Welcome to the Trinity Admissions Test Solutions Thread.This is the place to discuss, post problems, or ask any questions you may have regarding the exam before your interview!
Here we are going to post solutions to all of the sample Trinity College Cambridge Admissions Tests available online, in the hope to help those preparing for interviews in the coming years. These tests will be useful preparation for all Cambridge Colleges, and Oxford for any mathematical interview.
We do ask that any solutions are spoilers, and, if you are stuck with any questions to first ask for any hints or look at the hints given before the TSR solutions.
When preparing for your interview at Trinity College specifically, please remember that you are not expected to answer all of the questions, or necessarily to give complete answers to questions. You should just attempt those that appeal to you, and they will be used as a basis for discussion in the interview that follows. It has also been mentioned that they will most likely ignore any full solutions you have done in the test, and have you attempt the questions you have not done.
Once finished, each solution will be linked below to the relevant post!
SPECIMEN TEST I
1. Solution by Edothero
2. Solution by Zacken
3. Solution by Zacken
4. (1)Solution by Number Nine(2) DFranklin
5. Solution by Jordan\
6. Solution by 16Characters...
7. Solution by 16Characters...
8. Solution by 16Characters...
9. Solution by Joostan
10. Solution by Edothero;Part C by DFranklin
SPECIMEN TEST II
1. (1) Solution by Zacken(2) DFrankin
2. Solution by Zacken
3. Solution by Krollo
4. Solution by Zacken
5. Solution by MadChickenMan
6. Solution by Jordan\
7. Solution by Krollo
8. Solution by Renzhi10122
9. Solution by Joosten
10. Solution by Krollo /with some further comments (1) astruser (2) DFranklin
SPECIMEN TEST III
1. Solution by Joostan
2. Solution by Joostan
3. Solution by Hauss
4. Solution by ManChickenMan
5. Solution by Krollo
6. Solution by Joostan
7. Solution by Krollo
8. Solution by Euclidean
9. Solution by MadChickenMan
10. Solution by Krollo
SPECIMEN TEST IV
1. Solution by Krollo
2. Solution by 16Characters
3. Solution by Zacken
4.
5. Solution by Krollo
6. Solution by joostan
7. Solution by Krollo
8. Solution by Mop3476
9. Solution by Krollo
10. Solution by Krollo
Maths and Physics Paper I
We have also came across the Downing Admissions Paper,
where you can also find the solutions to below:
1.
2.
3.
4. Solution by Number Nine
5.
6.
7.
8.GOOD LUCK!
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Trinity Admissions Test Solutions watch

username1763791
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 26102015 13:57
Last edited by Zacken; 03042016 at 21:02. 
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 26102015 14:01
Specimen Test 1, Question 2:
Hints:Spoiler:Full solution:Show
Draw a diagram.
Label the angles the lines make with the positive xaxis, how can you relate these angles to the tangents of the slope?
Spoiler:Show
Let the angle that makes with the positive axis be and the angle that makes be , then we know that and .
A quick sketch shows us that and hence, applying the tangent to both sides yields:
, since we only want the acute angle, then we need , so we have, finally:
We note that our lines cannot be perpendicular since and that and cannot be vertical.
Last edited by Zacken; 26102015 at 14:08. 
16Characters....
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 26102015 14:02
So we post solutions for them to be put in? If so I have one for Q2. Also I am not a Trinity applicant, but do you think these questions are worthwhile doing to prepare for my St John's interview (if I get one)?
Edit: Whoops Zacken beat me to it anyway :) 
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 26102015 14:05
(Original post by 16Characters....)
So we post solutions for them to be put in? If so I have one for Q2. Also I am not a Trinity applicant, but do you think these questions are worthwhile doing to prepare for my St John's interview (if I get one)?
Edit: Whoops Zacken beat me to it anyway :) 
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 26102015 14:07
(Original post by 16Characters....)
So we post solutions for them to be put in? If so I have one for Q2. Also I am not a Trinity applicant, but do you think these questions are worthwhile doing to prepare for my St John's interview (if I get one)?
Edit: Whoops Zacken beat me to it anyway :)
Also, If you're solution is any different in the approach it'll still be useful to post it! 
16Characters....
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 26102015 14:17
(Original post by Jordan\)
Definitely  these will be helpful for any college interview. I myself have applied to Pembroke, and Zacken^ for Kings I believe
Also, If you're solution is any different in the approach it'll still be useful to post it!
Speciment Test 1, Question 6
Hints
Spoiler:Show
For the first part consider the necessary condition for the object to decelerate..
The second part can either be approached using energy considerations or a combination of Newton's Second Law and the SUVAT equations.
SolutionSpoiler:Show
Let m be the mass of the packing case. For the sliding packing case to eventually come to a rest there must be a resultant force acting up the slope. Hence resolving parallel up the slope:
Since I can safely divide by without altering the direction of the inequalities hence
The work done by the force up the slope in decelerating the packing case is equal to the initial KE of the object, where u is the intitial speed. Hence:
Last edited by 16Characters....; 26102015 at 14:46. 
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 26102015 14:41
Specimen Test 1, Question 3:
Hints:Spoiler:Full solution:ShowIntegration by parts, although I'm sure there's a more elegant method that I can't spot. 
Number Nine
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 26102015 14:44
Specimen Test 1 Question 4
Hint:Spoiler:Solution:ShowTry finding numbers that aren't multiples of 2,3,5 for numbers not quite as big as 6000Spoiler:ShowUsing 2 x 3 x 5 = 30 and 6000 ÷ 30 = 200:
Numbers that aren't multiples of 2,3,5 between 1 and 30 are as follows
1, 7, 11, 13, 17, 19, 23, 29
There are 8 numbers here, 8 * 200 = 1600
Therefore there are 1600 numbers between 16000 that aren't multiples of 2, 3 and 5 
DFranklin
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 26102015 14:50
(Original post by Number Nine)
Specimen Test 1 Question 4
Hint:Spoiler:Solution:ShowTry finding numbers that aren't multiples of 2,3,5 for numbers not quite as big as 6000Spoiler:ShowUsing 2 x 3 x 5 = 30 and 6000 ÷ 30 = 200:
Numbers that aren't multiples of 2,3,5 between 1 and 30 are as follows
1, 7, 11, 13, 17, 19, 23, 29
There are 8 numbers here, 8 * 200 = 1600
Therefore there are 1600 numbers between 16000 that aren't multiples of 2, 3 and 5
Spoiler:ShowSince 2,3,5 are prime, removing multiples of one number doesn't affect the proportion of multiples of the others. So the answer is simply 6000 * (1/2) * (2/3) * (4/5) = 1600. 
16Characters....
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 26102015 15:11
Speciment Test 1, Question 8
HintSpoiler:SolutionShowAgain conservation of energy, but rotational KE must be considered as opposed to regular KE.Spoiler:ShowLast edited by 16Characters....; 26102015 at 15:12. 
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 26102015 15:19
SPECIMEN TEST 1, QUESTION 5
HintsSolutionLast edited by username1763791; 26102015 at 15:23. 
tiny hobbit
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 26102015 15:19
(Original post by Jordan\)
[center]
When preparing for your interview at Trinity College specifically, please remember that you are not expected to answer all of the questions, or necessarily to give complete answers to questions. You should just attempt those that appeal to you, and they will be used as a basis for discussion in the interview that follows. 
DFranklin
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 26102015 15:20

