Algebra Math 151 - Page 2 of 2 Index Number:.............. ATTEMPT ALL QUESTIONS. 1. (a) Let 21 = V3 + i and z2 = a + bi be complex numbers. Suppose that Argument 2 2 7 7 21 12 find Argument(22). (b) Let the map f : C - C be defined by f(z) = 1_ 22 . Find If (2) | if z = 1 + 2i. (c) Solve the equation z - 12 = (9-2z), giving the answer in the form a + iy where I, y ER. (d) Simplify the expression z = i-2i (e) Find the cubic root of 1. (f) For z = 22 + 4i and - = 6 - 8i, find the modulus and argument of w. w (g) The following complex numbers are given, 21 = 2 - 2i, and 23 = a + bi, if |2123) = 16, find the modulus of 23. (h) Given that z = 2eliz satisfies the equation 24 = a(1 + v37). Find the value of a (i) Evaluate (1 + i)8 3 (j) Find the values of x and y in the equation + 2ti 2 - 1 -2;),yER 2. Using mathematical induction, (a) Prove that 13 + 23 + 33 + ... + n3 = _ n2 (n + 1)2 (b) Show that n3 + 2n is divisible by 3