To solve this problem, we need to analyze the given rate law for the reaction:$\newline$Rate$= k[A]^2[B] \newline$This indicates that the reaction is second order with respect to $A$ and first order with respect to $B. \newline$Let's consider the initial rate of the reaction when the concentration of $A$ is $[A]_0$ and $B$ is [B]_0:} \newline Initial rate$= k[A]_0^2[B]_0 \newline$Now, if the concentration of $A$ is tripled, the new concentration of $A$ becomes $3[A]_0. \newline$The concentration of $B$ remains constant at $[B]_0. \newline$The new rate of the reaction is given by:$\newline$New rate$= k(3[A]_0)^2[B]_0 \newline$Simplify the expression:$\newline$New rate$= k \cdot 9[A]_0^2[B]_0 \newline$This shows that the new rate is 9 times the initial rate.$\newline$Therefore, when the concentration of $A$ is tripled, the initial rate increases by a factor of nine.$\newline$This corresponds to Option 3.