JEE MAIN - Mathematics (2021 - 17th March Evening Shift - No. 4)
Let S1, S2 and S3 be three sets defined as
S1 = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}
S2 = {z$$\in$$C : Re((1 $$-$$ i)z) $$ \ge $$ 1}
S3 = {z$$\in$$C : Im(z) $$ \le $$ 1}
Then the set S1 $$\cap$$ S2 $$\cap$$ S3 :
S1 = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}
S2 = {z$$\in$$C : Re((1 $$-$$ i)z) $$ \ge $$ 1}
S3 = {z$$\in$$C : Im(z) $$ \le $$ 1}
Then the set S1 $$\cap$$ S2 $$\cap$$ S3 :
has exactly three elements
is a singleton
has infinitely many elements
has exactly two elements
Explanation
Let, z = x + iy
_17th_March_Evening_Shift_en_4_1.png)
S1 $$ \equiv $$ (x $$-$$ 1)2 + y2 $$ \le $$ 2 ..... (1)
S2 $$ \equiv $$ x + y $$ \ge $$ 1 ..... (2)
S3 $$\equiv$$ y $$ \le $$ 1 .... (3)
$$ \Rightarrow $$ S1 $$\cap$$ S2 $$\cap$$ S3 has infinitely many elements.
S1 $$ \equiv $$ (x $$-$$ 1)2 + y2 $$ \le $$ 2 ..... (1)
S2 $$ \equiv $$ x + y $$ \ge $$ 1 ..... (2)
S3 $$\equiv$$ y $$ \le $$ 1 .... (3)
$$ \Rightarrow $$ S1 $$\cap$$ S2 $$\cap$$ S3 has infinitely many elements.
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