3
3 3i. You are given the complex number w (i) Find arg w and w 2i . (ii) On an Argand diagram, shade the region representing complex numbers z which satisfy both of these inequalities.
(iii)
2i ! 2
and
1 π 2
! arg z ! 23 π.
Indicate the point on your diagram which corresponds to w. Given that z satisfies both the inequalities in part (ii), find the greatest possible value of z w .
Exercise 11F
z
P3 11
[MEI, part]
4
The complex number w is given by w = 1 i 33. 2 2 (i) Find the modulus and argument of w. (ii) The complex number z has modulus R and argument θ, where − 13 π " θ "− 13 π. State the modulus and argument of wz and the modulus z and argument of w . (iii) Hence explain why, in an Argand diagram, the points representing z, wz z and w are the vertices of an equilateral triangle. (iv) In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number 4 2i. Find the complex numbers represented by the other two vertices. Give your answers in the form x + iy, where x and y are real and exact. [Cambridge International AS & A Level Mathematics 9709, Paper 3 Q10 November 2008]
5 (i) (ii) (iii)
Solve the equation z2 – 2iz – 5 = 0, giving your answers in the form x + iy where x and y are real. Find the modulus and argument of each root. Sketch an Argand diagram showing the points representing the roots. [Cambridge International AS & A Level Mathematics 9709, Paper 3 Q3 June 2005]
6 (i) (ii) (iii) (iv)
Solve the equation z2 (2 3)iz 4 0, giving your answers in the form x + iy, where x and y are real. Sketch an Argand diagram showing the points representing the roots. Find the modulus and argument of each root. Show that the origin and the points representing the roots are the vertices of an equilateral triangle. [Cambridge International AS & A Level Mathematics 9709, Paper 3 Q7 June 2009]
7
The complex numbers 2 (i)
i and 3
i are denoted by u and v respectively.
Find, in the form x iy, the complex numbers (a) u v, u (b) , showing all your working. v 295