Hey there! Sign in to join this conversationNew here? Join for free
    Offline

    3
    ReputationRep:



    Can someone help me, I stop following the example when you get 1/3 ... I dont get the last 3 lines of the this.
    Offline

    13
    ReputationRep:
    (Original post by Proflash)



    Can someone help me, I stop following the example when you get 1/3 ... I dont get the last 3 lines of the this.
    You have that \tan (\arctan 3 + 2 \arctan 2) = \tan \left( \arctan \frac{1}{3} + k \pi \right), k \in \mathbb{Z} as the tan graph is periodic with integer multiples of \pi

    But, if you take principle values, you know if \tan \theta = 3 \Rightarrow \frac{\pi}{4} < \theta < \frac{\pi}{2} (because \tan \frac{\pi}{4} = 1 and \lim_{\theta \to \frac{\pi}{2}} \tan \theta = \infty so the value of \tan \theta = 3 has to be sandwiched between these two. The same goes for \tan \phi = 2

    As \frac{\pi}{4} < \theta < \frac{\pi}{2}, \frac{\pi}{4} < \phi < \frac{\pi}{2} \Rightarrow \frac{2 \pi}{4} < 2 \phi < \pi \Rightarrow \frac{3}{4} \pi < \theta + 2 \phi < \frac{3}{2} \pi (adding the lower and upper bounds)

    Thus, from the given range, we have that k = 1 \Rightarrow \theta + 2 \phi = \pi + \arctan \frac{1}{3}

    \arctan \frac{1}{3} = \alpha \Rightarrow \tan \alpha = \frac{1}{3} \Leftrightarrow \cot \alpha = 3 \Rightarrow \alpha = arccot 3
    Offline

    0
    ReputationRep:
    (Original post by Proflash)



    Can someone help me, I stop following the example when you get 1/3 ... I dont get the last 3 lines of the this.
    In the line that says BUT.... they have basically ammended the range:

     \theta+2\phi where \theta =0.25\pi and  \phi=0.25\pi you end up with \frac{1}4\pi+2(\frac{1}4\pi{}){}  =(\frac{3}{4} \pi)

    Also for the end of the range \theta =0.5\pi and \phi=0.5\pi so end of range is: \frac{1}2\pi+2(\frac{1}{2}\pi)=(  \frac{3}{2} \pi)

    \therefore (\frac{3}{4})\pi< \theta+ 2\phi <(\frac{3}{2})\pi

    Felix has the rest covered :P
    Offline

    16
    ReputationRep:
    (Original post by Proflash)



    Can someone help me, I stop following the example when you get 1/3 ... I dont get the last 3 lines of the this.
    Here is an alternative way you may like to approach the question, using complex numbers:

     (1+3i)(3-i)(1+2i)^2 = (6+8i)(-3+4i)

     = (-50)

    So we have:

     (1+3i)(3-i)(1+2i)^2 = -50

    Take arguments of both sides:

     arg[(1+3i)(3-i)(1+2i)^2] = arg(-50)

     arctan3+2arctan2-arctan\frac{1}{3} = \pi

     arctan3 + 2arctan2 = \pi +arccot3
    Offline

    9
    ReputationRep:
    Can someone help me solve this:
    e^{-x}=10

    I tried to take logs both sides but I don't seem to get the answer in the textbook which says

    \ln (0.1)

    Thanks


    • Thread Starter
    Offline

    18
    ReputationRep:
    (Original post by Westeros)
    Can someone help me solve this:
    e^{-x}=10

    I tried to take logs both sides but I don't seem to get the answer in the textbook which says

    \ln (0.1)

    Thanks


    e^(-x) = 10
    ln10 = -x
    x = -ln10

    Power rule.

    Spoiler:
    Show
    x = ln(10^-1)
    x = ln(1/10)
    x = ln(0.1)


    Make sense?
    Offline

    9
    ReputationRep:
    (Original post by L'Evil Fish)
    e^(-x) = 10
    ln10 = -x
    x = -ln10

    Power rule.

