Write the Context free Grammar What will be the context-free grammar of $B= A_1.A_2 $
when $A_1 = \{0^n1^n|n\geq 0\}$
$A_2 = \{1^n0^n|n \geq 0\}$
Also verify using a string.
if  the Grammar for
$A_1$: $S_1 \rightarrow 0S_11 | \varepsilon$
$A_2$: $S_2 \rightarrow 1S_20 | \varepsilon$
I am stuck with the value of $S \rightarrow$ ?
Your grammars for $A_1$ and $A_2$ seem correct. Since you want to compute the concatenation of $A_1$ and $A_2$, you just need to express this fact with the start symbols of $A_1$ and $A_2$.
$S \rightarrow S_1S_2$ is a way to do it.
You can check by computing the derivation that creates the word $001110$.