doubt of intersection of ROC Z Transform
In order to answer this question we would need to know the ROCs of $H(z)$ and $Y(z)$. The pole radii only determine the limits of the possible ROCs. E.g., for $H(z)$ there are three different possible ROCs:

$$\begin{align}1.\quad&amp;|z|&gt;\frac{3}{2}\\
2.\quad&amp;\frac32&gt;|z|&gt;\frac14\\
3.\quad&amp;|z|&lt;\frac14\end{align}\tag{1}$$

For $Y(z)$, there are two possible ROCs:

$$\begin{align}1.\quad&amp;|z|&lt;2\\
2.\quad&amp;|z|&gt;2\end{align}\tag{2}$$

Note that each of these ROCs corresponds to a different time domain sequence.

Certain combinations of the ROCs in $(1)$ and $(2)$ overlap, others don't. Below are shown the combinations that result in an overlap of ROCs, and the resulting combined ROC. The first number is the number of the ROC of $H(z)$ given in $(1)$, the second number is the ROC of $Y(z)$ given in $(2)$:

$$\begin{align}1+2:\quad&amp;|z|&gt;2\\
1+1:\quad&amp;2&gt;|z|&gt; \frac32\\
2+1:\quad&amp;\frac32&gt;|z|&gt;\frac14\\
3+1:\quad&amp;|z|&lt;\frac14\end{align}\tag{3}$$

All other combinations of ROCs result in no overlap.