to have a go at them?

You can also look at a past Trinity Maths Interview Quiz at

http://www.trin.cam.ac.uk/show.php?dowid=4

I would imagine it to be near impossible that you would get a questions which you have seen before otherwise the whole point of the test will be wasted.

Thanks (deserve reputation for this!)
9. For the first one, write the consecutive numbers
n(n+1)(n+2)(n+3), then if n is odd, n+1, and n+3 are multiples of 2. Either n+1 or n+3 is a multiple of 4. n or n+2 must be a multiple of 3.

n - even. Same arguements.

2) A great trick i read. The highest power of a prime into a factorial is found as follows.
Sum from i=1 to infinity of [20/2^i]
Where [x] is the greatest integer function.

Lol, except it doesnt seem to work.
10. (Original post by integral_neo)
I would imagine it to be near impossible that you would get a questions which you have seen before otherwise the whole point of the test will be wasted.

Thanks (deserve reputation for this!)
I beg to differ. I had one question that I had seen and done a couple of times before:

If f(n) is the nth fibonnacci number, prove that f(n+1)f(n-1) - (f(n))^2 = (-1)^n.
11. I mistyped 2 in my calculator to check, it is infact correct, 18 is the highest power.
12. how would you find the answer to question (3) the definite integral one
13. For 5, if you differentiate the first equation, then find k such that y = kx is a tangent to the circle, then all k greater than that, satisfy those conditions, because we have a line crossing a circle with centre (2,0)

Actually, we will have 2 solutions to the tangent, so k will be above one number or less than some other number.
14. (Original post by integral_neo)
how would you find the answer to question (3) the definite integral one
I think you might want to try the substitution t = tan(x/2), correct me if this doesn't lead anywhere.
15. (Original post by JamesF)
For 5, if you differentiate the first equation, then find k such that y = kx is a tangent to the circle, then all k greater than that, satisfy those conditions, because we have a line crossing a circle with centre (2,0)

Actually, we will have 2 solutions to the tangent, so k will be above one number or less than some other number.
there is no doubt that ur right. However, a gemetrial approach was needed to solve this, at my interview, i suggested algebriac manipulation and the interviwer responded it can be solves far more easily using geometry. anyone try that
16. (Original post by theone)
I think you might want to try the substitution t = tan(x/2), correct me if this doesn't lead anywhere.
It doesnt work, however ur right that a substitution is needed.

i obviously know the answer as they told me during the interview..
17. (Original post by integral_neo)
It doesnt work, however ur right that a substitution is needed.

i obviously know the answer as they told me during the interview..
Are you sure it doesn't work, maybe i've got a problem with the limits or something but I get 2, which is the right answer...
18. (Original post by theone)
Are you sure it doesn't work, maybe i've got a problem with the limits or something but I get 2, which is the right answer...
Yeah 2 is the right answer but they used the substitution of t = cosx
19. didnt know that tan (x/2) will work as well.. how did u realise this?
20. (Original post by integral_neo)
Yeah 2 is the right answer but they used the substitution of t = cosx
Interesting, I thought the standard way to approach these types of question was the t-substitution... I can't get their substitution to work any easier...

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