If I'm using kit lens on a DX body will I get sharper image if using tripod for exposure time of 1/90sec or less ?
The general rule is that a tripod will help make the image more stable somewhere in between 1/focal length to 1/5*focal length depending on how stable you are. There really isn't any disadvantage (shot wise) to using a tripod if you have one handy and setup for the shot you need.
The main reason not to use one is if the advantage you would gain isn't worth carting it to where ever you are or if you need to rapidly be able to change shooting positions (in which case a monopod may be a better option.) If you are going to be shooting from a static position and portability isn't a concern, use a tripod if you have one.