I actually looked at it, got sick of this paper by about Q3, so thats all I did. But here's some hints up to there:
Q2: a)
i: You are given that AB = AG, and since ABCD is a ||ogram, then AB=AG=CD. So you can prove that ACGD is a rectangle, as its diagonals are equal, and thus its cyclic...
ii: Since its cyclic, then AD must be an arc of the circle, and you know a property about angles subtended off the same arc...
2. c)
i. To show this thing is in SHM, you have to prove that <sup>d<sup>2</sup>x</sup>/<sub>dt<sup>2</sup></sub> is in the form -n<sup>2</sup>X. You have an x, and take -4 out the front and it gives you that form straight away, where X = (x-1/4)
ii. Use the formula v<sup>2</sup> = n<sup>2</sup>(a<sup>2</sup> - X<sup>2</sup>). You know X from part i, you are given when v = 0 in the question, sub in, get a = 1.
T = 2pi/n, and you know n, so sub in again.
iii. You could do it a number of ways, but using the standard for for velocity, where v = -ansin(nt) would probably work, since you just found all these things in part ii.
3.
i. Answer is 9!/(4!x2!) = 7560
ii. You need the number of arrangements where it goes like this:
EEEE RR P S V.
So this is like having 5 different letters, so the answer (i think), is the number of ways you can arrange these '5' letters, divided by the answer in part i.
so: 5!/(7560) = 1/63 (this treats all e's and r's as the same letter.)
This is where i got sick of it and stopped:
but for b. iii: at t=0, x=1. lnx > 0 at x > 1, and if x < 1 (but >0) then lnx is negative... think about that.
c) didnt even look at it, looks too long.
Q4, c: ii: Looks like you will need cosine rule
Q5: a) Draw triangles.
Q6: a) Similar triangles...
Q6:
b) use midpoint formula, get your x and y coords, and shove in the result you came up with in i. then solve these x and y coords to give you a locus where you have eliminated p's and q's (which will be done because you put in the part i result)
c) Generally when dividing a polynomial, the remainder has to have a degree of less than the bit you are dividing by, so thus it has form ax + b, as other bit is x^2-7x+12
Other part of this: use remainder theorem, by writing the polynomial something like: p(x) = (x-4)(x-3).Q(x) + R(x), where R(x) is remainder ( i think you can do it like this... also look at the form of R(x) from the 1st bit.)
7. Didnt do any of this. Might look at it later...
