Puneet Narula - MSFT
I am guessing you are using MOSS because you have custom scopes.
SearchBoxEx control has a property
DropdownModeEx which you can set to HideScopeDD enum value(In UI Scope Dropdown section ->Drop down mode->Do not show scope drop down).
Since you want to do this for the SearchBox in the chrome which is in master page, the easiest thing to do is instead of putting the delegate control in the PlaceHolderSearchArea content place holder there put the following
<asp:ContentPlaceHolder id="PlaceHolderSearchArea" runat="server">
<SPSWC
earchBoxEx id="SearchBox" runat="server"
GoImageUrl="/_layouts/images/gosearch.gif"
GoImageUrlRTL="/_layouts/images/gosearch.gif"
DropdownModeEx=HideScopeDD
FrameType="None"
SearchResultsPageUrl="/SearchCenter/Pages/result.aspx"/>
</asp:ContentPlaceHolder>
You may have to change the results page url based on what you have. Also for the complete list of properties on SearchBoxEx you can use the following link
http://msdn2.microsoft.com/en-us/library/microsoft.sharepoint.portal.webcontrols.searchboxex_members.aspx
You will have to register the SPSWC tag prefix using the following at the top of master page
<%@ Register Tagprefix="SPSWC"
Namespace="Microsoft.SharePoint.Portal.WebControls"
Assembly="Microsoft.SharePoint.Portal, Version=12.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c" %>