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Thread: Cambridge Prelim MX1 Textbook Marathon/Q&A

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by rand_althor View Post
    Thanks for the corrections. Is there any difference if it is rather than ?




    Last edited by InteGrand; 30 Jul 2015 at 11:41 PM.
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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    a) Find the area bounded by the curve y = ax^2 + bx + c and the x - axis between x = h and x = - h, where y > 0 for - h < and equal to x < and equal to h.

    b) Hence show that if y = y0, y1, y2 when x = -h, 0 and h respectively, then the area is given by 1/3 h (y0 + 4y1 + y2) .

    I can do part a), need help with b).

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    a) Find the area bounded by the curve y = ax^2 + bx + c and the x - axis between x = h and x = - h, where y > 0 for - h < and equal to x < and equal to h.

    b) Hence show that if y = y0, y1, y2 when x = -h, 0 and h respectively, then the area is given by 1/3 h (y0 + 4y1 + y2) .

    I can do part a), need help with b).
    If you substitute the point (0,y1) into the equation of the parabola, you get y1 = c.

    So y = ax^2 + bx + y1

    Substituting the other points (-h, y0) and (h, y2) gives:
    y0 = ah^2 - bh + y1
    y2 = ah^2 + bh + y1

    By alternately adding and subtracting these equations, you can get expressions for a and b in terms of y0, y1, y2.

    Substitute these expressions into your answer to part (i) to get the answer.
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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Consider the function G(x) = Integral from x to 0 of g(u) du, where g(u) = 4 - 4/3 u , for 0 < equal to u < 6,
    u - 10, for 6 < equal to u < equal to 12,

    a) Sketch a graph of g(u)
    b) Find the stationary points of the function y = G(x), and determine their nature.
    c) Find those values of x for which G(x) = 0
    d) Sketch the curve y = G(x), indicating all important features.
    e) Find the area bounded by the curve y = G(x) and the x axis for 0 < equal to x < equal to 6.


    For part a) Do you just work out the intercepts with the axis in the two different sections and also the endpoints x = 0, 12 and also where the two parts meet, ie, at x = 6

    Have no clue about part b) as there are two different parts. Also find stationary points means when first derivative is 0. So is that the derivative of the integral???
    Last edited by appleibeats; 31 Jul 2015 at 11:36 PM.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    Consider the function G(x) = Integral from x to 0 of g(u) du
    You mean from 0 to x, right? (i.e. 0 on the bottom and x on the top, which is technically read as from '0 to x')

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by InteGrand View Post
    You mean from 0 to x, right? (i.e. 0 on the bottom and x on the top, which is technically read as from '0 to x')
    Yes that is what I meant. Sorry for the confusion.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    Consider the function G(x) = Integral from x to 0 of g(u) du, where g(u) = 4 - 4/3 u , for 0 < equal to u < 6,
    u - 10, for 6 < equal to u < equal to 12,

    a) Sketch a graph of g(u)
    b) Find the stationary points of the function y = G(x), and determine their nature.
    c) Find those values of x for which G(x) = 0
    d) Sketch the curve y = G(x), indicating all important features.
    e) Find the area bounded by the curve y = G(x) and the x axis for 0 < equal to x < equal to 6.


    For part a) Do you just work out the intercepts with the axis in the two different sections and also the endpoints x = 0, 12 and also where the two parts meet, ie, at x = 6

    Have no clue about part b) as there are two different parts. Also find stationary points means when first derivative is 0. So is that the derivative of the integral???






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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread











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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    I understand that there is a maximum at x = 3 and minimum at x = 10 and I have used a table to show that. Now which formula do I sub in these values to find the y intercept of the coordinates.

    Also, still confused on part c), 'Find the values of x for which G(x) = 0 , why is it you only use 4 - 4/3 u, for 0 < equal to u < 6 and not the other section??

    Furthermore, in your second post, where G(6) comes from and hence (24 - 24)

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    For the y-coordinates sub into one of these formulas:
    Quote Originally Posted by InteGrand View Post
    You should use both those formulas to find when G(x)=0, so you get 2 y-intercepts - y=0 for the first and y=6 for the second.
    Last edited by rand_althor; 1 Aug 2015 at 10:57 PM.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Can anyone do this?
    Find the exact gradient, with rational denominator, of the normal to the parabola y^2 = 12x at the point where x=4 in the first quadrant.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    I used implicit differentiation and got the answer, but is there any way I can square root both sides and take the positive answer (y=root12x instead of y=-root12 because it states "at the point x=4 in first quadrant".)

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    LOL nevermind, I got it thanks

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    8b from 11G

    Graph of y = 5x and y = 5

    Shaded area is the area within the y axis, y = 5x and y = 5

    Find the volume of the solid generated by rotating the region about the x axis ..

    I get the answer 25 pi / 3

    However the answer in the book say 50 pi / 3

    What is the correct answer??

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Take the difference volumes and you should get

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Why can't you just take the integral as usual , ie V = pi integral 0 to 1 25x^2 dx

    Why do you need to take the differences of volume??

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Because the region you are rotating is bounded by the y-axis (not x-axis), so you are taking the the difference of volumes when rotating around the x-axis. In this case you are findig the volume of solid of revolution of the rectangle bounded by (which is a cylinder with volume ) and subtracting the volume of the solid of revolution bounded by the x-axis, x = 1 and y = 5x. A diagram will greatly help you visualise what's going on.
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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    We can also do it without using integration by subtracting the volume of a cone (using the cone volume formula) from the cylinder volume.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Also similarly for 8c

    ii) volume of the solid generated by rotating region about the y axis

    region is the area between x axis, x = y^2 and x = 4

    My working:

    V = pi integral from 0 to 2 8 - y^4 dy

    then integrate normally.

    What is wrong with my working??

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    Also similarly for 8c

    ii) volume of the solid generated by rotating region about the y axis

    region is the area between x axis, x = y^2 and x = 4

    My working:

    V = pi integral from 0 to 2 8 - y^4 dy

    then integrate normally.

    What is wrong with my working??
    What is the reason for your working?

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    I am trying to find the volume. Not sure where I have gone wrong. The answer is 128 pi / 5 u^3 and I am getting way off.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    I am trying to find the volume. Not sure where I have gone wrong. The answer is 128 pi / 5 u^3 and I am getting way off.
    What's the reasoning for your working, as in, why do you think you've done it right in the first place?

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    I realised my mistake. Really stupid of me. I finally got the answer.

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    c)i ) A sphere of radius r is generated by rotating the semicircle y = (r^2 - x^2)^ 1/2 about the x asix. Show that the volume of the sphere is given by 4/3 pi r^3

    I can do this first part

    Not sure about part ii)

    ii) A spherical cap of radium r and heigh h is formed by rotating the semicircle y = (r^2 - x^2)^1/2 between x = r and x = r - h about the x - axis . Show that the volume of the cap is geven by 1/3 pi h^2( 3r - h)

    I am quite certain it has something to do with part i) but am not sure how to use part i) .

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    Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by appleibeats View Post
    c)i ) A sphere of radius r is generated by rotating the semicircle y = (r^2 - x^2)^ 1/2 about the x asix. Show that the volume of the sphere is given by 4/3 pi r^3

    I can do this first part

    Not sure about part ii)

    ii) A spherical cap of radium r and heigh h is formed by rotating the semicircle y = (r^2 - x^2)^1/2 between x = r and x = r - h about the x - axis . Show that the volume of the cap is geven by 1/3 pi h^2( 3r - h)

    I am quite certain it has something to do with part i) but am not sure how to use part i) .
    doesn't need part i) :P
    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
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