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# STEP Prep Thread 2016 (Mark. II) Watch

1. (Original post by sweeneyrod)
What would I get with this?

1 - full
2 - did everything up to showing that the minimum distance was whatever (at least I think I got it right, I ended up with T being x = -a).
3 - did part (i), and showed that Q had a factor of (x +1) and that its degree was one more than P's.
7 - showed the first part in what seemed like a bit of a dodgy way, basically just quoting the roots of unity expression from the formula booklet
8 - did (i) and (ii), but verified g(x) by subbing it in rather than deriving it.
12 - full
If the point on QR is (-2a,0) you have a full on q2

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2. (Original post by physicsmaths)
If the point on QR is (-2a,0) you have a full on q2

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Which point? (I've kind of forgotten the question). The last part I did was "show some point lies on a line T" and I found T to be x = -a (I think). Then it said "show the maximum distance from that point to the origin is something", but I didn't write anything for that.
3. Thanks to all you guys estimating, a rough idea of mine would be greatly appreciated!

1- (i) good, couldn't get (ii) and my induction was less than sound in (iii), had a bit of a factorial brain fart.
2- (i) found normals and such using parametric equations but couldn't get the required answer.
3- (i) was good, explanation for (ii) was dodgy but had all the right parts: Q(x) was a factor of (x+1) and degQ(x)=degP(x)-1
4- (i) fine, (ii) subbed in x=e^-2y and after some manipulation got the desired result, but think my sum from infinity to negative infinity was wrong.
6- Intro, (i) and (ii) good, started considering derivatives of the curves for (iii) but got a bit rushed, (iv) think I had y=1/sqrt(a^+1) or something like that
8- (i) and (ii) good, got some reasonable stuff down from h(x) but only considered h(h(x)) not h(h(h(x)))

Thinking 1/2 boundary approximately
4. (Original post by sweeneyrod)
Which point? (I've kind of forgotten the question). The last part I did was "show some point lies on a line T" and I found T to be x = -a (I think). Then it said "show the maximum distance from that point to the origin is something", but I didn't write anything for that.
Ok. So, one was show that QR passes through a point independant of P which is (-2a,0) and now let OP and QR meet at T which led to (-a,-2/p or something similar) then it said show that the distance of T from x acis is less then a/root2 which i did by saying |p|>2root2 or somehing similar by using discriminant as points exist. Last part was probs 5 marks at most so dw tbh.

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5. Bit of a random request, but can people help me develop my memory on questions 1,3,5 and 8?
Q1 was something to do with In/In+1 = 2n-1(b-a^2)/2n or something like that
Q3 can't remember
Q8 was the f(x), g(x) and h(x) one...

I got partials on all 4 of these, want to know how far I actually got
6. (Original post by physicsmaths)
Ok. So, one was show that QR passes through a point independant of P which is (-2a,0) and now let OP and QR meet at T which led to (-a,-2/p or something similar) then it said show that the distance of T from x acis is less then a/root2 which i did by saying |p|>2root2 or somehing similar by using discriminant as points exist. Last part was probs 5 marks at most so dw tbh.

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Cool, thanks.
7. How did people do the end of Q3? Here was my argument (not very sure of it hahah)

Show Q(x) has a factor of (x+1), so (Q(x))^2 has a factor of (x+1)^2. Then the numerator has to have a factor of (x+1) less than the denominator (for it to simplify to 1/(x+1)), and everything in the numerator has a Q(x) term besides Q'(x)P(x), so that has an (x+1) term.

