NSAA Specimen Paper Worked Solutions

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    NSAA Specimen Paper Worked Solutions
    Hi there! As some of you are considering applying to Cambridge (or already decided to apply), you'll no doubt be looking towards preparing for the pre-interview assessments. I'm not sure how helpful this will be but Cambridge only provides the answers and so some of you might find it difficult in knowing how to get to that answer if you are stuck on a particular question. So I hope that having some worked solutions would be beneficial in your preparation. Good luck!

    Note: More information on the NSAA and links to all the original NSAA files as well as general information about Natural Sciences at Cambridge can be found here: http://www.undergraduate.study.cam.a...tural-sciences
    More information on the Admissions Assessments themselves can be found here: http://www.undergraduate.study.cam.a.../pre-interview

    Note: I do not study Biology and therefore will not be able to provide worked solutions to the Biology sections. If someone is willing to write up some worked solutions to those sections, you will be given full credit.

    Note: This is a WIP. Expect more solutions to all the different sections in the upcoming few weeks. Also, there are different ways to solve these questions and my solutions are far from optimal. If you have a better way to solve them, please do share!

    Section 1:

    Part A - Mathematics:
    Q1-4:
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    Q5-8:
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    Q9-12:
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    Q13-16:
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    Q17-18:
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    Part B - Physics:
    Q19-21:
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    Q22-25:
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    Q26-29:
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    Q30-31:
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    Q32-34:
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    Q35-36:
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    Part C - Chemistry:
    Q37-39:
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    Q40-42:
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    Q43-47:
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    Q48-51:
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    Q52-54:
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    Part D - Biology: (Sorry )
    Part E - Advanced Mathematics and Advanced Physics:
    Thank you Zacken for your (Partial) Solutions
    Q73-75:
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    Q76-78:
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    Q79-82:
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    Q83-88:
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    Q89-90:
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    (Original post by NeverLucky)
    .
    Are you also applying for NatSci this year? Really great that you're putting this up, as a friend and I are going to be starting a little study group next month to practice for the NSAA. Really feel like I am gonna die during that test though, especially in advanced mathematics and advanced physics..
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    (Original post by NeverLucky)
    x
    I can do the chemistry ones and send them to you if you don't have the time.

    Also could probably give the biology ones a decent go as I did cells in 1A.
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    (Original post by alow)
    I can do the chemistry ones and send them to you if you don't have the time.

    Also could probably give the biology ones a decent go as I did cells in 1A.
    I've already done the Chemistry ones. I just need to write it out all in neat etc. Thanks for the offer tho!

    Although, if you could do the Biology ones, that'll be great! You'll be given full credit of course.
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    (Original post by NeverLucky)
    I've already done the Chemistry ones. I just need to write it out all in neat etc. Thanks for the offer tho!

    Although, if you could do the Biology ones, that'll be great! You'll be given full credit of course.
    I'll do the bio ones tomorrow.
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    What are good scores for the papers do you think?
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    (Original post by metrize)
    What are good scores for the papers do you think?
    bit hard to say isn't it, since no one has actually sat it yet.
    I'd estimate an average of about 75% over the 3 sections you are sitting would be quite strong for the MCQ's?
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    (Original post by metrize)
    What are good scores for the papers do you think?
    (Original post by k.russell)
    bit hard to say isn't it, since no one has actually sat it yet.
    I'd estimate an average of about 75% over the 3 sections you are sitting would be quite strong for the MCQ's?
    I've found the questions to not be that incredibly challenging. The problem is the time limit you're given. It works out to be around 25 minutes for each section of the MCQ's which isn't that much considering some of the questions do require a fair amount of thought and insight. I would say a good score would be around 50% for the MCQ's.
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    (Original post by NeverLucky)
    I've found the questions to not be that incredibly challenging. The problem is the time limit you're given. It works out to be around 25 minutes for each section of the MCQ's which isn't that much considering some of the questions do require a fair amount of thought and insight. I would say a good score would be around 50% for the MCQ's.
    to be honest with you, I think 50% is probably around average. I'm not saying that I think I will get 75%, I just think that will be what the highest performing students will score (maybe I'm wrong)
    btw, do you want me to do the bio questions? have gone through and solved all of them now but haven't got a neat write up
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    (Original post by k.russell)
    to be honest with you, I think 50% is probably around average. I'm not saying that I think I will get 75%, I just think that will be what the highest performing students will score (maybe I'm wrong)
    btw, do you want me to do the bio questions? have gone through and solved all of them now but haven't got a neat write up
    Fair enough, while I wouldn't say 50% would be average, I think the highest performing students can get above 75-80%. I'd say 50% would make you a competitive candidate. Regardless, Cambridge look at the application holistically and they interview a large majority of candidates anyway. A bad score in the assessment can be negated by other parts of your application.