    Spoiler:
    Show
    x = ln(10^-1)
    x = ln(1/10)
    x = ln(0.1)


    Make sense?
    Thank you *dumb moment*
    Offline

    1
    ReputationRep:
    Do unis care about the average UMS of AS maths more than the average of AS and A2 combined together?
    • Thread Starter
    Offline

    18
    ReputationRep:
    (Original post by Westeros)
    Thank you *dumb moment*
    Np :yy:
    Offline

    11
    ReputationRep:
    (Original post by MAyman12)
    Do unis care about the average UMS of AS maths more than the average of AS and A2 combined together?
    If you've done both before you apply then both.
    And if by Universities you mean Oxbridge (maybe warwick/imperial).
    Most unis don't look at UMS, only grades.
    Offline

    1
    ReputationRep:
    (Original post by joostan)
    If you've done both before you apply then both.
    And if by Universities you mean Oxbridge (maybe warwick/imperial).
    Most unis don't look at UMS, only grades.
    Yes I meant Oxbridge. You mean that I don't need to worry that my average UMS in AS maths is >95 if I've applied after finishing my A Level?

    If I got A*A*A would they care about my UMS?
    I feel really stupid
    Offline

    11
    ReputationRep:
    (Original post by MAyman12)
    Yes I meant Oxbridge. You mean that I don't need to worry that my average UMS in AS maths is >95 if I've applied after finishing my A Level?

    If I got A*A*A would they care about my UMS?
    I feel really stupid
    UMS is far less important than grades.
    Cambridge go by module scores - I imagine an average of 90+ is sufficient - they've got their own methods of selection like STEP + interview etc.
    Oxford prefer GCSEs, though there's also the MAT to do.
    EDIT: Malicious and petty negging, really? :erm:
    Offline

    1
    ReputationRep:
    (Original post by joostan)
    UMS is far less important than grades.
    Cambridge go by module scores - I imagine an average of 90+ is sufficient - they've got their own methods of selection like STEP + interview etc.
    Oxford prefer GCSEs, though there's also the MAT to do.
    So by modules scores you mean A B C etc...? And about STEP, should I prepare for it if I'm applying for NatSci (Physics)?
    Offline

    13
    ReputationRep:
    (Original post by MAyman12)
    Oh okay take your time:yy: Sorry, if I'm asking too much
    Here's the best that I've managed to come up with to show that the recurring radical converges and even then, I'm not sure if my logic is correct/ rigorous enough...

    Spoiler:
    Show
    \sqrt{1 + \sqrt{1 + \sqrt{1 + ... }}}

    Define the sequence x_{n+1} = \sqrt{1 + x_{n}}, x_{0} = \sqrt{1}

    Clearly the sequence is strictly increasing (that is x_{n+1} > x_{n} \ \forall n \in \mathbb{N}).

    Now, x_{0} = \sqrt{1} = 1 \Rightarrow 1 \leq x_{n}

    Now, we will prove that x_{n} \leq \phi \ \forall n \in \mathbb{N} by induction.

    x_{0} = 1 \leq \phi so it holds true for the base case of n = 0.

    We assume it is true for x_{k}, x_{k} \leq \phi \Rightarrow x_{k} + 1 \leq \phi^{2} as \phi^{2} = \phi + 1

    \Rightarrow x_{k+1} \leq \phi as x_{k+1} = \sqrt{1 + x_{k}} hence the result is true \forall n \in \mathbb{N} by the principle of mathematical induction, hence:

    1 \leq x_{n} \leq \phi hence a limit exists for x_{n} as n \to \infty which gives the required recurring radical in question, and we can proceed to verify it to be \phi

    As I've said, my knowledge of analysis is limited so if anyone who knows more about it than I do could check over my logic, I'd be appreciative.
    Offline

    16
    ReputationRep:
    (Original post by joostan)
    UMS is far less important than grades.
    Cambridge go by module scores - I imagine an average of 90+ is sufficient - they've got their own methods of selection like STEP + interview etc.
    Oxford prefer GCSEs, though there's also the MAT to do.
    (Original post by MAyman12)
    Do unis care about the average UMS of AS maths more than the average of AS and A2 combined together?
    Several things:

