To enable the scope picker next to search input box I started off with the 2007 approach of writing a custom feature and giving it higher priority as mentioned in this MSDN article. Even after setting the search control’s DropDownModeEx property to “ShowDD” it wasn’t displaying the dropdown. After some struggle I stumbled up on the search settings option under site collection settings. As shown in the screenshot below, it gives us an option to show or hide the scope picker next to the search input box.
Hope this helps.
Hope this helps.