If p , 1 , q are in AP and p , 2 , q are in GP, then which of the...

Since $p$, $1$, $q$ are in AP, we have $p + q = 2(1) = 2$.

Since $p$, $2$, $q$ are in GP, we have $pq = 2^2 = 4$.

For $p$, $4$, $q$ to be in HP, the middle term must be the harmonic mean of $p$ and $q$, which is given by $\dfrac{2pq}{p+q}$.

Substituting the values of $p+q$ and $pq$, we get $\dfrac{2(4)}{2} = 4$.

Thus, $p$, $4$, $q$ are in HP. Statement I is correct.

By the definition of harmonic progression, if $p$, $4$, $q$ are in HP, their reciprocals $\dfrac{1}{p}$, $\dfrac{1}{4}$, $\dfrac{1}{q}$ must be in AP.

Thus, Statement II is also correct.

Both statements I and II are correct.

Answer: Both I and II