After merriage lady can apply passport with her maiden name

No issues.  She does not have to legally change her name after marriage for any reasons.

There can be  implications although she can apply for the same  and get it . 

Citing a few reasons for eg . 

1. while applying for a VISA for eg say in the gulf the difference in the maiden name may become difficult to secure one . 

2. While residing with her in the gulf if cops come to your residence on complains that you are not husband and wife and you have to show your marriage certificate which may not be attested and her passport maiden name is different it would lead to complications to convince them .

yes, but you have to show your husband name.

what is the status of her marraige ( living with husband or seprated) ?