STEP III 2007 question 1 solution

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Solution

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Substituting into the given equation:

Substituting into (*):

as required.

Solution by generalebriety.

Alternative

We are asked to find in terms of . Use De Moivre:

Then is just the ratio of the real and imaginary parts of ; the most difficult part at this point is simply writing out the answer - excuse me if I don't bother LaTeXing it.

Comparing with also lets you more or less write down the relation between and the coefficients of the quartic with roots as well.

Of course, this approach generalises nicely to arbitrary numbers of as well.

Alternative by DFranklin.