Posted by dennis on September 13, 2002 at 00:59:57:
...I was lying. Here're 2 irritating buggers from my last math test. Can anyone help?
1. Let w be the root of the equation z^5 = 1 such that 0 < arg(w) < pi/2. Show that
w^2 + w^3 = 2cos(4pi/5)
and hence show that (and this is the bit I couldn't do)
cos(pi/5)cos(2pi/5) = 1/4
2. Let A, B and C be the roots of the equation
x^3 + px + q = 0
where p and q are real, and q is not 0 (i.e. q =/= 0). Show that
(A - B)^2 = C^2 + (4q)/C
indicating clearly where you need to use q =/= 0.
(i) Show that (A-B)^2, (B-C)^2 and (C-A)^2 are the roots of the equation
y^3 + 6py^2 + 9p^2y + 27q^2 + 4p^3 = 0
(ii) Solve the equation
[27(42^2) - 4(43^3)]t^3 + 9(43^2)t^2 - 6(43)t + 1 = 0
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