Easy to remove Search scopes from Out Of the Box Search Scope Drop down list using JQuery.
Use below JQuery for remove Scopes from Dropdown list.
$(document).ready(function(){
$(window).error(function(){
alert("Error");
});
$("select[id$='SBScopesDDL'] option:contains('Scope2')").remove();
$("select[id$='SBScopesDDL'] option:contains('This Site:')").remove();
});