The objective of DRX is reduce the power consumption at UE so that UE's battery will run long time compare to the situation when DRX is not being used. There are mainly two kinds of DRX. The one which is used during RRC connected mode is known as connected mode DRX (C-DRX) and the other one that is used in Idle mode is known as Idle mode DRX.
In case of idle mode DRX, eNodeB would not be aware of UE's location and whenever any data arrives at core network for that UE, network triggers paging message. One foremost important point here is Paging message is not triggered immediately from the network , actually paging message is sent by network at the defined paging frame and paging occasion.
Now coming to your question.
After receiving the paging message from network, UE establishes radio connection with the eNodeB and enters into the RRC connected mode. Again in RRC connected mode, if there is no data transmission for the configured time interval, UE enters into C-DRX mode. In this stage, UE will be in RRC Connected and C-DRX mode.
An eNodeB runs user inactivity timer for an UE when it observes that there is no data transmission happened for a particular UE for the configured time internal. Once that timer gets expired, eNodeB releases radio connection with UE and UE enters into RRC-IDLE mode. In this stage, UE will be in RRC Idle and DRX mode.
Answer is Yes. In case of C-DRX mode, UE will be in RRC Connected mode.