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AQA A2 Mathematics MPC4 Core 4 - 9th June 2015 [Discussion & unofficial markscheme] Watch

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    Excellent, so I've still got a good chance of getting an A* ! I was really disappointed with myself once I realised my below personal-average performance may have jeopardised my chances to achieve an A*, so it's very reassuring to get some affirmation regarding the grade boundaries - even if they are estimates.

    Anyway, thank you for the unofficial mark schemes/discussion threads for C3 and C4 (they were very helpful, honestly I wouldn't have had much of a shot with vectors otherwise - given my limited revision on them until 3 days prior to the exam ^^), I wish you good luck for results day and I hope you get the grades for your firm university choice !
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    (Original post by datpr0)
    Excellent, so I've still got a good chance of getting an A* ! I was really disappointed with myself once I realised my below personal-average performance may have jeopardised my chances to achieve an A*, so it's very reassuring to get some affirmation regarding the grade boundaries - even if they are estimates.

    Anyway, thank you for the unofficial mark schemes/discussion threads for C3 and C4 (they were very helpful, honestly I wouldn't have had much of a shot with vectors otherwise - given my limited revision on them until 3 days prior to the exam ^^), I wish you good luck for results day and I hope you get the grades for your firm university choice !
    Don't worry about it dude, it's summer I'll be back to hear people's thoughts on the actual boundaries on 12th August!

    From the sounds of it me and you will end up with similar UMS in each of the exams so hopefully we both get the A* we wanted.

    Good luck


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    (Original post by CD223)
    Don't worry about it dude, it's summer I'll be back to hear people's thoughts on the actual boundaries on 12th August!

    From the sounds of it me and you will end up with similar UMS in each of the exams so hopefully we both get the A* we wanted.

    Good luck


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    I think i got similar in both too so hopefully we will all get an A*!! Good luck guys!
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    (Original post by HennersPD)
    I think i got similar in both too so hopefully we will all get an A*!! Good luck guys!
    It would be nice, but it's not worth worrying about for the next 7 weeks! Good luck to you too.


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    (Original post by CD223)
    It would be nice, but it's not worth worrying about for the next 7 weeks! Good luck to you too.


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    That's true! Thank you! hopefully see you at Bath in September
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    (Original post by CD223)
    *** OFFICIAL AQA CORE 4 MPC4 JUNE 2015 EXAM DISCUSSION THREAD ***
    UNOFFICIAL MARK SCHEME
    1. It is given that f(x) = \dfrac{19x - 2}{(5 - x)(1 + 6x)} can be expressed as \dfrac{A}{(5-x)} + \dfrac{B}{(1 + 6x)} , where A and B are integers.

    (a) Find the values of A and B. [3 marks]

    A = 3

    B = -1

    \therefore \dfrac{3}{(5-x)} + \dfrac{-1}{(1 + 6x)}

    (b) Hence show that \displaystyle \int_0^4 f(x)\ dx\ = k \ln5, where k is a rational number. [6 marks]

    \displaystyle \int_0^4 f(x)\ dx\ = \frac{8}{3} \ln5

    2.
    (a) Express 2 \cos x - 5 \sin x in the form R \cos (x + \alpha), where R > 0 and 0 < \alpha < \frac{\pi}{2}, giving your value of \alpha, in radians, to three significant figures. [3 marks]

    2 \cos x - 5 \sin x = \sqrt{29} \cos (x + 1.19)

    (b) (i) Hence find the value of x in the interval 0 < x < 2 \pi for which 2 \cos x - 5 \sin x has its maximum value. Give your value of x to three significant figures. [2 marks]

    x = 5.09

    (ii) Use your answer to part (a) to solve the equation 2 \cos x - 5 \sin x + 1 = 0 in the interval 0 < x < 2 \pi , giving your solutions to three significant figures. [3 marks]

    x = 0.568, 3.34

    3.
    (a) The polynomial f(x) is defined as f(x) = 8x^3 - 12x^2 - 2x + d, where d is a constant. When f(x) is divided by (2x + 1), the remainder is -2. Use the Remainder Theorem to find the value of d. [2 marks]

    d = 1

    (b) The polynomial g(x) is defined by g(x) = 8x^3 - 12x^2 - 2x + 3.
    (i) Given that x = \frac{-1}{2} is a solution of the equation g(x) = 0, write g(x) as a product of three linear factors. [3 marks]

    g(x)  = (2x + 1)(2x - 3)(2x - 1)

    (ii) The function h is defined by h(x) = \dfrac{4x^2 - 1}{g(x)} for x > 2. Simplify h(x), and hence show that h is a decreasing function. [4 marks]

    h(x) = \dfrac{1}{(2x - 3)}

    EITHER
    h'(x) = -2(2x - 3)^{-2}\ \therefore for x > 2, h'(x) < 0 therefore h(x) is a decreasing function.

    OR

    Plot a graph of h(x). h(x) \rightarrow 0 when x \rightarrow \infty.

    4.
    (a) Find the binomial expansion of (1 + 5x)^{\frac{1}{5}} up to and including the term in x^2. [2 marks]

    1 + x - 2x^2

    (b) (i) Find the binomial expansion of (8 + 3x)^{\frac{-2}{3}} up to and including the term in x^2. [3 marks]

    \dfrac{1}{4} + \dfrac{1}{16} x + \dfrac{5}{256} x^2

    (ii) Use your expansion from part (b)(i) to find an estimate for \sqrt[3]{\frac{1}{81}}, giving your answer to four decimal places. [2 marks]

    = 0.2313

    5. A curve is defined by the parametric equations x = \cos 2t, y = \sin t.
    The point P on the curve is where t = \frac{\pi}{6}.