We assume P and Q have no common factors (given), so for Q'(-1)P(-1) = 0, we need Q'(x) to have a factor of (x+1). So Q(x) and Q'(x) have a factor of (x+1), so it's a repeated root and Q(x) has a factor of (x+1)^2

So (Q(x))^2 has a factor of (x+1)^4, so the numerator has a factor of (x+1)^3, and so of (x+1)^2. Again we consider the Q'(x)P(x) term cuz we know Q(x) has a factor of (x+1)^2, so Q'(x) has a factor of (x+1)^2

So Q(x) has a factor of (x+1) and Q'(x) has a factor of (x+1)^2, so Q(x) has a factor of (x+1)^3

repeating the argument we get that (x+1)^n is a factor of Q(x) which is clearly not possible for arbitrarily large n

I also found that the order of p was one more (or less, i forgot which) than the order of Q and given the rest of the question i'm sensing there may have been an easier way....
8. Should this be enough for a 1

Q1: Did part 1 and 3
Made the substitution x+a =√b+a^2tanu and managed to get an expression to the power of n but not required result for part 2

Q2: did part i, worked out the required lines in part ii but not much more

Q3: first part complete and half or so of the second

Q5: did part i ii and iv

Q8 did all but last part. Evaluated hh(x) but didn't think to do hhh(x)

Q12 did first part, wrote down a poisson with mean n and narrowly ran out of time while I tried to manipulate it to achieve result

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9. Can someone estimate my marks please,
1.) For ii.) Couldnt get the desired result missing an a^2 In+1 term but did part i.) And iii.)
2.) Did part i.)
3.) Last part slightly iffy lacked generality in argument.
4.) Couldnt get part i.) Saw method of differences but couldnt get result. Subbed in e^-2y and got final result.
7.) Got the first Z^n+1 result and got close to result for even n using first result.
8.) Got all of the results but not sure if used the correct method. For part ii.) I subbed in the g(x) and showed it satisfied the equation.

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10. (Original post by jweo)
How did people do the end of Q3? Here was my argument (not very sure of it hahah)

Show Q(x) has a factor of (x+1), so (Q(x))^2 has a factor of (x+1)^2. Then the numerator has to have a factor of (x+1) less than the denominator (for it to simplify to 1/(x+1)), and everything in the numerator has a Q(x) term besides Q'(x)P(x), so that has an (x+1) term.

We assume P and Q have no common factors (given), so for Q'(-1)P(-1) = 0, we need Q'(x) to have a factor of (x+1). So Q(x) and Q'(x) have a factor of (x+1), so it's a repeated root and Q(x) has a factor of (x+1)^2

So (Q(x))^2 has a factor of (x+1)^4, so the numerator has a factor of (x+1)^3, and so of (x+1)^2. Again we consider the Q'(x)P(x) term cuz we know Q(x) has a factor of (x+1)^2, so Q'(x) has a factor of (x+1)^2

So Q(x) has a factor of (x+1) and Q'(x) has a factor of (x+1)^2, so Q(x) has a factor of (x+1)^3

repeating the argument we get that (x+1)^n is a factor of Q(x) which is clearly not possible for arbitrarily large n

I also found that the order of p was one more (or less, i forgot which) than the order of Q and given the rest of the question i'm sensing there may have been an easier way....
I let Q(x) = x + 1, I don't know if you can do that, but I did. Whilst also have (x+1)/(x+1)^2 like you. I then said p(x) = ax^2 + bx + 1 and when I compared coefficients, they were inconsistent, so I said p(x) and Q(x) don't exist in this case. Again, I don't know if you can assume Q(x) = x + 1, so I may be wrong there.
11. (Original post by student0042)
I let Q(x) = x + 1, I don't know if you can do that, but I did. Whilst also have (x+1)/(x+1)^2 like you. I then said p(x) = ax^2 + bx + 1 and when I compared coefficients, they were inconsistent, so I said p(x) and Q(x) don't exist in this case. Again, I don't know if you can assume Q(x) = x + 1, so I may be wrong there.
I don't think you can assume Q(x) = x + 1, all you know is that Q(x) = R(x)(x +1) (I didn't do that part fully though).
12. (Original post by student0042)
I let Q(x) = x + 1, I don't know if you can do that, but I did. Whilst also have (x+1)/(x+1)^2 like you. I then said p(x) = ax^2 + bx + 1 and when I compared coefficients, they were inconsistent, so I said p(x) and Q(x) don't exist in this case. Again, I don't know if you can assume Q(x) = x + 1, so I may be wrong there.
Yeah i think they wanted it to be more general :/
13. (Original post by jweo)
Yeah i think they wanted it to be more general :/
I thought so. I did it like this first and then thought it should be general, but I didn't have much time left. I may get a few marks.
14. (Original post by jweo)
Thank you Zacken That difference between a 2 and a 1 is gonna be a crucial one - from what you said it seems like a 78 which should definitely be a 1, but ofc we never know :/ (those are higher than I guessed so clearly I was biased or you're too kind hahah)

