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Yes, this test is terrible.
Also, the interpretation of "life, liberty, and the pursuit of happiness is "life, liberty, and property". I would imagine that many people got that wrong.
Question #33 - D. tax per person equals government spending per person
This is obviously not true. The government has lots of things they can sell, buy, etc.
is where to start when it comes to deriving PDEs. The heat equation and the wave equation fall easily out of vector analysis, as do a number of other familiar PDEs. I'd start with a vector analysis book.