Solution

Given distance from router $R$ to its neighbors:

$R\to X=3$

$R\to Y=2$

$R\to Z=5$

For router $P$:

Via $X$: $3+7=10$

Via $Y$: $2+6=8$

Via $Z$: $5+5=10$

Minimum distance from $R$ to $P$ is $8$ via $Y$.

So, statement C is true.

For router $Q$:

Via $X$: $3+4=7$

Via $Y$: $2+6=8$

Via $Z$: $5+8=13$

Minimum distance from $R$ to $Q$ is $7$ via $X$.

So, statement B is true.

Correct statements are B and C only.

Correct option is:

$\boxed{(a)\ \text{B and C only}}$