isi-entrance 2015 Q6

isi-entrance · India · UGB 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements
Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(a) $A = B$
(b) $A \subset B$ and $A \neq B$
(c) $B \subset A$ and $B \neq A$
(d) none of the above.
(c) $z ^ { 2 } = w ^ { 2 } \Rightarrow z = \pm w \Rightarrow B \subseteq A$. But $| i | = 1$ and $i ^ { 2 } \neq 1$. 
Let $\mathbb { C }$ denote the set of all complex numbers. Define
$$\begin{aligned}
& A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\
& B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\}
\end{aligned}$$
Then,\\
(a) $A = B$\\
(b) $A \subset B$ and $A \neq B$\\
(c) $B \subset A$ and $B \neq A$\\
(d) none of the above.
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