A) \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{4}\]
B) \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{2}\]
C) \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{4}\]
D) \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{2}\]
Correct Answer: C
Solution :
Equation of ray PQ \[\arg (z+1)=\frac{\pi }{4}\] Equation of ray PR \[\arg (z+1)=-\frac{\pi }{4}\] Shaded region is \[\frac{-\pi }{4}<\arg (z+1)<\frac{\pi }{4}\] \[|\arg (z+1)|<\frac{\pi }{4}\]; \[|PQ|=\sqrt{{{(\sqrt{2})}^{2}}+{{(\sqrt{2})}^{2}}}=2\] |PA| =2; |PR| = 2 so, arc QAR is of a circle of radius 2 unit with centre at \[P(-1,0)\]. All the points in the shaded region are exterior to this circle\[|z+1|=2\].
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