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

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    *** 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:
    Attached Images
  1. File Type: pdf AQA Core 4 Revision Notes.pdf (271.2 KB, 2028 views)
  2. File Type: pdf C4 revision notes 2.pdf (363.4 KB, 6981 views)
  3. Attached Files
  4. File Type: docx Core 4 Revision Sheet.docx (78.6 KB, 937 views)
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    How is everyone preparing? Does anyone have access to papers before 2005?

    I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.


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    (Original post by CD223)
    How is everyone preparing? Does anyone have access to papers before 2005?

    I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.


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    hows your preperation going for c3 and c4? have you finsihed all the content?
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    (Original post by madmenace)
    hows your preperation going for c3 and c4? have you finsihed all the content?
    Hey! Not bad thanks yeah we've finished the contents for both and I'm about to move onto M2. How about you?
    Getting around 63-65 in each paper. Desperately want to get that up to about 72 before the exam so I can get at least an A overall


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    (Original post by CD223)
    Hey! Not bad thanks yeah we've finished the contents for both and I'm about to move onto M2. How about you?
    Getting around 63-65 in each paper. Desperately want to get that up to about 72 before the exam so I can get at least an A overall


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    Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already

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    (Original post by madmenace)
    Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already

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    Ah I'm scared too! What do you have left of C4? I only finished it like 2 weeks ago, so we're starting M2 tomorrow I believe because the last few weeks has been mock period! What are you aiming for grade wise?


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    Vectors, but I've nearly finished.... Some of the harder parametric equations I'm still getting used to.. I've got 10 exams in total :-( I want to get AAA minimum... but need to hurry up so i can start past papaer season lol

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    (Original post by madmenace)
    Vectors, but I've nearly finished.... Some of the harder parametric equations I'm still getting used to.. I've got 10 exams in total :-( I want to get AAA minimum... but need to hurry up so i can start past papaer season lol

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    Oh nice! Yeah parametric questions can be hard. I need AAA for my uni offer haha. I've got 8 exams :/ what other subjects do you do?


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    Physics and chemistry.... no further maths;-) Chemistry is the killer

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    (Original post by madmenace)
    Physics and chemistry.... no further maths;-) Chemistry is the killer

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    Ah I've heard chemistry is hard. AQA physics?


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    I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?
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    (Original post by Tiwa)
    I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?
    Is your issue with the content or timing? Because I'd say nail the content before even worrying about the 90 minutes! Try Solomon papers for a variety of extra practice. Exam solutions and various other videos on YouTube are great. Have you seen Jack Brown's videos?


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    (Original post by CD223)
    Is your issue with the content or timing? Because I'd say nail the content before even worrying about the 90 minutes! Try Solomon papers for a variety of extra practice. Exam solutions and various other videos on YouTube are great. Have you seen Jack Brown's videos?


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    I think it's about the content mostly. Sometimes when I do past papers, I sometimes forget the method of answering a questions. I know of Exam solutions. I haven;t heard of Solomon papers, though. I'll take it into account. Thank you so much!
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    (Original post by Tiwa)
    I think it's about the content mostly. Sometimes when I do past papers, I sometimes forget the method of answering a questions. I know of Exam solutions. I haven;t heard of Solomon papers, though. I'll take it into account. Thank you so much!
    Yeah the method varies much more at A2 compared to AS aha. Sure, any time!


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    Does anyone know what standard integrals we have to know? There's some in my textbook that refer to sin and tan in a fraction but I wasn't sure if this was still relevant to today's syllabus.


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    Hey, yeah it's those 2 you have to be able to recognise....
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    (Original post by CD223)
    How is everyone preparing? Does anyone have access to papers before 2005?

    I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.


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    C3 didn't exist before 2005. Before that, C1 - 4 was in 3 modules, P1 - 3.
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    (Original post by madmenace)
    Hey, yeah it's those 2 you have to be able to recognise....
    Should I memorise them?


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    (Original post by tiny hobbit)
    C3 didn't exist before 2005. Before that, C1 - 4 was in 3 modules, P1 - 3.
    Ah thanks for clearing that up! I'll attempt all from 2005 onwards


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    (Original post by CD223)
    Should I memorise them?


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    What textbook do you use? My one explains how they come about ...but yeah I just remembered the two
 
 
 
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