Q77 drives the lower jet bumper on Taxi. It's one of the 6 "special solenoids".
There are many reasons this transistor could have fried in the past.
The resistors make the circuit work. They look OK other than the blackened R103. It may still be OK. Measure it's resistance and compare to R97.

The only real problem I see on the board is the "carbonized" PCB. Carbon is a conductor. I think I'd use a DMM to measure resistance between parts that are NOT connected to ensure that the carbon isn't connecting them incorrectly.
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Chris Hibler - CARGPB #31
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Not diodes....resistors. You can spot check them with your meter while in the board and just compare like resistors to see if they read the same values. To be more accurate, tho, you would need to unsolder one leg of the resistor as you could be reading some parallel resistance while it is still connected to other circuits. But since each transistor bank is similar and the resistor is likely a pulldown/pullup resistor to ground or a 5V supply, you'll probably be accurate enough comparing them in board.

If you want to know what the exact value should be, each resistor is color coded and you can use a color code decoder like http://www.digikey.com/en/resources/conversion-calculators/conversion-calculator-resistor-color-code-4-band

Or check your manual schematic...it should list the ohm measurement of each resistor

Tolerance is listed on chart he linked for you. Gold is 5% so those values are close enough and not your problem.

10% tolerance is normal. So a good 500 ohm resistor could be between 450 and 550 ohms and still be within range.

As Chris said carbon is conductive. All of that carbon needs to be removed from the board, the area needs to be rebuilt and then you can see how things work. Unless you have a lot of board repair experience I would let a pro (like Chris) clean the board up for you. Even if you get it working for now it is not going to be reliable.

The carbon is not just on the surface and will have to be physically removed. You will likely have to go 1/4 to 1/2 way through the board to remove it all (I use a Dremel with a cutter ball). I've seen cases where the board has been turned to carbon all the way through.

This process will destroy traces, pads and through-holes and remove part of the board. The board will have to be filled and the damaged circuits repaired. This is not for the faint of heart, and you need to know what you're getting into before starting.

Here's an example of similar damage.

https://pinside.com/pinball/forum/topic/fixing-a-hole-in-a-pcb

Here is a driver board I repaired. Didn't think to take a picture before I bead blasted and removed the components. Definitely recommend bead blasting if you have access to a blast cabinet with glass beads. Two different holes burnt through the board. First picture is after components have been added back to the board. Never can figure out how to make the pictures show in the order I want.

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