Good luck with your cambridge offer too!
Sorry, I miscounted your marks. You've definitely 100% got a 1.
15. (Original post by sweeneyrod)
What would I get with this?

1 - full
2 - did everything up to showing that the minimum distance was whatever (at least I think I got it right, I ended up with T being x = -a).
3 - did part (i), and showed that Q had a factor of (x +1) and that its degree was one more than P's.
7 - showed the first part in what seemed like a bit of a dodgy way, basically just quoting the roots of unity expression from the formula booklet
8 - did (i) and (ii), but verified g(x) by subbing it in rather than deriving it.
12 - full
20 + 16/17 + 13 + Q7 + 8 + 20 = 1 for sure. (I did the same thing in Q8 fml)
16. (Original post by Zacken)
20 + 16/17 + 13 + Q7 + 8 + 20 = 1 for sure. (I did the same thing in Q8 fml)
Need the paper m8?

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17. (Original post by Insight314)
Need the paper m8?

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If you've got it, you could post it up on here if you want. That'd be helpful.
18. how much for q7 part ii? and q12 part i?
19. (Original post by Zacken)
20 + 16/17 + 13 + Q7 + 8 + 20 = 1 for sure. (I did the same thing in Q8 fml)
Thanks! Q8 was a bit ambiguous, it didn't really specify how to show g(x) was what it was, so hopefully they'll be generous.
20. (Original post by smartalan73)

Q1- did part i fully (the substitution) but thats it
Q2- completed (thought this was the easiest question on the paper, surprised more people didn't do it, or maybe I'm just better with parabolas than I thought)
Q3- completed
Q5- only had like 5 mins left so wrote out the binomial expansion and a few inequalities in the hope it might be worth some marks
Q6- did up to part ii fine, attempted part iii but don't feel very confident about it and don't think i really proved the conditions were N & S
Q8- did all but the last part however i proved the equations by taking what they said g(x) was and putting it into the equation to prove it worked, which i thought would be ok as they didn't tell you specifically how to show it, but i'm sure there was probably a 'better' way that also made the last part doable

I'm actually quite disappointed with myself, I got the solution to Q2 in under half an hour and got all excited like I could be on to a really good score, then i just kept starting questions and not seeing them through. Poor question choice and lack of time management let me down I think.
5 + 20 + 20 + 4 + 8 + 10 = 67, should be a low but definite 1.

(Original post by Cnoel)
Thanks to all you guys estimating, a rough idea of mine would be greatly appreciated!

1- (i) good, couldn't get (ii) and my induction was less than sound in (iii), had a bit of a factorial brain fart.
2- (i) found normals and such using parametric equations but couldn't get the required answer.
3- (i) was good, explanation for (ii) was dodgy but had all the right parts: Q(x) was a factor of (x+1) and degQ(x)=degP(x)-1
4- (i) fine, (ii) subbed in x=e^-2y and after some manipulation got the desired result, but think my sum from infinity to negative infinity was wrong.
6- Intro, (i) and (ii) good, started considering derivatives of the curves for (iii) but got a bit rushed, (iv) think I had y=1/sqrt(a^+1) or something like that
8- (i) and (ii) good, got some reasonable stuff down from h(x) but only considered h(h(x)) not h(h(h(x)))

Thinking 1/2 boundary approximately
14 + 2 + 16 + 16 + 15 + 15 = 78 = definite 1.

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