    Also, thanks for the offer. @alow said he'll do it but he hasn't said anything since then so if you want to do, that'll be great. I won't be able to update anything until next week anyway cos I don't have a laptop atm and trying to format something on a phone is the worst thing ever.
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    (Original post by NeverLucky)
    Fair enough, while I wouldn't say 50% would be average, I think the highest performing students can get above 75-80%. I'd say 50% would make you a competitive candidate. Regardless, Cambridge look at the application holistically and they interview a large majority of candidates anyway. A bad score in the assessment can be negated by other parts of your application.

    Also, thanks for the offer. @alow said he'll do it but he hasn't said anything since then so if you want to do, that'll be great. I won't be able to update anything until next week anyway cos I don't have a laptop atm and trying to format something on a phone is the worst thing ever.
    yep, I'll have them all finished by tomoz and can email them to you or something
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    (Original post by NeverLucky)
    Fair enough, while I wouldn't say 50% would be average, I think the highest performing students can get above 75-80%. I'd say 50% would make you a competitive candidate. Regardless, Cambridge look at the application holistically and they interview a large majority of candidates anyway. A bad score in the assessment can be negated by other parts of your application.
    I completely agree with this assessment, but just thought I'd post this as reassurance for people's peace of mind! The Cambridge website states that:

    "Experience with similar assessments and from trials indicates that, on average, typical applicants to the most highly selective undergraduate courses (who are by definition academically very able) will gain approximately half of the available marks. The best applicants will score more highly, but only relatively few are expected to gain more than 80% of the available marks."

    Hope this helps
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    Posting so I can follow this thread.
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    I made my own there (link here http://www.thestudentroom.co.uk/show...8#post67543266) and I was told to come comment here. My problem is basically in my opinion one specimen paper for each section is too less. I need more material to prepare for this test from. If anyone could tell me where I should prepare these subjects from: maths, physics, chemistry, advanced maths and advanced physics. I don't take bio so I'm not worried about that. But if you guys could tell me a source of past papers or questions at equal or slightly higher difficulty than this pre interview test that would be so awesome. Thank you!!
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    And also please if the source of questions/past papers has the same course and topics as the entry test that would be good because then it's just a waste of time finding relevant questions. Again thanks!
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    Not sure what the point of using x = \frac{1}{2} was for Q7, you should be able to pick out \frac{1}{x} immediately.
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    Section 1: "Advanced" Mathematics:

    73. Given that a^xb^{2x}c^{3x} = 2, where a,b,c are positive reals, then x=

    Rewrite a^x b^{2x} c^{3x} = a^x (b^2)^x (c^3)^x = (ab^2 c^3)^x = 2, so taking logs gives \displaystyle x \log (ab^2c^3) = \log 2 \Rightarrow x = \frac{\log 2}{\log (ab^2c^3)} .

    Alternatively, but really the same thing:
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    Take \log of both sides to get \log(a^x b^{2x} c^{3x}) = \log a^x + \log b^{2x} + \log c^{3x} via addition rule, then power rule gives x \log a + 2x \log b + 3x \log c , factoring out an x gives x(\log a + 2\log b + 3 \log c).

    So all in all, we have x(\log a + 2 \log b + 3 \log c) = \log 2 \Rightarrow x = \frac{\log 2}{\log a + 2 \log b + 3\log c} = \frac{\log 2}{\log (ab^2 c^3)} via the addition and power rule again.

    74. Particle P has a fixed mass of 2 kg and particle Q has a fixed mass of 5 kg. The two particles are moving in opposite directions along a straight line on a smooth plane. Particle P has a speed of 3 ms^-1 and particle Q has a speed of r ms^-1. The particles collide directly. After the collision the direction of each particle is reversed. The speed of P is now 1 ms^-1 and the speed of Q is halved. What is the value of r?

    Conservation of momentum before collision: 2 \times 3 - 5 \times r = -2 \times 1 + 5 \times \frac{r}{2}, solving this gives 12 - 10r = 5r-4 \Rightarrow 16 = 15r \Rightarrow r = \frac{16}{15}

    75. Which one of the following is largest in value?

    \tan \frac{3\pi}{4} = \tan \left(\frac{3\pi}{4} - \pi \right) = \tan \left(-\frac{\pi}{4} \right) = -1

    \log_{10} 100 = \log_{10} 10^2 = 2 \log_{10} 10 = 2

    \sin^{10} \frac{\pi}{2} = 1^{10} = 1

    \log_2 8 < \log_{2} 10 < \log_2 16 \Rightarrow 3 < \log_2 10 < 4

    0 < (\sqrt{2} - 1)^{10} < 1 since \sqrt{2} - 1 < 1 so (\sqrt{2} - 1)^{10} < 1.

    77. The sum of the roots of the equation 2^{2x} - 8 \times 2^x + 15 = 0 is?

    Let u = 2^x then you get the quadratic with equation u^2 - 8u + 15 = 0. Call the roots of the original equation \alpha and \beta, you want \alpha + \beta. You know that 2^{\alpha} 2^{\beta} = 15 since that's the product of the roots of the quadratic, and so 2^{\alpha + \beta} = 15 \Rightarrow \alpha + \beta = \frac{\log_{10} 15}{\log_{10} 2} .

