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Official thread FP1, 1st Febraury 2010 watch

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  • View Poll Results: How do you rate this paper (on a scale of 1-10 with 10 being the best)?
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    Let's just remember the nine questions.
    topic-wise...

    1. Complex Numbers
    2. Proof by induction? i think..
    3. Interval Bisection and Newton Raphson i think
    4.
    5.
    6.
    7.Co-ordinate systems
    8. Proof by induction and summation
    9. Matrices
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    :tumble:

    What happened to remembering what came up?
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    1. Complex Numbers

     z_{1} = 2+8i , z_{2} = 1-i

    Find:

    a)  \frac{z_{1}}{z_{2}}

    A: -3+5i

    b)  |\frac{z_{1}}{z_{2}}|

    A:  \sqrt{34}

    c)  arg( |\frac{z_{1}}{z_{2}}|)

    A: 2.11

    2. Proof by induction? i think..


    3. Interval Bisection and Newton Raphson

     f(x) = x^3 -6x - \frac{11}{x^2} < Need help remembering this properly

    There is a root between 1.3 and 1.4.

    a) Using Interval Bisection find an approximation to the root within an interval of 0.025.

    b) Taking the initial value of 1.4, apply Newton Raphson once to find an approximation to the root


    4.
    5.
    6.
    7.Co-ordinate systems
    8. Proof by induction and summation

    a) \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2

    Prove true for all positive integers.

    A: (Credit to Khodu)

    Prove that n = 1 is true

    Test:  (1)^3 = 1

    0.25(1)^2(1+1)^2 = 0.25(4)= 1

    Therefore true for n=1

    Assume true for n = k

    S(k)= 0.25^2(k+1)^2

    Prove true for n= k+1

    S(k+1)= S(k) + (k+1)th term

    =  [ 0.25^2(k+1)^2 ] + (k+1)^3

    = 0.25(k+1)^2[k^2+4(k+1)]

    =0.25(k+1)^2(k+2)^2

    Therefore true for n=k+1 if n=k is true. True for n=1, therefore by induction true for all positive integers.


    9. Matrices

    Copy and paste! And correct me if I am wrong. I'll be trying to remember more and will be editing my post
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    I don't know what came up. I block out exams as soon as they're over :p:

    Seriously though, I have no idea.
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     z_{1} = 2+8i , z_{2} = 1-i

    Find:

    a)  \frac{z_{1}}{z_{2}}

    ans= -3 + 5i

    b)  |\frac{z_{1}}{z_{2}}|

    \sqrt 34

    c)  arg( |\frac{z_{1}}{z_{2}}|)

     2.11

    2. Proof by induction? i think..
    3. Interval Bisection and Newton Raphson i think
    4.
    5.
    6.
    7.Co-ordinate systems
    8. Proof by induction and summation

    a) \displaystyle\sum_{r=1}^n  = r^3 =\frac{1}{4}n^2(n+1)^2
    Prove true for all positive integers.

    Prove that n = 1 is true

    {1}^3  = 1

    \frac{1}{4}(1)^2(1+1)^2 = \frac{1}{4}(4)= 1

    Therefore true for n = 1

    Assume true for n = k

    \displaystyle\sum_{r=1}^k  \frac{1}{4}k^2(k+1)^2

    Prove true for n= k+1

    \displaystyle\sum_{r=1}^{k+1}  =\displaystyle\sum_{r=1}^k   + (k+1)^{th} term

    = \frac{1}{4}k^2(k+1)^2 + (k+1)^3

    = \frac{1}{4}(k+1)^2[k^2+4(k+1)]

    = = \frac{1}{4}(k+1)^2[k^2+4k+4]

    = \frac{1}{4}(k+1)^2(k+2)^2

    Therefore true for n = k+1 if n = k is true. True for n = 1, therefore by induction true for all positive integers.

    9. Matrices


    My first ever attempt with latex, do not laugh lol
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    I got a bit too excited that it was my last exam and totally forgot to take my calculator in with me, woops.
    I had to use some really old model but once I found the shift button I coped.
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    (Original post by ClaireW2708)
    I don't know what came up. I block out exams as soon as they're over :p:

    Seriously though, I have no idea.
    Sorry I haven't a clue how to use latex. But that took me a while to type up lol.
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    (Original post by melon1)
    I got a bit too excited that it was my last exam and totally forgot to take my calculator in with me, woops.
    I had to use some really old model but once I found the shift button I coped.

    That's like my worst nightmare lol. Man, this girls phone went off... But it was her alarm and her phone was switched off.. But they still had to write it on her paper, I doubt they'd disqualify her though...
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    (Original post by Khodu)
    Sorry I haven't a clue how to use latex. But that took me a while to type up lol.
    latex isnt too hard to learn quickly.
    http://www.thestudentroom.co.uk/wiki/LaTex
    if it helps.
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    Khodu, i wrote it in latex for you , and I put it into my post, obviosuly giving you credit :P

    Keep the questions coming guys!

    EDIT: Someone take over for me now. I have to go :P
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    (Original post by Khodu)
    That's like my worst nightmare lol. Man, this girls phone went off... But it was her alarm and her phone was switched off.. But they still had to write it on her paper, I doubt they'd disqualify her though...
    yeh the first question where you needed arctan took me ages as I was trying out every button there was, lol, I learnt my lesson though won't be doing that again

    Bad times, I'm always uber paranoid and check all my pockets, hopefully not, atleast there's always the summer if they do disqualify her from this exam.
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    (Original post by 2710)
    Khodu, i wrote it in latex for you , and I put it into my post, obviosuly giving you credit :P

    Keep the questions coming guys!

    EDIT: Someone take over for me now. I have to go :P

    :rofl::rofl::rofl::rofl::rofl::rofl::rofl:

    I've been working my butt off trying to fix it.

    Check lol. I'm so long.

    You joker. I don't know why i find it so funny. Maybe I'm just high because I had a cup of tea
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    Question 6 (I think)

    Given that two roots of the equation  x^3 - 12x^2 + ax + b are  2, 5+2i find:

    (a) The third root. Answer 5-2i.

    (b) Find the values of a and b. Answer a=49, b=-58.

    (c) Display the three roots on an Argand diagram.

    I think that this was the question but I'm not 100% sure that the values are correct.
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    Can anyone upload the proper paper and stuff?
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    (Original post by Patrick.Swaddle)
    Can anyone upload the proper paper and stuff?
    I don't know if anyone actually has it. We're just trying to build it up from memory.
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    Question 4 or 5, Matrices.

    Given Matrix A, which I think looked like this \begin{pmatrix}  a & -5 \\ 2 & a + 4 \\  \end{pmatrix}. But I'm not sure about the -2 and 5.

    (a) Find Det A. Det A = a(a+4) +10 = a^2 + 4a +10

    (b) Prove that Matirix A is non-singular for all values of a.

    b^2 -4ac = -24

     -24 &lt; 0 \Longrightarrow a\not=0 and therefore the Matrix is non-singular.

    (c) Find A^{-1} given that a=0.

    A^{-1} = \frac{1}{10}\begin{pmatrix}  4 & 5 \\ -2 & 0 \\  \end{pmatrix}
    • PS Helper
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    PS Helper
    (Original post by Zacula)
    ********... This is just a myth Cambridge managed to create so as to be able to make more money.

    A clever student, does not need to go to a "name". Clever students go to non Ox-Bridge unis!

    yes, I totally agree with you. the only reason i asked was because i found it a bit unusual!!

    cheers
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    bam! repped. top bloke
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    Did anyone give the old syllabus FP1 exam?
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    no one fancy doing a mark scheme then????
 
 
 
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