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# Trinity Admissions Test Solutions watch

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1. Test 5, Question 2 (Maths and Physics, Paper 1)

Hints:
Spoiler:
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Look at squares modulo 7 and split it into cases for different values of c^2

Solution:
Spoiler:
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(we neglect 0 for obvious reasons in this question)

Since we then have

Case 1: yields
Case 2: yields
Case 3: yields

In all cases, we have as required.
2. (Original post by Zacken)
Test 5, Question 2 (Maths and Physics, Paper 1)

Hints:
Spoiler:
Show
Look at squares modulo 7 and split it into cases for different values of c^2
Solution:
Spoiler:
Show

(we neglect 0 for obvious reasons in this question)

Since we then have

Case 1: yields
Case 2: yields
Case 3: yields

In all cases, we have as required.
Can you link me to it.
3. Trinity Test 4, Question 9

Hint
Spoiler:
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Consider using dtheta and dT, and try resolving radially and tangentially.

Solution
Spoiler:
Show

Diagram attached. I've replaced some symbols for each of typing.

and since 0.5dA is small, we can approximate thus:

since dT*dA is negligibly small.

Resolving tangentially:

since dA is small, and mu is 0.25. Letting dA, dT ->0, this gives us a differential equation which may be solved by separating the variables:

And since A=0 -> T=1:

So T = 10^6 -> A = 4*13.6 = 54.4
As this is the angle in radians, M = 54.4/2pi = 9 turns or so.
4. (Original post by Krollo)
Trinity Test 4, Question 9

Hint
Spoiler:
Show

Consider using dtheta and dT, and try resolving radially and tangentially.

Solution
Spoiler:
Show

Diagram attached. I've replaced some symbols for each of typing.

and since 0.5dA is small, we can approximate thus:

since dT*dA is negligibly small.

Resolving tangentially:

since dA is small, and mu is 0.25. Letting dA, dT ->0, this gives us a differential equation which may be solved by separating the variables:

And since A=0 -> T=1:

So T = 10^6 -> A = 4*13.6 = 54.4
As this is the angle in radians, M = 54.4/2pi = 9 turns or so.
Damn my mechanics is bad, don't even understand this mate.

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5. (Original post by physicsmaths)
Damn my mechanics is bad, don't even understand this mate.

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Neither do I

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6. (Original post by Krollo)
Trinity Test 4, Question 9

Hint
Spoiler:
Show

Consider using dtheta and dT, and try resolving radially and tangentially.

Solution
Spoiler:
Show

Diagram attached. I've replaced some symbols for each of typing.

and since 0.5dA is small, we can approximate thus:

since dT*dA is negligibly small.

Resolving tangentially:

since dA is small, and mu is 0.25. Letting dA, dT ->0, this gives us a differential equation which may be solved by separating the variables:

And since A=0 -> T=1:

So T = 10^6 -> A = 4*13.6 = 54.4
As this is the angle in radians, M = 54.4/2pi = 9 turns or so.
I'll take your word for it

I just scribbled the method I would use to solve it, didn't have the courage to try it myself
7. (Original post by Krollo)
Neither do I

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When is your interview? If i have no mechanics or stats i will bang it, but no mechanics and stats is unlikely

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8. (Original post by physicsmaths)
When is your interview? If i have no mechanics or stats i will bang it, but no mechanics and stats is unlikely

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14th, I think you're a bit later? I'm sure you'll be absolutely fine. I wouldn't mind applied myself, to be honest, but each to their own. :-)
9. (Original post by Krollo)
14th, I think you're a bit later? I'm sure you'll be absolutely fine. I wouldn't mind applied myself, to be honest, but each to their own. :-)
I would not mind pure geometry 😃! But yh mechanics i can work with sometiems and the same with probability but the topics are iffy for me. Yeh I am on the 18th!

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10. 15rootpi/8 for the maths and CompSci Q1 Test 1?

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11. (Original post by physicsmaths)
15rootpi/8 for the maths and CompSci Q1 Test 1?

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Yep (I think this was solved a page or so back)
12. (Original post by Krollo)
Yep (I think this was solved a page or so back)
Ah so those solutions are up aswell, Q2 is so quick.

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13. Anybody know how to do Q3 without using jensens equality?
This is how I have done it,

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14. (Original post by physicsmaths)
Anybody know how to do Q3 without using jensens equality?
This is how I have done it,

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Yeah, I had to use Jensen's. Although, Jensen's in the case of two is *fairly* intuitive. Btw, the next part is actually Holder's inequality.
Btw, I think you overcomplicated it, using weights of 1/p and 1/q, you dont have to do much manipulation.
15. (Original post by Renzhi10122)
Yeah, I had to use Jensen's. Although, Jensen's in the case of two is *fairly* intuitive. Btw, the next part is actually Holder's inequality.
Btw, I think you overcomplicated it, using weights of 1/p and 1/q, you dont have to do much manipulation.
Yh I figured it out after that I was using the p,q weirdly but I used p,q for the sake of the powers haha, , Yh holders is the one with 3 sequences right and the sum of powers on the least side right? And when it is two sequences it is just Cauchy right?

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16. (Original post by Renzhi10122)
Yeah, I had to use Jensen's. Although, Jensen's in the case of two is *fairly* intuitive. Btw, the next part is actually Holder's inequality.
Btw, I think you overcomplicated it, using weights of 1/p and 1/q, you dont have to do much manipulation.
Would they really expect everyone to know jensens inequality? That is trinity for you

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17. (Original post by physicsmaths)
Yh I figured it out after that I was using the p,q weirdly but I used p,q for the sake of the powers haha, , Yh holders is the one with 3 sequences right and the sum of powers on the least side right? And when it is two sequences it is just Cauchy right?

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Nah, more sequences is just more cauchy, i think. Holder's is where the powers arent 1/2 anymore (square root cauchy, you get square roots on the larger side), they are 1/p and 1/q where 1/p+1/q=1
18. (Original post by physicsmaths)
Would they really expect everyone to know jensens inequality? That is trinity for you

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Honestly, probably not. Only about max 20 or so people applying would know it, though some might spot the trick for the 2 variable case. I guess it would be like the new material kind of question, where they tell you the inequality in interview, and ask you to use it.
19. (Original post by Renzhi10122)
Nah, more sequences is just more cauchy, i think. Holder's is where the powers arent 1/2 anymore (square root cauchy, you get square roots on the larger side), they are 1/p and 1/q where 1/p+1/q=1
Yh that is what i meant, when the powers are 1/2 and it is 2 sequences holders becomes cauchy schwarz.
I would cautious of such questions if they ask me some crazy indepth questions about Holders inequality ill be like 'sorry i only know the basic proofs mate'. Atleast I am fairly certain I won't need this in my 30 minute test 😂.

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