

1. Find cos x + cos(x + 2π/3) + cos(x + 4π/3) and sin x + sin(x + 2π/3) + sin(x + 4π/3).


2. Draw the graph of the functions y =  x^{2}  1  and y = x +  x^{2}  1 . Find the number of roots of the equation x +  x^{2}  1  = k, where k is a real constant.


3. Let O be a point not in the plane p and A a point in p. For each line in p through A, let H be the foot of the perpendicular from O to the line. Find the locus of H.


4. Define the sequence of positive integers f_{n} by f_{0} = 1, f_{1} = 1, f_{n+2} = f_{n+1} + f_{n}. Show that f_{n} = (a^{n+1}  b^{n+1})/√5, where a, b are real numbers such that a + b = 1, ab = 1 and a > b.

