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Edexcel FP2 Official 2016 Exam Thread - 8th June 2016

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Original post by A Slice of Pi
Heres a quick question you might want to try for extra practice...

Use the method of differences to show that

cosec2+cosec4+cosec8+…+cosec2n=cot⁡1−cot⁡2n\mathrm{cosec}\hspace{3pt}2 + \mathrm{cosec}\hspace{3pt}4 + \mathrm{cosec}\hspace{3pt}8 + \ldots + \mathrm{cosec}\hspace{3pt}2^n = \cot 1 - \cot 2^n

(Hint: Think about trig identities linking cosec and cot)


Ah, your hint gave it away. :tongue:

It's not hard to prove that tan⁡x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\sin x}{1+\cos x} from which you argue that cot⁡x2=csc⁡x−cot⁡x\cot \frac{x}{2} = \csc x - \cot x and hence csc⁡x=cot⁡x2−cot⁡x\csc x = \cot \frac{x}{2} - \cot x.

So your sum is: cot⁡1−cot⁡2+cot⁡2−cot⁡4+⋯+cot⁡2n−1−cot⁡2n\cot 1 - \cot 2 + \cot 2 - \cot 4 + \cdots + \cot 2^{n-1} - \cot 2^n which telescopes down to cot⁡1−cot⁡2n\cot 1 - \cot 2^n.
I feel bad for stealing the above question, so I'll contribute one:

Unparseable latex formula:

\displaystyle[br]\begin{equation*} \sum_{n=1}^m 2^n \tan (2^n \theta) \end{equation*}

https://8dedc505ac3fba908c50836f59059ccce5cd0f1e.googledrive.com/host/0B1ZiqBksUHNYdHIxUkJmdndfMlE/June%202013%20(Withdrawn)%20MA%20-%20FP2%20Edexcel.pdf

Q5b - i get the 1/2 but get completely lost trying to find what n terms wouldnt cancel?
Anyone wanna help me answer these questions. I cant do anything until I get these cleared out and time is running out. So reposted. @Zacken can you help you seem tobe able to solve anything
Inequalities:
-Do you need to know how to draw the 1/x , x / xsquared graphs etc. If so, can anyone explain to me how to draw the graphs once you get the x values and then how do you work out the inequalities for those type of graphs.
-How do you do questions 1 and 2 on page 9 (I know how to get the x values for 1 but cant do the rest and I cant do 2 at all)

Complex numbers:
- Page 49 question 6. How to draw the locus without solving the equation for those questions when theres a number outside the |z|.
- Page 59 question 1b??

Polar coordinates:
-On pages 133 and 134 when sketching the curves, the choose different values for theta. How do you know which values to choose.
- Page 143 question 4. How do you solve it when its r squared.

Thank you
(edited 7 years ago)
Reply 744
Avatar for lkara
lkara
For questions like this one does it matter whether you give your answers with theta between -pi and pi or between 0 and 2pi?
root q.png
Original post by lkara
For questions like this one does it matter whether you give your answers with theta between -pi and pi or between 0 and 2pi?
root q.png


No.
Original post by fpmaniac
Anyone wanna help me answer these questions. I cant do anything until I get these cleared out and time is running out. So reposted. @Zacken can you help you seem tobe able to solve anything
Inequalities:
-Do you need to know how to draw the 1/x , x / xsquared graphs etc. If so, can anyone explain to me how to draw the graphs once you get the x values and then how do you work out the inequalities for those type of graphs.
-How do you do questions 1 and 2 on page 9 (I know how to get the x values for 1 but cant do the rest and I cant do 2 at all)


You should be able to sketch rational functions. It's not hard, all you need to note is that 1x\frac{1}{x} is asymptotic to the y-axis for small values of x and then curves downwards steeply before levelling out and moving down towards the x-axis asymptotically for large values of xx. That's the behaviour in the first quadrant; behaviour in the third quadrant is basically the same although mirrored.

1x2\frac{1}{x^2} is pretty much the same except it's in the first and second quadrant, i.e: symmetric in the y-axis.

I don't own a textbook so can't help without pictures or something.
Original post by imnoteinstein


Not easy to explain :tongue: other than looking at what they've done.

the 'middle term' for r = k is cancelled out by combining the 'last term' in r = k-1 and the 'first term' in r = k +1, so you work through up to r = 4 and you see that the only term at the start that isn't cancelling out is 1/2.

Then you work towards the end (starting from n-2).

So for r=n-2 everything cancels out (as you have the 'first term' below it and the 'last term' above it so the middle term is cancelled out, the 'first term' in r=n-2 is used up in cancelling out the previous 'middle term' and the last one is used in cancelling out the 'middle term' in r = n-1.



Similar logic in r=n-1 for the first and second term, but the third term is only half of the middle term in r = n so it only 'half cancels' it.

which is why you get 1/(n+1) - 2/(n+1) = -1/(n+1) as one of the terms in the summation.



and then for r=n the 'middle term' half cancels (as before) and the first one is used up, and the last one does not do anything as there needs to be something below it.
Original post by Zacken
You should be able to sketch rational functions. It's not hard, all you need to note is that 1x\frac{1}{x} is asymptotic to the y-axis for small values of x and then curves downwards steeply before levelling out and moving down towards the x-axis asymptotically for large values of xx. That's the behaviour in the first quadrant; behaviour in the third quadrant is basically the same although mirrored.

