Reference no: EM133056222
Question 1: Solve the differential equation,
(1 + cos(x) + 2sin(x)) dy/dx + (sin(x)) - 2cos(x))y = 4sin(x) - 8cos(x).
Question 2:
(a) By using the Maclaurin series for ln(1 + x) or otherwise, find the Maclaurin series for ln (1/1 - 3x3) up to and including the term in x12.
(b) Hence find,
lim x->0 In (1-1/(1-3x2)/ x3
(c) Verify your answer in part (b) using l'Hopital's rule.
Question 3:
(a) Use the binomial theorem to expand (cos(θ) + isin(θ))3. Give your answer in the form of a +bi.
(b) Use de Moivre's theorem and the result from (a) to show that cos(3θ) ≡ 4cos3(θ) - 3cos(θ).
Let w be one of the non-real solutions of the equation z7 = 1.
(c) Show that, ∑n=06 ωn = 0
(d) Hence find the value of cos (2Π/7) + cos (4Π/7) + cos (6Π/7).
Mr. Pettifor solves the equation 8m3 + 4m2 - 4m - 1 = 0 using his calculator and finds the following solutions rounded to three significant figures.
m = -0.901
m = -0.223
m = 0.623.
(e) Find cos (2Π/7).
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