Year 13 Maths Help Thread

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    (Original post by Palette)
    Engineering sounds very cut-throat- I believe it's the most competitive course at Cambridge.

    Is it mechanical engineering you're interested in?
    Yeah it probably is, I think nat sci is the only one with more applicants.

    And yeah mechanical
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    Why is the derivative of e^3x -> 3e^3x?
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    (Original post by jamestg)
    Why is the derivative of e^3x -> 3e^3x?
    You know that derivative of e^x = e^x
    use chain rule with x goes to 3x
    d(3x)/dx = 3

    d(e^3x)/d(x)
    = d(3x)/dx * d(e^3x)/d(3x)
    =3*e^3x
    =3e^3x
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    (Original post by ValerieKR)
    You know that derivative of e^x = e^x
    use chain rule with x goes to 3x
    d(3x)/dx = 3

    d(e^3x)/d(x)
    = d(3x)/dx * d(e^3x)/d(3x)
    =3*e^3x
    =3e^3x
    Thanks! The bit I put in bold, is that just a principle we have to know then?
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    (Original post by jamestg)
    Thanks! The bit I put in bold, is that just a principle we have to know then?
    You have to know that - it comes from the infinite series of e^x being equal to

    1+x+x^2/(2!)+x^3/(3!)+x^4/(4!) + ...
    Note that if you take the derivative term by term the first term disappears and the rest 'shift' so that you end up with the same sequence (e^x) again
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    (Original post by jamestg)
    Thanks! The bit I put in bold, is that just a principle we have to know then?
    Yes, that's one of the definitions of e^x.
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    (Original post by Zacken)
    Yes, that's one of the definitions of e^x.
    If only we did the exponentials chapter before differentiation...
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    Is this a good question to leave up to Year 13 students?

    Can we use matrices to see how many solutions the following simultaneous equations have?

    x-2y+3z=1
    2x+3y-z=4
    4x-y+5z=6.

    [Question adapted from MAT 1997].
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    (Original post by Palette)
    Is this a good question to leave up to Year 13 students?

    Can we use matrices to see how many solutions the following simultaneous equations have?

    x-2y+3z=1
    2x+3y-z=4
    4x-y+5z=6.

    [Question adapted from MAT 1997].
    For those who study 3x3 matrices it is, and that's in one of the later FP modules, so not sure how many that would be. Quite a straight forward one in which case by the looks of it.
    Spoiler:
    Show
    Let M=\begin{bmatrix} 1 & -2 & 3 \\2 & 3 & -1 \\4 & -1 & 5 \end{bmatrix}.

    Therefore M \begin{bmatrix} x \\y \\z \end{bmatrix}=\begin{bmatrix} 1  \\4 \\6 \end{bmatrix}

    \det{(M)}=0 therefore there are no unique solutions. This means there are either an infinite amount of solutions, or none.

    Display the system in an augmented matrix form (not sure about the latex for this), do some manipulation and you should find that 0x+0y+0z=0 therefore the system is consistent and there are an infinite amount of solutions.

    Fun stuff.
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    (Original post by Palette)
    Is this a good question to leave up to Year 13 students?

    Can we use matrices to see how many solutions the following simultaneous equations have?

    x-2y+3z=1
    2x+3y-z=4
    4x-y+5z=6.

    [Question adapted from MAT 1997].
    It's fine for yr13 because it can be done with awkward algebra (the way i'd do it, remember to consider any cases with denominators (which there will be) equaling 0 differently though)
    You can use matrices.
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    Ok, the Bernouilli polynomial one is ready to cook, though I wish Siklos had used the identity sign instead of the equals sign in \frac{\mathrb{d}B_n}{\mathrb{d}x  }=nB_{n-1}(x). I will type it up for marking soon. The key to the last part is 'adding zero creatively'.
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    (Original post by Palette)
    Is this a good question to leave up to Year 13 students?

    Can we use matrices to see how many solutions the following simultaneous equations have?

    x-2y+3z=1
    2x+3y-z=4
    4x-y+5z=6.

    [Question adapted from MAT 1997].
    Surely the answer to your question is just a simple yes.
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    (Original post by B_9710)
    Surely the answer to your question is just a simple yes.
    I should have then added the multiple choice similar to the one in the actual MAT question:

    a) One solution b) two solutions c) three solutions d) no solutions e) infinitely many solutions.

    Or I could have phrased it as 'find the number of solutions to the simultaneous equations using matrices'
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    http://files.physicsandmathstutor.co...hapter%202.pdf
    s2 question 9c exercise 2c
    Why is the (pX>n)>0.99 not the probability its greater or equal to as if they have a number of boats equal to the amount demanded is that not sufficient to meet all demands so why is it just greater than n as i thought it cud equal n too if n are demanded?
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    (Original post by youreanutter)
    http://files.physicsandmathstutor.co...hapter%202.pdf
    s2 question 9c exercise 2c
    Why is the (pX>n)>0.99 not the probability its greater or equal to as if they have a number of boats equal to the amount demanded is that not sufficient to meet all demands so why is it just greater than n as i thought it cud equal n too if n are demanded?
    dodgy link
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    Let f(x+1)-f(x)=(n+1)x^n and F'(x)=f(x). Is F(x+1)-F(x)=x^{n+1}+c?
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    (Original post by Palette)
    Let f(x+1)-f(x)=(n+1)x^n and F'(x)=f(x). Is F(x+1)-F(x)=x^{n+1}+c?
    Yes
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    (Original post by ValerieKR)
    Yes
    I'm worried that the presence of the 'c' completely derails my proof by induction for the Bernouilli polynomials STEP question which is really annoying although I think I may get around that by using another property.
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    (Original post by Palette)
    I'm worried that the presence of the 'c' completely derails my proof by induction for the Bernouilli polynomials STEP question which is really annoying although I think I may get around that by using another property.
    you're on the right lines
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    Q) "The diagram shows a particle of mass 0.5 kg resting on a rough plane inclined at 30 (degrees) to the horizontal. The coefficient of friction between the particle and the plane is 0.4. A of force of magnitude PN, acting directly up the plane, is just sufficient to prevent the particle sliding down the plane. Find the value of P."
 
 
 
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