TPJC/P2/4

(a) The complex number w is given by

(i) Find the modulus and argument of w, giving your answer in exact form.

(ii) Without using a calculator, find the smallest positive integer value of n for which is a real number.

(b) The complex number z is such that

(i) Find the value of z in the form , where and .

(ii) Show the roots on an argand diagram.

(iii) The roots represented by and are such that . The locus of all points z such that intersects the line segment joining points representing and at the point P. P represents the complex number p. Find, in exact form, the modulus and argument of p.