    (a) Find the gradient at P. [3 marks]

    \dfrac{dy}{dx} = \dfrac{-1}{2}

    (b) Find the equation of the normal to the curve at P in the form y = mx + c. [3 marks]

    y = 2x - \frac{1}{2}

    (c) The normal at P intersects the curve again at the point Q (\cos 2q, \sin q). Use the equation of the normal to form a quadratic equation in \sin q and hence find the x-coordinate of Q. [5 marks]

    = \dfrac{-1}{8}

    6. The points A and B have coordinates (3, 2, 10) and (5, -2, 4) respectively.
    The line l passes through A and has equation r = \begin{bmatrix} 3 \\ 2 \\ 10 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ 1 \\ -2 \end{bmatrix}

    (a) Find the acute angle between the line l and the line AB. [4 marks]

    60^{o}

    (b) The point C lies on l such that angle ABC is 90^{o}. Find the coordinates of C. [4 marks]

    C = (15, 6, 2)

    (c) The point D is such that BD is parallel to AC and angle BCD is 90^{o}. The point E lies on the line through B and D and is such that the length of DE is half that of AC. Find the coordinates of the two possible positions of E. [4 marks]

    E = (11, 0, 0)
    E = (23, 4, -8)

    7. A curve has equation y^3 + 2e^{-3x}y - x = k, where k is a constant. The point P (\ln2, \frac{1}{2}), lies on this curve.

    (a) Show that the exact value of k is q - \ln2, where q is a rational number. [1 mark]

    k = \dfrac{1}{4} - \ln2

    (b) Find the gradient of the curve at P. [6 marks]

    \dfrac{dy}{dx} = \dfrac{11}{8}

    8.
    (a) A pond is initially empty and is then filled gradually with water. After t minutes, the depth of the water, x metres, satisfies the equation

    \dfrac{dx}{dt} = \dfrac{\sqrt{4 + 5x}}{5 (1 + t)^2}

    Solve this differential equation to find x in terms of t. [7 marks]

    x = \frac{1}{5} [(\frac{-1}{2} (1 + t)^{-1} + \frac{5}{2})^2 - 4]

    (b) Another pond is gradually filling with water. After t minutes, the surface of the water forms a circle of radius r metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.

    (i) Write down a differential equation, in the variables r and t and a constant of proportionality, which represents how the radius of the surface of the water is changing with time. (You are not required to solve your differential equation.) [3 marks]

    \dfrac{dr}{dt} = \dfrac{k}{\pi r^2}

    (ii) When the radius of the pond is 1\ \text{metre}, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.

    r = 3\ \text{metres}
    ________________________________ ________________________________ __________

    [TOTAL: 75 marks]
    ________________________________ ________________________________ __________

    CORE 3 EXAM DISCUSSION THREAD:
    http://www.thestudentroom.co.uk/show....php?t=3041463

    MECHANICS 2 EXAM DISCUSSION THREAD:
    http://www.thestudentroom.co.uk/show....php?t=3047357
    ________________________________ ________________________________ _____________

    RESOURCES:
    For 3bii) I simplified it to h(x)=1/2x-3 then said as x gets larger, h(x) gets more negative so it is a decresing function... would I get any marks? Also for the last differential question, I forgot to put +C and didn't mention it but did everything else correctly... how many marks would I get?
    Thanks!
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    Thoughts?

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    Grade Boundaries published:

    Core 3
    A* 64 A 57 B 52

    Core 4
    A* 63 A 57 B 52

    http://filestore.aqa.org.uk/over/sta...-JUNE-2015.PDF

    Almost exactly as I predicted. Core 4 1 mark lower than previously thought.
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    Core 4 is high this year. It shouldve been 62 imo. It was 58 last year what a massive increase


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    To be honest though I'm pretty happy across both papers. Desperate for an A* overall.


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    Can someone please PM me the full UMS cap mark for this exam when the converter becomes available?


    I'm at work now until 3pm


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    Yeah I'm desperate for an A* aswell. Would really really help. Hopefully


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    For 100 UMS:

    Core 3
    71

    Core 4
    69
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    (Original post by Anon123hahaha)
    Yeah I'm desperate for an A* aswell. Would really really help. Hopefully


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    Sweet! What's your offer and for where?


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    (Original post by CD223)
    Sweet! What's your offer and for where?


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    Whatd you think 65 on both C3 AND C4 will be in UMS? thats what i think ive got :/
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    (Original post by HennersPD)
    Whatd you think 65 on both C3 AND C4 will be in UMS? thats what i think ive got :/
    A* and enough to get into bath


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    (Original post by CD223)
    A* and enough to get into bath


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    I hope so :/ i just hope my other subjects have gone well
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    (Original post by CD223)
    A* and enough to get into bath


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    I hope so :/ i just hope my other subjects have gone well
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    (Original post by HennersPD)
    I hope so :/ i just hope my other subjects have gone well
    What else did you do?


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    Anyone got the MS?
 
 
 
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