Can anyone kindly say which of the following two is right?

1.I cannot imagine him getting the first mark.

or

2.I cannot imagine he getting the first mark.

It's the first one. This is a great question.

What about:

1.I cannot imagine his getting the first mark.

That is also a possibility, but it's a bit unwieldy. In this example "getting" is a gerund; whereas, in your example, it is a present participle.

Stick with your first one!

Thanks Buddy. Actually I thought if we are using 'His', then probably we need to frame like 'I cannot imagine his chance/possibility of getting the first mark'. Perhaps, I am not sure about this formation.

in your question we never can use he
but we can also use him and his
that grammatically "his" is more better but in speech "him" is common too.
and if we want to use he we can change it into:
I can not imagine (that) he get the first mark.

the first on is correct....