JEE Main202117 Mar 2021Evening ShiftMathematicsComplex NumberActual
Let S 1 , S 2 and S 3 be three sets defined as S 1 = z ∈ ℂ : | z - 1 | ≤ 2 , S 2 = z ∈ ℂ : R e ( ( 1 - i ) z ) ≥ 1 and S 3 = z ∈ ℂ : I m ( z ) ≤ 1 . Then, the set S 1 ∩ S 2 ∩ S 3
Options
- Ais a singleton
- Bhas exactly two elements
- Chas infinitely many elements
- Dhas exactly three elements
Correct answer
C. has infinitely many elements
Step-by-step solution
We know that the complex number z satisfying z - z 0 = r represents a circle with centre z 0 and radius r units. Hence, for S 1 = | z - 1 | ≤ 2 ,   z lies on and inside the circle of radius 2 units and centre ( 1 ,   0 ) . For S 2 , let z = x + i y Now, ( 1 - i ) ( z ) = ( 1 - i ) ( x + i   y ) ⇒ ( 1 - i ) ( z ) = x + i y - i x - i 2 y ⇒ ( 1 - i ) ( z ) = x + i y - i x + y ⇒ R e ( ( 1 - i ) z ) = x + y ⇒ x + y ≥ 1 . And, for S 3 , again let z = x + i y , ⇒ y