Do I need to test mobile and desktop landing pages separately?

Check your traffic split first
Look at your analytics to see what percentage of your PPC traffic comes from mobile versus desktop. If mobile is less than 20% of traffic, combining tests is probably fine. If mobile is 50% or higher, you should test each separately.
Why they often need different tests
Mobile has different constraints: smaller screen, touch interaction, shorter attention span, and higher abandonment. A desktop page optimized for mouse clicks and desktop reading patterns might have poor mobile UX. Desktop users might scroll through content; mobile users abandon after two scrolls.
The practical approach
Start with a responsive design that works acceptably on both. Run your initial A/B test combining mobile and desktop traffic. If the test result is inconclusive or one variation wins by a tiny margin, segment the results afterward. If mobile converts at 1% while desktop converts at 4%, you now know they need separate optimization.
When to separate from the start
If you're testing radically different page layouts, test desktop first. Once you have a winner, adapt that design specifically for mobile and test mobile variants separately.
Mobile-specific tests
Once you have a solid desktop page, test mobile elements: button size, form field count, text length, and tap target spacing specific to small screens.