(a) Let f (z) = z' - 1 be a complex valued polynomial. (i) By using de Moivre's Theorem, write the five solutions of f (z) = 0 in the form z = cis ( ) , i = 1, 2, 3, 4, 5 and a, b E N [5] (ii) The complex numbers z, are graphed on the Argand diagram and connected to make a regular pentagon inscribed in a unit circle as shown below: 2 2 O.8- 0.6 0.4 0.2 1 -0,8 -0.6 -0.4 -0.2 0 0.2 0,4 0.6 0.8 -0.2 -0.4- .0.6 O.N (b) (i) Calculate the lengths of the line segments Z122, 2123, Z124, 2125 . [4] (ii) Let P be the product of the lengths of the line segments found in (b)(ii).State the relationship between z2 and zs and z3 and z4 and deduce that the pairs of line segments z1Z2 and z125 are equal. Do the same for line segments z1 23 and z1 74. 13] (iii) Hence, using the factorisation of f (z), show that 1+ z2+ 23 + 24 = (22 - 2cos - . z+ 1 ) (22 - 2 cos 5" . z+ 1) [6] Let P(z) = z2 - 2 cos . z + 1 22 - 2cos . z + 1). 4 7 Show that P = P(1) and state its value. [4] [Maximum mark: 22]