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OCR (non mei) FP2 Monday 27th June 2016

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Original post by duncanjgraham
Sub in really big postive X values eg X=100 - see where the curve approaches

Sub in really negative X values eg X= -100- see where the curve approaches

Sub in an X value extremely close to one side of a known asymptote , eg if an asymptote is X=1 sub in X=1.001 and see where the curve is going

Sub in an X value extremely close to the OTHER side of the known asymptote , eg for the above you'd sub in X=0.009 and this gives you a very very very good idea of the nature of the curve either side of the asymptotes

You then find points of intersections and any stationary points, and this is enough information to then intuitively join up the curve correctly


Thank you so much. I will try to implement these steps from now.
Original post by tangotangopapa2
June 2013, Q 8) A line (x + 2y = 2) cuts curve with polar equation (r=1+cos theta ) is two parts. Find the ratio of areas above and below the line enclosed by the curve.
Mark Scheme does not seem to be helpful either. It simply states points of intersection as (0,1) and (2,0) out of nowhere and then proceeds in alien fashion. Could someone explain to me? I don't even know where to begin.
The Cartesian equation of curve is x2 + y2 - x = sqrt( x2 + y2).


I'm out atm, if it hasn't been answered by the time I'm back I'll try work through it


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Original post by tangotangopapa2
Thank you so much. I will try to implement these steps from now.


I hope it helps, also if the graph has an unknown constant like y= kx/(X-1)(X-3)... When you're using similar steps to what I gave you, just kinda let k=1 so it doesn't interfere with you're graph drawing

(You can then put k back into anything important like coordinates AFTER you've got the shape of the graph correct)

This is what I do anyways
Original post by tangotangopapa2
June 2013, Q 8) A line (x + 2y = 2) cuts curve with polar equation (r=1+cos theta ) is two parts. Find the ratio of areas above and below the line enclosed by the curve.
Mark Scheme does not seem to be helpful either. It simply states points of intersection as (0,1) and (2,0) out of nowhere and then proceeds in alien fashion. Could someone explain to me? I don't even know where to begin.
The Cartesian equation of curve is x2 + y2 - x = sqrt( x2 + y2).


I was very confused on this question yesterday and I'm still not sure how the mark scheme has done it. The way I did it was convert x+2y=2 into polar coordinates and draw it with the other graph since it cuts it at theta= 0 and pi/2 when you draw it like that, then it becomes a lot clearer which areas you need to find!
Original post by becksreills
I was very confused on this question yesterday and I'm still not sure how the mark scheme has done it. The way I did it was convert x+2y=2 into polar coordinates and draw it with the other graph since it cuts it at theta= 0 and pi/2 when you draw it like that, then it becomes a lot clearer which areas you need to find!


Thank you, but how do you convert x+2y=2 to polar coordinates. (Also this line does not pass through the origin.)
Original post by tangotangopapa2
Thank you, but how do you convert x+2y=2 to polar coordinates. (Also this line does not pass through the origin.)


x=rcos(theta) and y=rsin(theta) so sub those in and rearrange to r=f(theta)
It doesn't pass through the origin, but the value of r at theta=0 is the same as the value of r at theta=0 for the other graph so it cuts it there
How do I do 7iii. I know the solution but could somebody explain why the answer is what it is.. I have found the point in polar coordinates of P, but can't see how this helps me find the tangent...image.jpg
Original post by becksreills
x=rcos(theta) and y=rsin(theta) so sub those in and rearrange to r=f(theta)
It doesn't pass through the origin, but the value of r at theta=0 is the same as the value of r at theta=0 for the other graph so it cuts it there


Thank you
So, by substituting x=rcos(theta) and y=rsin(theta) and rearranging to r=f(theta), I got r = 2/(cos(theta) + 2 sin (theta)). Then equating 'r' with 'r' of r=cos(theta)sin2(theta). This equation is very hard to solve. (I cant :tongue:) but as you said, theta = 0 and pi/2 seem to satisfy the equation. Now how do we find the required areas.