    79. For any real numbers a,b and c where a \geq b, consider

    (1) -b \geq -a

    (2) a^2 + b^2 \geq 2ab

    (3) ac \geq bc

    (1) is correct, since it is equivalent to b \leq a by multiplying by -1 on both sides and flipping the inequality, and we know that b \leq a holds.

    (2) (a-b)^2 \geq 0 \iff a^2 - 2ab + b^2 \geq 0 \iff a^2 + b^2 \geq 2ab, so it's true (trivial AM-GM).

    (3) false for c = -1 (or any other negative real).

    80. A white billiard ball of mass 0.20 kg is travelling horizontally at 3.0 ms^-1 and hits a red billiard ball of the same mass which is at rest. After the collision the white ball continues in the same direction with a speed of 1.0 ms^-1. What is the speed of the red ball immediately after the collision?

    Conservation of momentum: 0.2 \times 3 = 0.2 \times 1 + 0.2 \times v \iff 3 = 1 + v \iff v= 2'

    81. The sequence a_n is given by the rule a_1 = 2 and a_{n+1} = a_n + (-1)^n for n \geq 1. Find \displaystyle \sum_{n=1}^{100} a_n.

    Write down the first few terms: a_1 = 2, a_2 = 1, a_3 = 2, a_4 = 1, \cdots - in general, all the odd terms are 2 and all the even terms are 1 so the overall sum is 50 \times 2 + 50 \times 1 = 100 + 50 = 150.

    83. How many real roots does the equation x^4 - 4x^3 + 4x^2 - 10 = 0 have?

    Re-arrange this to get x^2(x^2 - 4x + 4) = 10 \Rightarrow x^2(x-2)^2 = 10 \Rightarrow (x(x-2))^2 = 10 \Rightarrow x(x-2) = \pm \sqrt{10} which gives rise to two different quadratics: x^2 - 2x - \sqrt{10} and x^2 - 2x + \sqrt{10}, from which the first has two real roots and the second has none (easy to see by checking x^2 - 2x + 1 = (x-1)^2) so since \sqrt{10} > 1 the conclusion follows.

    85. The variables x and y and the constants a and b are real and positive. The variables x and y are related. A graph of \log y versus \log x is drawn. For which one of these relationships will this graph be a straight line?

    You want \log y = c + m\log x from which it is easy to see that y = ax^b is the one that works, since \log y = \log ax^b = \log a + b \log x just as we wanted.

    86. The track for a tram is straight and horizontal. A tram is travelling along the track at a velocity of 12.0 ms^-1 when the brakes are applied. Because of this, the tram decelerates to rest at a constant rate of 1.50ms^-2. What is the distance traveled by the tram over the total time for which it is decelerating?

    Use SUVAT: s = s, u = 12, v = 0 (to rest), a = -1.5, t = ? so we have 0 = u + at \Rightarrow 0 = 12 - 1.5t \Rightarrow t = 8 so s = ut + \frac{1}{2}at^2 = 12 \times 8 - \frac{1.5 \times 8^2}{2} = 96 - 48 = 48

    87. For what values of the non-zero real number a does the quadratic equation ax^2 + (a-2)x = 2 have distinct real roots?

    We need the discriminant to be >0 so (a-2)^2 + 8a = a^2 + 4a + 4 = (a+2)^2 > 0 which is true for all a \neq -2.

    88. Resolve horizontally to get T_a \cos 60^{\circ} = 5 \Rightarrow T_a = 10 and so, resolving horizontally gives T_v \sin 60^{\circ} = T_b \Rightarrow T_b = \frac{10\sqrt{3}}{2} \times \frac{1}{2} = 5 \sqrt{3}

    89. okay i'm bored now
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    (Original post by Student1256)
    I made my own there (link here http://www.thestudentroom.co.uk/show...8#post67543266) and I was told to come comment here. My problem is basically in my opinion one specimen paper for each section is too less. I need more material to prepare for this test from. If anyone could tell me where I should prepare these subjects from: maths, physics, chemistry, advanced maths and advanced physics. I don't take bio so I'm not worried about that. But if you guys could tell me a source of past papers or questions at equal or slightly higher difficulty than this pre interview test that would be so awesome. Thank you!!
    You're not meant to be preparing for these tests, just turn up and do the best you can. That's what I did last year with my test at King's, turned out fine.
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    (Original post by Zacken)
    You're not meant to be preparing for these tests, just turn up and do the best you can. That's what I did last year with my test at King's, turned out fine.
    True. I would like to know what subject you appeared for and what kind of test was that at King's? What Marks did you get and by turned out fine I suppose you mean you're in Cambridge now, true? Also I'm sure you shouldn't just turn up like that without going over the whole syllabus provided on the site.
 
 
 
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