AP EAMCET · Maths · Complex Number
Let \(z\) satisfy \(|z|=1, z=1-\bar{z}\) and \(\operatorname{Im}(z)>0\)
Statement-I : z is a real number
Statement-II : Principal argument of z is \(\frac{\pi}{3}\).
Then
- A Statement-I is true, Statement-II is true and Statement-II is a correct explanation of statement-I
- B Statement-I is true, Statement-II is true, but Statement-II is not a correct explanation of statement-I
- C Statement-I is false, Statement-II is true
- D Statement-I is true, Statement-II is false
Answer & Solution
Correct Answer
(C) Statement-I is false, Statement-II is true
Step-by-step Solution
Detailed explanation
Let \(z = x + iy\). \(z = 1 - \bar{z} \implies x+iy = 1-(x-iy) \implies x+iy = 1-x+iy \implies 2x=1 \implies x = \frac{1}{2}\). \(|z|=1 \implies x^2+y^2=1 \implies (\frac{1}{2})^2+y^2=1 \implies y^2 = 1-\frac{1}{4} = \frac{3}{4}\).…
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