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Q. If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three sets such that $\mathrm{A} \cap \mathrm{B}=\mathrm{A} \cap \mathrm{C}$ and $\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cup \mathrm{C},$ then :-
(1) B = C
$(2) \mathrm{A} \cap \mathrm{B}=\phi$
(3) A = B
(4) A = C
[AIEEE- 2009]
Ans. (1)
$\mathrm{n}(\mathrm{A})=3, \mathrm{n}(\mathrm{B})=6, \mathrm{n}(\mathrm{A} \cap \mathrm{B})=3(\text { maximum })$
so $\mathrm{n}(\mathrm{A} \cup \mathrm{B})=\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})-\mathrm{n}(\mathrm{A} \cap \mathrm{B})$
$3+6-3$
$\mathrm{n}(\mathrm{A} \cup \mathrm{B})=6$
so minimum no of elements in $\mathrm{A} \cup \mathrm{B}$ is $6 .$
Q. Two sets A and B are as under
$\mathrm{A}=\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R} \times \mathrm{R}:|\mathrm{a}-5|<1 \text { and }|\mathrm{b}-5|<1\}$
$\mathrm{B}=\left\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R} \times \mathrm{R}: 4(\mathrm{a}-6)^{2}+9(\mathrm{b}-5)^{2} \leq 36\right\} .$ Then
(1) $\mathrm{A} \subset \mathrm{B}$
(2) $\mathrm{A} \cap \mathrm{B}=\phi$ (an empty set)
(3) neither $\mathrm{A} \subset \mathrm{B}$ nor $\mathrm{B} \subset \mathrm{A}$
(4) $\mathrm{B} \subset \mathrm{A}$
[JEE-MAINS-2018]
Ans. (1)
Q. Let $\mathrm{S}=\{\mathrm{x} \in \mathrm{R}: \mathrm{x} \geq 0 \text { and } 2|\sqrt{\mathrm{x}}-3|+\sqrt{\mathrm{x}}(\sqrt{\mathrm{x}}-6)+6=0\} .$ Then $\mathrm{S}:$
(1) contains exactly one element.
(2) contains exactly two elements.
(3) contains exactly four elements.
(4) is an empty set.
[JEE-MAINS-2018]
Ans. (2)
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