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 26102015 15:24
Could anyone help me with question 10 of this paper:
https://share.trin.cam.ac.uk/sites/p...ampletest3.pdf
Do I assume that volume of the dust is equal to the volume of the spaceship? Do I also assume that the spaceship has unit length, so the volume equals the cross sectional area? And also my answer has the constant velocity,'u', in it is it not meant to?
Thank you. 
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 26102015 15:24
(Original post by tiny hobbit)
They may also just ignore any complete solutions that you have done and ask you about the questions you hadn't attempted. This is what happened to my son (several years back). 
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 26102015 15:39
(Original post by DFranklin)

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 26102015 15:45
(Original post by Gifted)
Could anyone help me with question 10 of this paper:
https://share.trin.cam.ac.uk/sites/p...ampletest3.pdf
Do I assume that volume of the dust is equal to the volume of the spaceship? Do I also assume that the spaceship has unit length, so the volume equals the cross sectional area? And also my answer has the constant velocity,'u', in it is it not meant to?
Thank you.
Last edited by StrangeBanana; 26102015 at 15:53. 
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 26102015 15:55
(Original post by Jordan\)
Zacken^ for Kings I believe
What made you choose Pembroke? I hear the food's fantastic.
(Original post by Zacken)
xLast edited by StrangeBanana; 26102015 at 15:58. 
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 26102015 15:58
(Original post by StrangeBanana)
I don't think you need the length of the spaceship, or to assume anything about the volume of the dust, but, I do think you have to assume that dust only sticks to the front of the ship. I'm not sure if my solution is correct or not, but here it is:

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 26102015 16:01
Specimen Test 1, Question 7
HintSpoiler:SolutionShow
For the first part consider what each of the two possible cases means with regards to what set the other elements in the subset come from, and remember that
Total number of subsets = Number of subsets containing 1 + Number of subsets NOT containing 1
The second part is similar to the first but by considering another element in the set {1, 2,... , n}
Spoiler:Show
Consider a subset of S = {1,2,....,n}.There are possible subsets containing r elements (since this is the number of combinations of r elements).
Each possible subset can either contain 1 or not contain 1. If it contains 1 then the other (r1) elements must come from {2, 3, ... , n} of which there are combinations. If it does not contain 1 then all r elements come from {2, 3, ..., n}, of which there are combinations. Hence
Equally, a subset of S with r elements either contains 2 or does not contain 2. This gives rise to 4 possibilities:
 A given subset of S contains 1 and 2
 A given subset of S contains 1 but NOT 2
 A given subset of S contains 2 but NOT 1
 A given subset of S contains neither 1 or 2
In the first case the other (r2) elements of the subset come from {3, 4,...n} of which there are combinations.
In the second case the other (r1) elements of the subset come from {2, 3,..., n} of which there are combinations. The third case can be derived by replacing the "2" in {2, 3,..., n} with a "1" hence there are subsets for the third case.
In the final case all r elements of the subset come from {3, 4,..., n}, of which there are combinations.
Hence the total number of subsets is
Last edited by 16Characters....; 26102015 at 20:39.
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