Hiya
I agree with the Yaw damper one, the indicator only shows Rudder movement caused by the Yaw Damper.
The Autosynch one, looks like part of a multi-part question, because failed autosynch could prevent AP engagement, and it is inhibited when the AP is engaged, so both answers are correct.
Where on earth did you get that awful fuel question from?
2. Fuel quantity in aircraft tanks is measured by capacitor contents gauges. These sensors measure.
a. The condensor charge
b. Height of the fuel
Okay be careful with these fuel guaging questions. The sensors are capacitors and they will have a certain capacitance or charge depending on the Dielectric value of what's in the tank, but on their own they can only measure volume or height of fuel. What makes the system measure mass or weight of fuel is the compensators fitted in the reference loop. These feed the system effectively with density information, so, what you are looking for in the question is, is it a "Compensated" system or is it just an accurate volume sensing system, which is not affected by attitude, acceleration etc.
If the question does not specify, then assume Compensated since most aircraft will be.
Okay I'm off to lie down now, but feel free to message me if you need to.
My specialist subjects are drinking, bad jokes, Autoflight, Warning and Recording, Engine Instruments, DC Electrics, Piston Engines and Gas Turbines, but I'll have a crack at any Systems stuff.
PFD