I have a year old DELL INSPIRON N5010laptop which in turn turns ON for 2 minutes and then shuts off.
So I opened up the laptop cleaned dust between fan frills applied new thermal paste (Arctic Silver) and installed the Processor.
With above also I am seeing the issue.
When I closely examined the heat-sink I see some liquid leakage on the copper heat-sink, so is my heat sink went bad? do I need to buy a heat-sink will that fix my problem? Or is it problem with my mother board?
Are you sure it's a liquid leak or simply condensation? The heatsink is a separate part so if you go to http://www.dell.com then follow the link to your own country's local Dell site, click Support and Drivers then input the Service Tag from the label on the bottom of the laptop, you can follow the trail to the specific part you need. You might also find one on e-Bay, maybe by buying a whole breaker for spares.
If the heatsink is not fitted then the processor will destroy itself within a second, so always fit a heatsink. Some processors may throttle back the clock frequency to protect themselves if you are lucky but it is not good to do this.
The liquid could be refrigerant from the heat pipe, if it is leaking then the processor will overheat very quickly, and you will need a new heatsink. A new heasink will cost around $30