(i) Let x+iy be a root of the quadratic equation $z^2 +pz+ 1 = 0$ , where p is a real number.
Show that $x^2 − y^2 + px + 1 = 0$ and $(2x + p)y = 0$.
Show further that either $p = −2x$ or $p = −(x^2 + 1)/x$ with $x\neq 0 $.

Hence show that the set of points in the Argand diagram that can (as p varies) represent
roots of the quadratic equation consists of the real axis with one point missing and a
circle. This set of points is called the root locus of the quadratic equation.

Hello,

I cannot manage to make a good diagram in any of the two situations. I have the hints and answers pdf, but there it is only written the equation after taking moments.

If you could describe (or draw) how to find the distance from A and G to the point of contactt, it would be very helpful.

Thank you

I have been stuck on part iv) of this question for a while now.

I have tried doing proof by contradiction by writing e as the fraction c/d where c and d are co-prime, but have not progressed from there.
I rearranged the equation in part iii) with c/d as the subject
I replaced An with 1/(n+p) where p is positive, since we know that 0

Another geometry question...

I really have no idea how to start problem #8 (other than drawing a picture). The hint said I would need to use some circle theorems, but I can't think of any that would help.

Thanks!

Maybe I'm reading it wrong, but how come S1, S2, S3 are real when S1 = z1 + z2 + z3 = r(cos(theta) + isin(theta)) + r^2(cos(2theta) + isin(2theta)) + .... ?

Thank you!

Paper I 2016 Q4 defines the signed curvature K (kappa) in terms of f'(x) and f''(x).

It then says "use this definition to determine all curves for which the signed curvature is a non-zero
constant".

I got this completely wrong. This was my reasoning:

I thought of K as a function and called it g(x)
I figured if g(x) is constant then g'(x) = 0
(Incidentally if it's non-zero then by the definition give f''(x) is not equal to zero, but I thought I'd look out for that later.)

Probably missing something obvious but,

Why is P(D n E) = (2/3)(1/6)(1/6) .... etc?

I get the concept but not why it's (2/3)(1/6)(1/6) and not (1/6)(1/6)(1/6) for the first (and following) terms as 4 then 5 then 6 would be (1/6)^3?

Thanks

In this binomial expansion question, I was able to get the fraction approximation for i)
However I am not clear on why the error is given by the first neglected term and how I would use this to find the error of my approximation of root(11)?
I got the first neglected term as -1/16 (x)^3, and I subbed in x=1/100 into that term. I am not sure if that is the error or whether I need to use that term divided by approximately 80000/24000 to get the error in this approximation?

Thank you

I'm not exactly sure how to get to y=g(x) or y=h(x)?

I mean, I can sketch y in both cases, and I get that it's the arc tan function, but I'm just not 100% sure how to get there

I kind of get the gist but I'd like a 100% solid understanding if you know what I mean?

Thanks in advance!

Looking at the solutions of TSR

I'm probably missing something obvious but,

Why does the theta dot = -u/(a rt 3) mean that velocity in the y direction is u/rt 3?

Thanks