For a ∈ ℂ , let A = { z ∈ ℂ : Re ( a + z ¯ ) > Im ( a ¯ + z ) } and B…

For a∈ℂ, let A={z∈ℂ:Re(a+z¯)>Im(a¯+z)} and B={z∈ℂ:Re(a+z¯)<Im(a¯+z)}. Then among the two statements:
(S1) : If Re(a),Im(a)>0, then the set A contains all the real numbers
(S2) : If Re(a),Im(a)<0, then the set B contains all the real numbers,
  1. Only S2 is true
  2. only S1 is true
  3. Both are true
  4. Both are false

Solution

Let a=x1+iy1, a¯=x1-iy1

And z=x2+iy2, z¯=x2-iy2      

Now,

a+z¯=x1+x2+iy1-y2

and a¯+z=x1+x2+iy2-y1

Now according to the question,

Re(a+z¯)=x1+x2 and

Im(a¯+z)=y2-y1

A=z:x1+x2>y2-y1=z:x1+y1+x2>y2
  B=z:x1+x2<y2-y1=z:x1+y1+x2<y2

If  y2=0 and x1,y1>0 then

A={z:x2>-x1+y1}

A covers a part of negative real axis and therefore, does not contain whole real axis

Similarly if y2=0, and x1,y1<0, then 
B={z:x2<-x1+y1}
∴ B covers part of positive real axis and therefore does not cover whole real axis.

Hence, both are false.

Asked in: JEE Main 2023 (11 Apr Shift 2)

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