1x2\frac{1}{x^2} is pretty much the same except it's in the first and second quadrant, i.e: symmetric in the y-axis.

I don't own a textbook so can't help without pictures or something.


1. Solve the inequality: |x² - 7| < 3(x+1) ....... if the 3 is outside the bracket do you treat it like enlargement
2.
(x² /|x| + 6) < 1 How do you solve it


3. How have they calculated the asymptotes for
y = 7x/3x+1


Complex numbers
1. how do you sketch the locus without solving it first: |z+3| = 3|z-5|
2. For the transformation w=z+4+3i sketch on an argand diagram the locus of w when z lies on the half line arg z = pi/2 ,
3. " " " " w = z-1+2i when z lies on the line y=2x

Polar:
1. Find the polar coordinates of the points on r²=a²sin2θ where the tangent is perpendicular to the initial line. ( dont know how to solve with r² so help me how to start pls.)
2. Sketching curves:
For r=a(1+cosθ) Thhey have chosen θ to be 0 , pi/2, pi, 3pi/2, and 2pi
But for r=sin3θ they considered 0≤θ≤pi/3 , 2pi/3≤θ≤pi and 4pi/3≤θ≤5pi/3 and they chose the values of θ to be 0, pi/6 and pi/3. How do you know which values to choose an which inequalities to consider.

Thanks
(edited 7 years ago)
Original post by Zacken
Ah, your hint gave it away. :tongue:

It's not hard to prove that tan⁡x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\sin x}{1+\cos x} from which you argue that cot⁡x2=csc⁡x−cot⁡x\cot \frac{x}{2} = \csc x - \cot x and hence csc⁡x=cot⁡x2−cot⁡x\csc x = \cot \frac{x}{2} - \cot x.

So your sum is: cot⁡1−cot⁡2+cot⁡2−cot⁡4+⋯+cot⁡2n−1−cot⁡2n\cot 1 - \cot 2 + \cot 2 - \cot 4 + \cdots + \cot 2^{n-1} - \cot 2^n which telescopes down to cot⁡1−cot⁡2n\cot 1 - \cot 2^n.


How would you go about proving
tan⁡x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\sin x}{1+\cos x}

?
Original post by Nikhilm
How would you go about proving
tan⁡x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\sin x}{1+\cos x}

?


The obvious way: tan⁡x2=12sin⁡xcos⁡x2cos⁡x2=sin⁡x2cos⁡2x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\frac{1}{2}\sin x}{\cos \frac{x}{2}\cos \frac{x}{2}} = \frac{\sin x}{2\cos^2 \frac{x}{2}} = \frac{\sin x}{1 + \cos x}
Can anyone help me with a few questions:

1a. Find the value of z which satisfies both |z+2| = |2z-1| and arg z = pi/4
b. Hence shade in the region R on an Argand diagram which satisfies both |z+2| ≥ |2z-1| and pi/4≤arg≤ pi

2. Given that |z+1-i| = 1 find the greatest and least values of |z-1|
3. Find the ccartesian equation of the locus of z when z-z* = 0 ...... what does the star mean
Original post by Zacken
The obvious way: tan⁡x2=12sin⁡xcos⁡x2cos⁡x2=sin⁡x2cos⁡2x2=sin⁡x1+cos⁡x\tan \frac{x}{2} = \frac{\frac{1}{2}\sin x}{\cos \frac{x}{2}\cos \frac{x}{2}} = \frac{\sin x}{2\cos^2 \frac{x}{2}} = \frac{\sin x}{1 + \cos x}


Ah yeah, I wouldn't have thought about using half angles for that question
Original post by Nikhilm
Ah yeah, I wouldn't have thought about using half angles for that question


Why not? The presence of you going from csc⁡2\csc 2 to cot⁡1\cot 1 on the LHS should have been screaming half angles. As were the powers of 2.
Reply 754
Avatar for Inges
Inges
3
Does anyone have any good sources for the general way to deal with plane transformations?

Z plane to W plane stuff?
Original post by Nikhilm
Ah yeah, I wouldn't have thought about using half angles for that question


I don't see how one can come up with that trig identity without working backwards lol
Original post by Pyslocke
I don't see how one can come up with that trig identity without working backwards lol


@Zacken


Ah yeah. They would probably ask you to prove something in a part (a) to give a hint on the approach
Original post by Pyslocke
I don't see how one can come up with that trig identity without working backwards lol


Original post by Nikhilm
@Zacken


Ah yeah. They would probably ask you to prove something in a part (a) to give a hint on the approach


At A-Level yeah. In general, it's pretty intuitive. Especially given the hint.
Anyone got tips on how to turn the equation into partial fractions the easier way? I still do it the C3/C4 way which takes much more time.
Original post by Zacken
At A-Level yeah. In general, it's pretty intuitive. Especially given the hint.


Yup hopefully! Btw where do you find all these obscure questions from?

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