    1 - Whilst Cambridge is the main university that asks for modular marks it is not the only one. I was asked to submit UMS by Durham, and I don't think it's unheard of for Imperial to ask.
    2 - For Cambridge an average of 90% in Maths is below average. I was told in all seriousness by a liaisons officer who came to our school that the average I applied with (97%) was "standard", which was borne out by the fact Maths doesn't have auto-pooling. This fits into point 3) - for Cambridge the UMS is very important, as there's a huge difference between 81% average and 100%.
    4 - Oxford preferring GCSE's is an urban myth, and I certainly don't think they put as much emphasis on them as some people think. Firstly, Oxford have the MAT scores which are significantly more important. In addition, Oxford interviews usually happen over a week (You have to stay at Oxford) and so they have more interview data to work with, whereas you're unlucky if you don't do all your Cambridge interviews in a single day.
    5 - Cambridge will care about every Maths module you've taken. I think (But do not know) that Pure and Mechanics take precedence; I believe you'd get away with 80 in D1 a lot easier than 80 in FP1, for example. I think that, if you've taken AS and A2 modules simultaneously at AS (e.g. your school teaches Maths the first year and Further the second) they are given equal precedence (Treated the same as people taking AS Maths and Further together), but if you're applying after completing A Levels then your 2nd year modules are expected to be higher to show a progression.
    • Thread Starter
    Offline

    18
    ReputationRep:
    (Original post by DJMayes)
    S.
    Do you think my average will be okay as I'm not applying for maths?

    Hopefully it'll be 92+%
    Offline

    1
    ReputationRep:
    (Original post by Felix Felicis)
    Here's the best that I've managed to come up with to show that the recurring radical converges and even then, I'm not sure if my logic is correct/ rigorous enough...

    Spoiler:
    Show
    \sqrt{1 + \sqrt{1 + \sqrt{1 + ... }}}

    Define the sequence x_{n+1} = \sqrt{1 + x_{n}}, x_{0} = \sqrt{1}

    Clearly the sequence is strictly increasing (that is x_{n+1} > x_{n} \ \forall n).

    Now, x_{0} = \sqrt{1} = 1 \Rightarrow 1 \leq x_{n}

    Now, we will prove that x_{n} \leq \phi \ \forall n by induction.

    x_{0} = 1 \leq \phi so it holds true for the base case of n = 1.

    We assume it is true for x_{k}, x_{k} \leq \phi \Rightarrow x_{k} + 1 \leq \phi^{2} as \phi^{2} = \phi + 1

    \Rightarrow x_{k+1} \leq \phi as x_{k+1} = \sqrt{1 + x_{k}} hence the result is true \forall n by the principle of mathematical induction, hence:

    1 \leq x_{n} \leq \phi hence a limit exists for x_{n} as n \to \infty which gives the required recurring radical in question, and we can proceed to verify it to be \phi

    As I've said, my knowledge of analysis is limited so if anyone who knows more about it than I do could check over my logic, I'd be appreciative.
    Awesome.

    Thank you, I'm sure that it took some time to write. It would be better If you used less symbols, as it confuses me

    How do you know such methods?
    Offline

    0
    ReputationRep:
    (Original post by DJMayes)
    Several things:

    1 - Whilst Cambridge is the main university that asks for modular marks it is not the only one. I was asked to submit UMS by Durham, and I don't think it's unheard of for Imperial to ask.
    2 - For Cambridge an average of 90% in Maths is below average. I was told in all seriousness by a liaisons officer who came to our school that the average I applied with (97%) was "standard", which was borne out by the fact Maths doesn't have auto-pooling. This fits into point 3) - for Cambridge the UMS is very important, as there's a huge difference between 81% average and 100%.
    4 - Oxford preferring GCSE's is an urban myth, and I certainly don't think they put as much emphasis on them as some people think. Firstly, Oxford have the MAT scores which are significantly more important. In addition, Oxford interviews usually happen over a week (You have to stay at Oxford) and so they have more interview data to work with, whereas you're unlucky if you don't do all your Cambridge interviews in a single day.
    5 - Cambridge will care about every Maths module you've taken. I think (But do not know) that Pure and Mechanics take precedence; I believe you'd get away with 80 in D1 a lot easier than 80 in FP1, for example. I think that, if you've taken AS and A2 modules simultaneously at AS (e.g. your school teaches Maths the first year and Further the second) they are given equal precedence (Treated the same as people taking AS Maths and Further together), but if you're applying after completing A Levels then your 2nd year modules are expected to be higher to show a progression.
    Just to ask for some input for my current scenario, for M1 I messed up badly (well not bad as in bad but you get the point) I got 63/75 (without method marks) which no doubt will probably be around 80-84ums. I also failed to complete S1 in time and as a result I couldn't have time to answer questions, dropping down to 66/75 which is give or take around 86-88 ums, will these two bad modules affect my overall chances?.

