State whether the following statements are true or false:
$(a)$ $A$ dimensionally incorrect equation can be correct.
$(b)$ $1 \ AU = 9.46 \times 10^{15} \ m$
$(c)$ The dimensional formula of force and stress are the same.
$(d)$ The dimensions of a physical quantity are different in different systems of units.

(NONE) False. $A$ dimensionally incorrect equation must be physically incorrect.
$(b)$ False. $1 \ AU = 1.496 \times 10^{11} \ m$. The value $9.46 \times 10^{15} \ m$ represents $1 \ \text{light year}$.
$(c)$ False. The dimensional formula of force is $[M^1 L^1 T^{-2}]$, while the dimensional formula of stress (force per unit area) is $[M^1 L^{-1} T^{-2}]$.
$(d)$ False. The dimensions of a physical quantity are independent of the system of units used; only the numerical value changes.