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Trinity Admissions Test Solutions

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Original post by JakeThomasLee
Specimen Paper III, Question 8:

Spoiler



I disagree with your answer. Squares do not have to be diagonal, they can be just slightly slanted.
Original post by Renzhi10122
I disagree with your answer. Squares do not have to be diagonal, they can be just slightly slanted.


Ah, yes you're right
Original post by JakeThomasLee
I currently have this integral, not too sure how I'm going to evaluate it, although it may just be completely obvious and I can't see it :colonhash:

Unparseable latex formula:

\RIghtarrow I=\displaystyle\int_0^{\frac{\pi }{2}} \dfrac{\cos(\theta)}{\sin(\theta )+\cos(\theta)} \ d\theta[br]



Use the same sub as for the other one. . .
I hope the others made it clear what I did :smile:
Original post by JakeThomasLee
I currently have this integral, not too sure how I'm going to evaluate it, although it may just be completely obvious and I can't see it :colonhash:

Unparseable latex formula:

\RIghtarrow I=\displaystyle\int_0^{\frac{\pi }{2}} \dfrac{\cos(\theta)}{\sin(\theta )+\cos(\theta)} \ d\theta[br]



Consider J=∫0π2sin⁡θsin⁡θ+cos⁡θ dx\displaystyle J = \int_0^{\frac{\pi}{2}} \frac{\sin \theta}{\sin \theta + \cos \theta} \, \mathrm{d}x then look at I+JI + J and I−JI-J.
Original post by JakeThomasLee
I currently have this integral, not too sure how I'm going to evaluate it, although it may just be completely obvious and I can't see it :colonhash:

Unparseable latex formula:

\RIghtarrow I=\displaystyle\int_0^{\frac{\pi }{2}} \dfrac{\cos(\theta)}{\sin(\theta )+\cos(\theta)} \ d\theta[br]



Can't you just divide top and bottom by cos (theta)

Giving 1/(1+tan)

Let u = tan

And using sec ^2 = tan ^2 +1

Gives,

1/(1+u)(1+u^2) which can be done by partial fractions?

Not the most elegant though


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Original post by Zacken
Consider J=∫0π2sin⁡θsin⁡θ+cos⁡θ dx\displaystyle J = \int_0^{\frac{\pi}{2}} \frac{\sin \theta}{\sin \theta + \cos \theta} \, \mathrm{d}x then look at I+JI + J and I−JI-J.


Deja vu, Siklos somewhere!


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Original post by I am Ace
Deja vu, Siklos somewhere!


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I will forever remember that STEP question.
Original post by I am Ace
Deja vu, Siklos somewhere!


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I better get reading... :o: :s-smilie:
Original post by physicsmaths
i don't agree with part 1, surely it is P(heads/ 3 heads) which is an 1/8?
then again if he is right can someone explain since i am **** at probability.
got it now my division of numbers was wrong


The probability of t−2,t−1t-2, t-1 and tt all being heads is 11 though, by definition tt. So the probability that s=t+1s=t+1 is just the probability the coin is flipped heads at t+1t+1.
Original post by MadChickenMan
The probability of t−2,t−1t-2, t-1 and tt all being heads is 11 though, by definition tt. So the probability that s=t+1s=t+1 is just the probability the coin is flipped heads at t+1t+1.


p( h|3 heads)=1/2 I had an error with numbers causing me to get 1/8


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Original post by joostan
Specimen Paper 2, Question 9:
Caveat - my probability is a bit weak and I'm not convinced about the last part.
Also succinct notation was a pain.

Spoiler



for part c why is it 48 and not 52-i for the number of cards in which to draw a king?
Is there a thread for the Natural Sciences solutions?
Original post by ak33m98
Is there a thread for the Natural Sciences solutions?


Don't think so. U might aswell make one then


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Original post by mattyd375
for part c why is it 48 and not 52-i for the number of cards in which to draw a king?


Because you want to pull a King from the pack, without pulling a Queen as this would change ii though as I said - I'm not entirely certain I'm right.
Original post by ak33m98
Is there a thread for the Natural Sciences solutions?


NatSci's aren't as cool as Mathmos, it seems. :wink:

jk, I don't think they have a thread - you might want to make one yourself, but it seems a little late for this years interview session.
Test 4, Question 5

Hint

You have two unknowns - so you need two equations. Use the trigonometry the question suggests to find them.


Solution

Spoiler

Test 5, Question 1 (Maths and Physics, Paper 1)

Hint:

Spoiler

Solution:

Spoiler

(edited 8 years ago)
Test 5, Question 5 (Maths and Physics, Paper 1)

Thoroughly enjoyed this question.

Hint:

Spoiler

Solution:
(a)

Spoiler

(b)

Spoiler

Original post by Zacken
Test 5, Question 5 (Maths and Physics, Paper 1)

Thoroughly enjoyed this question.

Hint:

Spoiler

Solution:
(a)

Spoiler

(b)

Spoiler



How did you manage to get Test 5 out of 4 numbered tests?
Original post by FutureOxford3
How did you manage to get Test 5 out of 4 numbered tests?


Zacken
(Maths and Physics, Paper 1)


...

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