4. (a) Prove that (-1+iv3) + (-1 - iv3) has either the value 2+1 or the...

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4. (a) Prove that (-1+iv3)" + (-1 - iv3)" has either the value 2+1 or the value -2" if n is any integer (positive, negative or zero). [39 marks] (b) The complex numbers z1 and z2 are connected by the relation 1 21 = 22 +-. 2 If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing zi describes the ellipse 1 %3D (1+ a²)2 (1– a²)² [17 marks) 5. (a) Find r, y, z and w if 6. 4. r+y 3 -1 2w z+ w 3 [7 marks) (b) и 4 4 -2 A -1 0 1 3 6 -1 show that A? = A+kI for some constant k, where I is the unit matrix of order 3. Hence find the inverse matrix A-1. [20 marks) (c) Show that for all values of 0 the determinant 1 sin 0 1 - sin 0 1 sin 0 -1 - sin 0 1 lies between 2 and 4 inclusive. State one value of 0 for which the determinant has the value 2, and one for which it has the value 4. (21 marks) 4. (a) Prove that (-1+iv3)" + (-1 - iv3)" has either the value 2+1 or the value -2" if n is any integer (positive, negative or zero). [39 marks] (b) The complex numbers z1 and z2 are connected by the relation 1 21 = 22 +-. 2 If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing zi describes the ellipse 1 %3D (1+ a²)2 (1– a²)² [17 marks) 5. (a) Find r, y, z and w if 6. 4. r+y 3 -1 2w z+ w 3 [7 marks) (b) и 4 4 -2 A -1 0 1 3 6 -1 show that A? = A+kI for some constant k, where I is the unit matrix of order 3. Hence find the inverse matrix A-1. [20 marks) (c) Show that for all values of 0 the determinant 1 sin 0 1 - sin 0 1 sin 0 -1 - sin 0 1 lies between 2 and 4 inclusive. State one value of 0 for which the determinant has the value 2, and one for which it has the value 4. (21 marks)