Further Mathematics: Post your doubts here!

[guigao1994]

Hey people,

Could anyone give me a help to solve the reduction formulae part of this question. I managed to prove the differentiation but I can't figure out how to apply that result to form the reduction formulae given:

I am also stuck in part (b) of this question. This one I have no idea whatsoever.

Thank you very much in advance!

 

[abcde]

Hey people,

Could anyone give me a help to solve the reduction formulae part of this question. I managed to prove the differentiation but I can't figure out how to apply that result to form the reduction formulae given:

I am also stuck in part (b) of this question. This one I have no idea whatsoever.

Thank you very much in advance!

AoA!
You've done the hard part of the first question. The rest is pretty simple. Check here:
I'll try doing the second question. Please give the link of the paper. (It might have something to do with series that I haven't yet studied.)

 

[guigao1994]

AoA!
You've done the hard part of the first question. The rest is pretty simple. Check here: View attachment 4920
I'll try doing the second question. Please give the link of the paper. (It might have something to do with series that I haven't yet studied.)

This is the link for the paper:

http://www.xtremepapers.com/CIE/Int.../9231 - Further Mathematics/9231_s07_qp_1.pdf

The ER mention something about doing a geometric progression and finding its sum. I managed to notice that the sum can be rewritten as the imaginary part of the sum of ((e^iO)/2)^n. O= theta. However when I apply the formula for the sum of geometric progression it doesn't work.

Thank you very much so far!

 

[OakMoon!]

I prefer converting sinnΘ to z form rather than e^iΘ. Even though both things are the same. Z seems a bit less complex. I tried to explain the whole concept using this question. But you may still have problems because this can really get tricky at times. Solve more such questions to get it clearly.



Make sure you save the picture and zoom to see the details.

 

[abcde]

^That was enjoyable. I wish I had the time to try it myself.

 

[guigao1994]

I prefer converting sinnΘ to z form rather than e^iΘ. Even though both things are the same. Z seems a bit less complex. I tried to explain the whole concept using this question. But you may still have problems because this can really get tricky at times. Solve more such questions to get it clearly.

View attachment 5004

Make sure you save the picture and zoom to see the details.

Thank you very much! It now makes sense!

 

[OakMoon!]

^That was enjoyable. I wish I had the time to try it myself.

 

[DragonCub]

I prefer converting sinnΘ to z form rather than e^iΘ. Even though both things are the same. Z seems a bit less complex.

Dude I'm not very familiar with the "Z form" you've mentioned. Can you tell me something about it?

 

[abcde]

Dude I'm not very familiar with the "Z form" you've mentioned. Can you tell me something about it?

Any complex number z = x + yi can be written as z = r(cos Θ + i sin Θ), where r is the modulus and Θ is the argument.
r = (x^2 + y^2)^1/2 and tan Θ = y/x.

 

[DragonCub]

Any complex number z = x + yi can be written as z = r(cos Θ + i sin Θ), where r is the modulus and Θ is the argument.
r = (x^2 + y^2)^1/2 and tan Θ = y/x.

Ha, I got it. Thanks abcde

 

[DragonCub]

Hei guys, I got stuck on a question about parametric equations. Here is the problem:



I really could not get through the integration part. Can anyone lend me a hand? Thanks in advance.

 

[abcde]

Hei guys, I got stuck on a question about parametric equations. Here is the problem:

View attachment 5251

I really could not get through the integration part. Can anyone lend me a hand? Thanks in advance.

Is the answer: pi (12 - pi^2)/2 ?

 

[DragonCub]

Is the answer: pi (12 - pi^2)/2 ?

Sorry I don't have the answer here... Can you show me how you worked out the result?

 

[OakMoon!]

Apply the following formula. Make sure you find the square root part separately and then add it to the integral. This will make things easier for you. Thank you!

 

[DragonCub]

Ahh... Is this the integration formula for surface area? Seems I have used the formula for volume by mistake. Thanks a lot dude.

 

[abcde]

Ahh... Is this the integration formula for surface area? Seems I have used the formula for volume by mistake. Thanks a lot dude.

Yes, that's the formula for finding the surface area of revolution about the x-axis. In case a question asks you to find the area of the surface generated when a curve is rotated about the y-axis, you simply substitute the y in the above formula with x and proceed as usual.

 

[DragonCub]

Yes, that's the formula for finding the surface area of revolution about the x-axis. In case a question asks you to find the area of the surface generated when a curve is rotated about the y-axis, you simply substitute the y in the above formula with x and proceed as usual.

And then it becomes exploiting the the techniques for integration?

 

[OakMoon!]

Yes, that's the formula for finding the surface area of revolution about the x-axis. In case a question asks you to find the area of the surface generated when a curve is rotated about the y-axis, you simply substitute the y in the above formula with x and proceed as usual.

Efficient.

 

[guigao1994]

Hello again everyone!

I don't know to to prove the following result:

It seems to be quite simple but I can't get to the given result.

Thank you very much!

 

[MW24595]

Hmm, well, the first thing would be to express the new roots in terms of another variable; let's take y.
So, y= (x/x-2), no? Further, y= 1- (2/x-2).
Therefore, x= 2+ (2/1-y). Simply substitute this expression for x into the original equation- ie x^3 - 3x^2+1= 0, and expand. You should, I reckon, end up with the final expression. It's a painfully tedious expansion, but it should get you the right answer.