- (i) (a) On the same Argand diagram sketch the loci given by the following equations.
$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$
(b) Shade on your diagram the region for which
$$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$
(ii) (a) Show that the transformation \(\quad w = \frac { z - 1 } { z } , \quad z \neq 0\),
$$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$
The region \(| z - 1 | \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
(b) Shade the region \(T\) on an Argand diagram.