Hi Ananth,
I stopped by the Mac Repair Shop near my house late this afternoon and explained your problem to the technician. He said that he has seen this behavior on a few occasions - and the fix is typically to replace the keyboard (which also includes the power button). In some cases the problem was fixed by removing the keyboard cable and cleaning the contacts with 90% Isopropyl alcohol and reinstalling the cable.
He also indicated that this problem has not been detectable even when using advanced diagnostic tools that Apple provides to repair shops.
I asked how one would go about buying a replacement power button - and he indicated that it is not possible to buy a power button by itself - to get the button - you would also have to buy a keyboard.
Apple only ships parts to repair shops. Some repair shops will sell parts to end users - but the end user must sign an agreement that there is NO Warranty on the replacement part if not installed by the repair shop that ordered it.
The technician indicated that sometimes the replacement keyboards have issues as well - and have to be sent back to Apple - so there is risk of getting a defective replacement part (with no warranty) if not installed by the repair shop.
Here in the US - Apple no longer repairs products that are at or beyond End of Life. End of Life products must be repaired at a repair shop. The 2010 MacBook is most likely End of Life.
He did not give me a price for replacing the keyboard because the shop was closing. If you want to email me your serial number (I think you can just click on my user name to do that) - I can have the repair shop look up an approximate price (although that would be a US price and could vary greatly in India) - just so you have an idea of what it might cost (it is somewhat expensive from what I understand). You might be better off putting the money toward a new and more modern MacBook Pro.
I hope this information is helpful and hopefully you will be successful at cleaning the keyboard cable contacts - or possibly find that the power button is simply sticking in a pressed position.
~Scott