In the resolution of the domain name \( \text{www.gate.org.in} \), the process involves resolving each part of the domain name through iterative queries to the respective DNS servers. We break down the process as follows:
1. The query for \( \text{www.gate.org.in} \) is first sent to the root DNS servers.
2. The root servers respond with a referral to the .in TLD DNS servers.
3. The .in TLD DNS servers respond with a referral to the gate.org DNS servers.
4. Finally, the gate.org DNS servers respond with the IP address for \( \text{www.gate.org.in} \).
Thus, there are 4 DNS query-response pairs involved in the complete resolution of the domain name.
Therefore, the number of DNS query-response pairs is:
\[
\boxed{4}.
\]