Clearly, finding total area enclosed by curve and area between theta = 0 and pi/2 won't give the required areas, as area of sector (i.e enclosed by curve and half lines theta = 0 and pi/2) is not what we are interested in. We are interested in areas cut by lines. I couldn't find a way to proceed.
Original post by Mathematicus65
How do I do 7iii. I know the solution but could somebody explain why the answer is what it is.. I have found the point in polar coordinates of P, but can't see how this helps me find the tangent...image.jpg


The line of symmetry is y=x. At P, (intuitively) the gradient of the tangent should be perpendicular to the line of symmetry. So the gradient at this point is -1. Now the equation of line passing through P and having gradient -1 is the required equation of line.
IN JUNE 2012 YOU ARE MADE TO FACTORISE A CUBIC EQUATION IN TWO DIFFERENT QUESTIONS - wtf? I don't even know how to do that **** methodically?

*edit* it's a paper with low grade boundaries and you don't lose too many marks for not being able to factorise the cubic , still annoying tho: P
(edited 7 years ago)
Original post by tangotangopapa2
The line of symmetry is y=x. At P, (intuitively) the gradient of the tangent should be perpendicular to the line of symmetry. So the gradient at this point is -1. Now the equation of line passing through P and having gradient -1 is the required equation of line.


Thank you!
Original post by tangotangopapa2
Thank you
So, by substituting x=rcos(theta) and y=rsin(theta) and rearranging to r=f(theta), I got r = 2/(cos(theta) + 2 sin (theta)). Then equating 'r' with 'r' of r=cos(theta)sin2(theta). This equation is very hard to solve. (I cant :tongue:) but as you said, theta = 0 and pi/2 seem to satisfy the equation. Now how do we find the required areas.

Clearly, finding total area enclosed by curve and area between theta = 0 and pi/2 won't give the required areas, as area of sector (i.e enclosed by curve and half lines theta = 0 and pi/2) is not what we are interested in. We are interested in areas cut by lines. I couldn't find a way to proceed.


You don't need to convert x+2y=2 in to polar form. Draw the polar function onto an x-y grid and you can see where it intersects.

Here's my diagram:


And here's my working if you're interested

Spoiler

(edited 7 years ago)
Original post by duncanjgraham
IN JUNE 2012 YOU ARE MADE TO FACTORISE A CUBIC EQUATION IN TWO DIFFERENT QUESTIONS - wtf? I don't even know how to do that **** methodically?

*edit* it's a paper with low grade boundaries and you don't lose too many marks for not being able to factorise the cubic , still annoying tho: P


I don't remember these questions? Which ones are you talking about?
Original post by Parallex
You don't need to convert x+2y=2 in to polar form. Draw the polar function onto an x-y grid and you can see where it intersects.

Here's my diagram:


And here's my working if you're interested

Spoiler



Thank you so, so much. Brilliant solution.
Original post by duncanjgraham
IN JUNE 2012 YOU ARE MADE TO FACTORISE A CUBIC EQUATION IN TWO DIFFERENT QUESTIONS - wtf? I don't even know how to do that **** methodically?

*edit* it's a paper with low grade boundaries and you don't lose too many marks for not being able to factorise the cubic , still annoying tho: P


Which question?


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Original post by marioman
I don't remember these questions? Which ones are you talking about?


june 2012 q 3ii and 8iii)
Original post by drandy76
Which question?


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june 2012 3ii and 8iii
Original post by tangotangopapa2
Thank you so, so much. Brilliant solution.


No problem. :P
Original post by duncanjgraham
june 2012 3ii and 8iii


Doesn't 3 ii reduce to a quadratic? At least that's what it looks like from a glance


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Original post by duncanjgraham
june 2012 3ii and 8iii


Thats really annoying but everytime you have cubic equation then try putting x=1,x=2 and x=-1. This should give you one of the factors. (They should not give cubics with hard to guess factors)

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