Question:

1. In a controversial school vote, only 19 percent of the teachers voted yes, whereas 95% of the students voted yes. If all the students and teachers voted, and 91% of the voters voted yes, what is the student to teacher ratio?
2. Find the domain of
   \[f\left( x \right)=\frac{2}{{{\left[ x \right]}^{2}}+\left[ x \right]-56}\]
   where \(\left[ x \right]\) is the greatest integer less than or equal to x .
3. Find all the positive integral solutions to \({{x}^{2}}y-{{y}^{3}}=105\).

(d) What is the value of \[\sqrt{17-12\sqrt{2}}+\sqrt{17+12\sqrt{2}}\] in its simplest form?

(e) A box contains two chips (one is blue and the other is red). You roll a regular die. If you roll a 2 or a 5, you get to blindly select a chip from the box. Otherwise, you don’t get to select a chip. You return the chips to the box. Each turn consists of rolling the die and drawing a chip, if allowed. The game continues until you get a blue chip. What is the probability that the game will not require a third turn?

(f) Henrietta and Tabitha are playing a game that involves tossing a coin several times. Henrietta wins a point for each head, and Tabitha wins a point for each tail. The winner of the game is the first to receive 6 or more points and to have at least a two-point lead over the opponent. If the score is now 5-5, what is the probability that Henrietta will win the game by the score 10 – 8?

(g) Let be a function defined by \(f\left( n \right)=2f\left( n-1 \right)+3f\left( n-2 \right)\), \(f\left( 1 \right)=1\) and \(f\left( 2 \right)=2\). Compute \(f\left( 5 \right)\)

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Deliverable: Word Document