    S2 went well and expecting about 95 ums, C2 I got 100ums so my maths average is looking about 95% whilst further maths about 91% (if FP1 is 98+). The main problem I see, is my mechanics, this is the only mechanics I have to show to Cambridge and I did fairly badly in it, I will apply regardless but I was wondering if you could use your advice to give me knowledge and apply it to my current situation.

    On a more bright note, expecting to get just about 90ums for economics, and an overall A for history. (which may or may not help)
    Offline

    1
    ReputationRep:
    (Original post by DJMayes)
    Several things:

    Spoiler:
    Show
    1 - Whilst Cambridge is the main university that asks for modular marks it is not the only one. I was asked to submit UMS by Durham, and I don't think it's unheard of for Imperial to ask.
    2 - For Cambridge an average of 90% in Maths is below average. I was told in all seriousness by a liaisons officer who came to our school that the average I applied with (97%) was "standard", which was borne out by the fact Maths doesn't have auto-pooling. This fits into point 3) - for Cambridge the UMS is very important, as there's a huge difference between 81% average and 100%.
    4 - Oxford preferring GCSE's is an urban myth, and I certainly don't think they put as much emphasis on them as some people think. Firstly, Oxford have the MAT scores which are significantly more important. In addition, Oxford interviews usually happen over a week (You have to stay at Oxford) and so they have more interview data to work with, whereas you're unlucky if you don't do all your Cambridge interviews in a single day.
    5 - Cambridge will care about every Maths module you've taken. I think (But do not know) that Pure and Mechanics take precedence; I believe you'd get away with 80 in D1 a lot easier than 80 in FP1, for example. I think that, if you've taken AS and A2 modules simultaneously at AS (e.g. your school teaches Maths the first year and Further the second) they are given equal precedence (Treated the same as people taking AS Maths and Further together), but if you're applying after completing A Levels then your 2nd year modules are expected to be higher to show a progression.
    I think I flunked my M1 exam this summer,though I was quite able at mechanics and was expecting a 100. But, I got really nervous in the exam and now I'm expecting 85 or something if lucky. Would they bother with that?

    I'm applying for Physics-NatSci, what UMS scores are they expecting from me in Maths and FM (I know that maths is extremely important for physics, actually it's impossible to do proper physics without it)?

    Thank you for explaining things.
    Offline

    1
    ReputationRep:
    (Original post by Robbie242)
    Just to ask for some input for my current scenario, for M1 I messed up badly (well not bad as in bad but you get the point) I got 63/75 (without method marks) which no doubt will probably be around 80-84ums. I also failed to complete S1 in time and as a result I couldn't have time to answer questions, dropping down to 66/75 which is give or take around 86-88 ums, will these two bad modules affect my overall chances?.

    S2 went well and expecting about 95 ums, C2 I got 100ums so my maths average is looking about 95% whilst further maths about 91% (if FP1 is 98+). The main problem I see, is my mechanics, this is the only mechanics I have to show to Cambridge and I did fairly badly in it, I will apply regardless but I was wondering if you could use your advice to give me knowledge and apply it to my current situation.

    On a more bright note, expecting to get just about 90ums for economics, and an overall A for history. (which may or may not help)
    We were both were screwed by M1
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Would you like to hibernate through the winter months?
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.