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(A, C) The equivalent resistance between points $A$ and $B$ remains unchanged when a resistance $R$ is connected between two points if the potential d

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$4$ and $7$

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(A, C) The equivalent resistance between points $A$ and $B$ remains unchanged when a resistance $R$ is connected between two points if the potential difference between those two points is zero. This is equivalent to the balanced condition of a Wheatstone bridge,where $\frac{P}{Q} = \frac{R}{S}$.For option $(a)$: Connecting points $2$ and $6$ creates a bridge with arms of resistances $(2r, r)$ and $(4r, 2r)$. Since $\frac{2r}{4r} = \frac{r}{2r} = \frac{1}{2}$,the bridge is balanced.For option $(b)$: Connecting points $3$ and $6$ creates a bridge with arms of resistances $(3r, r)$ and $(3r, 2r)$. Since $\frac{3r}{3r} 
eq \frac{r}{2r}$,the bridge is not balanced.For option $(c)$: Connecting points $4$ and $7$ creates a bridge with arms of resistances $(4r, 2r)$ and $(2r, r)$. Since $\frac{4r}{2r} = \frac{2r}{r} = 2$,the bridge is balanced.For option $(d)$: Connecting points $4$ and $6$ creates a bridge with arms of resistances $(4r, r)$ and $(2r, 2r)$. Since $\frac{4r}{2r} 
eq \frac{r}{2r}$,the bridge is not balanced.Thus,the equivalent resistance remains unchanged for options $(a)$ and $(c)$.

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