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1. Determine the remainders when dividing their squares by four

(2n + 1)^2 is not 4n^2 + 1. And I think he was more concerned with the remainder you get when you divide by 4, not the expression in terms of fractions.
2. Solving a trig equation

It's fine because you know that an x where cos(x) is zero can't possibly be a solution (otherwise you have 4 - sqrt(3) = 0).
3. Finding the angel between two vectors

It can't be 90. That would imply the dot product is zero (see the formula mentioned)
4. Question 19 - quadratic probility problem

You're getting there. What's the problem with n being 10.5. Look at your original assumptions. What are you tacitly assuming about the original n balls?
5. Question 19 - quadratic probility problem

Look, you're essentially supposed to say, "Suppose Bill is right. Suppose 7/(n + 7) = 2/5. Then such and such would follow." Why would the conclusion be a problem?
6. Question 19 - quadratic probility problem

Alternatively, you can note that you have 35/(2n + 14) = 1. 35 is odd. 2n + 14 is even. Strange, isn't it?
7. Question 19 - quadratic probility problem

Um...you're leaving out the equality part of the equation. Solve for n.