JEE Main202229 Jul 2022Evening ShiftMathematicsComplex NumberActual
Let S = z = x + i y : z - 1 + i ≥ z , z < 2 , z + i = z - 1 . Then the set of all values of x , for which w = 2 x + i y ∈ S for some y ∈ ℝ , is
Options
- A- 2 , 1 2 2
- B- 1 2 , 1 4
- C- 2 , 1 2
- D- 1 2 , 1 2 2
Correct answer
B. - 1 2 , 1 4
Step-by-step solution
Given, z - 1 + i ≥ z ,   z < 2   &   z + i = z - 1 Now solving z - 1 + i ≥ z put z = x + i y we get, x - y ≤ 1 so plotting we get, Now solving z < 2 we get, x 2 + y 2 < 4 on plotting the diagram we get, Now solving z + i = z - 1 we get x + y = 0 Now combine diagram of x + y = 0   &   x - y ≤ 1 will be, Now combining all diagram we get, Hence w = 2 x + i y ∈ S Now for w ∈ S , 2 x ≤ 1 2 so x ≤ 1 4 Now 2 x + y = 0 ⇒ y =