A) Statement I true: Statement II is true; Statement II is a correct explanation for statement I
B) Statement I is false; Statement II is true.
C) Statement I is true; statement II is false.
D) Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I.
Correct Answer: B
Solution :
Let\[z=x+iy,\overline{z}=x-iy\] Now, \[z=1-\overline{z}\] \[\Rightarrow \]\[x+iy=1-(x-iy)\] \[\Rightarrow \]\[2x=1\Rightarrow x=\frac{1}{2}\] Now,\[|z|=1\Rightarrow {{x}^{2}}+{{y}^{2}}=1\Rightarrow {{y}^{2}}=1-{{x}^{2}}\] \[\Rightarrow \]\[y=\pm \frac{\sqrt{3}}{2}\] Now, \[\tan \theta =\frac{y}{x}\](\[\theta \] is the argument) \[=\frac{\sqrt{3}}{2}\div \frac{1}{2}\](+ve since only principal argument) \[=\sqrt{3}\]\[\Rightarrow \]\[\theta ={{\tan }^{-1}}\sqrt{3}=\frac{\pi }{3}\] Hence, z is not a real number So, statement-1 is false and 2 is true.
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