I am a newer to GNS3. I am trying to connect a PC in a network. But i couldn't found any card on the host. Even i tried router as a host but it is not solving my prob. Pl let me know.

I have added router as a PC by not setting ip route. Pl find the same in attachment.

When i used to ping " 20.0.0.1" from PC - 10.0.0.10, it is pinging even default gateway is not configured.

Pl let me know how to sort out this issue.

There is lots of documentation on how to do this. You could start by looking in the GNS3 HOWTOs, or under the Documentation tab off the main GNS3 home page - in particular the Connection GNS3 to Real Networks article.

There is also a section in the book GNS3 Network Simulation Guide from PACKT publishing that takes you through a mini project called Connecting your GNS3 router to your LAN which would help you achieve what you are trying to do.
[Disclaimer. I am the author of the GNS3 Network Simulation Guide]

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GNS3 WorkBench-a VMware image of Ubuntu with GNS3 and VPCS installed and a collection of exercises/labs