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There are several possible reasons for the delayed start when cold(first start). The most common is fuel leakage from the carb. When the fuel in the carb bowl leaks out, there is none left in the bowl to start the car. Once you crank on it for a minute, it fills the bowl, and the car will then start. Then it will start OK the rest of the day. This is a perfect example of the expansion plugs on the bottom of the fuel bowl leaking. They leak slowly, so it would take an overnight sit to drain the bowl completely. It's VERY common on Quadrajets, and there are a couple of remedies for it.
The quickest is a rubber "pad", that goes between the base plate of the carb, and the center, or "main body" of the carb. It seals the plugs so they don't leak. Over time, however, this "fix" will cease to work, as the rubber pad will shrink, and get soft, due to it reacting with the chemicals in the fuel. It is a good, quick, patch, tho.
Another method to seal the plugs is to "expand" them again with a small mallet, or hammer. This will tap them farther up into the passage they seal, and create a new seal. This will last quite a while, but will also eventually start to leak again after a few years.
The
best solution is to first tap the plugs up a little with the small mallet, and then seal them with an epoxy such as JB Weld. This is a permanent fix.
Check out
Ken's Carb article for more info, and pics on working on the Quadrajet carb.

(click on the link, and then "Ken's Quadrajet")

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Joel Adams
C3VR Lifetime Member #56 
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"Money can't buy happiness -- but somehow it's more comforting to cry in a CORVETTE than in a Kia"