What instrument would fail if the pitot tube is clogged?

If the pitot tube is clogged, the airspeed indicator will no longer work properly.

If the ram air portion of the pitot tube is clogged, but the drain hole is open, the airspeed will indicate zero airspeed. Basically, the system would be displaying the same static pressure. However, if the ram air inlet and the drain hole of the pitot tube are clogged, the airspeed would increase when climbing and decrease when descending. This odd behavior is because the pressure in the tube is constant, but the static pressure would be changing. As such, the differences received would be inverse from what was expected.

Reference(s):

FAA-H-8083-25B Pilots Handbook of Aeronautical Knowledge pg. 8-10

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