[refref]

I'm assuming we are allowed to dicsuss it now,

1(a)

Find the minimum value of I(a) as a varies,
answer, 4/45

(b) Find the minimum distance between the origin and the circle
answer, 5

(c) What values of k is the equation have 4 solutions (included repeat roots)
answer, -1/4<k<1/4

(d) minimum value such that

answer> 199

(e) for what value of x is
answer, no solutions

(f) for what values of k does this quartic (I have it written down somewhere) have 4 distinct solutions?

answer, (I think) -5<k<0

(g) graph of sin(y) = sin(x)
answer, C, the bottom left one

(h) trapezium rule question on 2^x between 0 and 1, for n amount of strips
answer ,b 1/2n[1+ (1/{2^(1/n)-1}]

(i) For what values of n does divide ?
answer, B = 10 (I think, I didnt put this though, the other quadratic was kind of nasty).

(j) how many pairs of integers satisfy , answer, 2^9 - 1

(2)
(i) Find x_4 and x_5 (I forgot what the formula was)
(ii) Find A, B and C
B = 1/2, C = 1/2

(iii)
for y_n = I forgot the original recursion or w/e formula
find a formula for y_n for n>= 2 and explain your reasoning
I didnt explain my reasoning but I got -1 + n + n^2
(iiii)
Find the limit of x_n/y_n as n > infinity
Write it out, divide top and bottom by n^2, giving the limit as 1/2

(3)
f_n(x) = (x^(2n-1) - 1)^2

(i) sketch f_2(x) (I draw a quadratic like graph crossing (0, 1) and touching (1, 0)
(ii) sketch f_n(x) where n is a large integer, same graph but steeper gradient?
(iii) Integral of f_n(x) (between 0 and 1) <= 1 - A/(B + n) (assumed to be true in the question)

prove that (3n - 1)(B + n) <= An(4n - 1)

(Integrate the function with limits and you will get it in terms of n, cancel the 1's, add the fractions on the left, multiply over I think

The second part of this question was to prove that A<= 3/4. I'm pretty sure I did this in the exam but I forgot what I did (there was more to this question?)

(iii) Prove that A<=3/4
(iiii) Find the minimum value of B such that A<=3/4,


(4)
Parabola C, y = x^2, point (a, a^2) lies on C

(i) Find the normal to C at a

dy/dx = 2x, at point a, dy/dx = 2a

y - a^2 = (-1/(2a))(x - a)

(ii) The point P lies on the Y axis, find point P? Set x = 0, rearrange ( a = +-1/sqrt2

(iii) Find an expression for |QP|^2 in terms of a (pythagoras)

(iiii) minimise this expression.

(I didnt actually minimise it, I thought that a would be +-1/sqrt2

(iv) I really really didnt get this part

(5) (cant really write out the robot question)

(i) I drew a diagram, going all the way to the right and coming back down in a zig zag, then going back up the left. It will always come back up the left because of the even number of zig zags

(ii) Can the tour start anywhere? Yes, as long as it starts on the path

(iii) ROBOT For an even nxn grid, what is the value of f? For a 6x6 grid which is what I thought it meant, I put 36 wrong!, its n^2.

(iiii) Prove that r + 1 is always divisible by 4. I did it like this, but its slightly wrong, as I used the example I drew in (i)

For the first row, 1 right turn

For the next n-2 rows, 1 right turn then another 3 right turns to be facing south

For the n-1th row, 2 right turns

1 + 4(n-2) + 2
= 4n - 8 + 3
= 4n - 5 = r
r + 1 = 4n- 4 = 4(n-1)

I think this is slightly wrong because it assumed its always my example.

(iv) Can a tour exist when n is odd? I used my example again, so this part is wrong (I said because of an odd number of zig zags).

[euclidgeometry]

1b) (closest point on circumference of circle to origin) = 5

1h) trapezium rule ,b 1/2n[1+ (1/{2^(1/n)-1}]

[euclidgeometry]

for question 2:
yn= n^2 + n-1
and xn/yn tended towards 1/2

[meatball893]

Originally Posted by refref

(3)
f_n(x) = (x^(2n-1) - 1)^2

(i) sketch f_2(x) (I draw a quadratic like graph crossing (0, 1) and touching (1, 0)

I differentiated it and found a stationary point/point of inflection at (0,1)

[refref]

Originally Posted by meatball893

I differentiated it and found a stationary point/point of inflection at (0,1)

Yes, there is a point of inflection there, but I didnt draw this

Did the question ask you to point out any stationary points? If so I lost a few marks

[euclidgeometry]

for 4iv
R = (0,a) where a<0, i used R = (0,-1)
(0,-1) is 1 unit awayfrom 0,0. all other points on the cure y=x^2 are to the left/right on the y-axis (therefore the distance PR would have a x compotent) and all points on the curve are higher up than 0,0 and therefore wouild have a greater y compotent than there is for 0,-1. given this 0,0 is the closest point to r and there only point that close.

i have no idea what 1i was... it's really frustrating but i can tell you i put d for it.

[meatball893]

Originally Posted by refref

Yes, there is a point of inflection there, but I didnt draw this

Did the question ask you to point out any stationary points? If so I lost a few marks

No, it didn't.

[refref]

Haha, lets try remember what I was!

Ok I remember now.

[xmarilynx]

I'm impressed you guys can remember all the questions!

[meatball893]

Originally Posted by refref

Haha, lets try remember what I was!

Why, what were you? Are you something different now?

Sorry, couldn't resist

Originally Posted by xmarilynx

I'm impressed you guys can remember all the questions!

So am I actually; I can hardly remember any!

[euclidgeometry]

do be fair, i've been waiting for this for the last 2 and a half days.

[refref]

I've got 84 with out taking into account questions 2~5, which I would assume I would lose at least 10 marks on (for drawing the graph incorrectly maybe, for not actually proving the quadratic for y_n (dont want to use induction...), for not finding the minimum value of B on Q3, and for getting the last part on question 4 and 5 slightly wrong.

[euclidgeometry]

I!!!!!!!!!! i remember that question, that is such a forgettable question. I got 10 and 15 though.

[adamrules247]

Is this just for maths/something similar or a general test?

[refref]

Originally Posted by euclidgeometry

I!!!!!!!!!! i remember that question, that is such a forgettable question. I got 10 and 15 though.

x = 1, x = -1, put this into the equation giving you 2 quadratics. Solve the quadratics, 10 is the solution to both (I think this is right, I got it wrong anyway (I put no values of n))

Can someone do question 3 (iii) for proving A<=3/4, I forgot how I did it.

[euclidgeometry]

surely you should put in x = -1 and x =1 as x^2 -1 = (x-1)(x+1)?

[refref]

Typo, meant - 1

[Robbie10538]

What did people think of the relative difficulty of this paper?

[euclidgeometry]

argh, i think you're right. in the exam i just did x=1 and got 10 and 15. i couldn't be bothered to do the other equation with x =-1. that means 8/10 for me on the multiple choice.

[euclidgeometry]

relative difficulty- slightly harder than usual