Is there a way to run the Apple Hardware Test without an optical drive?
This answer Diagnosis or Hardware Test on a mid 2009 Macbook Pro with MacOS 10.9.2 discusses a GitHub page https://github.com/upekkha/AppleHardwareTest which has assembled various versions of the AppleHardwareTest which you can download and install in a specific place on your harddisk. This allows you to run the test without a working CRROM.

It worked for me on a Mid 2009 Macpook Pro.