Catrina selects three integers from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and then forms the six possible three-digit integers (leading zero allowed) they determine. For instance, for the selection 1, 3, and 7, she would form the integers 137, 173, 317, 371,713, and 731. Prove that no matter which three integers she initially selects, it is not possible for all six of the resulting three-digit integers to be prime.

A 1.50-kg, horizontal, uniform tray is attached to a vertical ideal spring of force constant 185 N/m and a 275-g metal ball is in the tray. The spring is below the tray, so it can oscillate up and down. The tray is then pushed down to point A, which is 15.0 cm below the equilibrium point, and released from rest. (a) How high above point A will the tray be when the metal ball leaves the tray? (This does not occur when the ball and tray reach their maximum speeds.) (b) How much time elapses between releasing the system at point A and the ball leaving the tray? (c) How fast is the ball moving just as it leaves the tray?

What is the product of 2x - 9 and 4x + 1? (A)

$$8 x ^ { 2 } - 38 x - 9$$

(B)

$$8 x ^ { 2 } - 34 x - 9$$

(C)

$$8 x ^ { 2 } + 34 x - 9$$

(D)

$$8 x ^ { 2 } + 38 x - 9$$

Determine whether each triangle Is a right triangle. Justify your answer. Show your work on a separate piece of paper.

$$135 \text { in. } 140 \text { in. } 175 \text { in. }$$

Multiply and simplify. All variables represent positive real numbers.

$$- 3 \sqrt { 11 } \sqrt { 33 }$$

Find the product.

$$\left( 2 x + \frac { 1 } { 3 } \right) ^ { 2 }$$

Graph f(x) = |x + 2| by creating a table of function values and plotting points. Give the domain and range of the function.

You are given the nonhomogeneous differential equation $y''-6y'+9y=f(t)$. Predict the form of $y_p$ for the $f(t)=te^{3t}$, remmembering to consider $y_h$. (You need not evaluate the coefficients.)

The directions for a homemade glass cleaner call for 1 part vinegar to 4 parts water. Give three ratios that are equivalent to the ratio $\frac { 1 } { 4 }$.

Estimate the quotient. $59 \div 6$

Solve each equation or inequality. Use interval notation to express solution sets of inequalities and graph these solution sets on a number line.

$$x ( x - 2 ) = 4$$

For any finite group $P$ let $d(P)$ be the minimum number of generators of $P$ (so, for example,$d(P)=1$ if and only if $P$ is a nontrivial cyclic group and $d(Q_8)=2$). Let $m(P)$ be the maximum of the integers $d(A)$ as $A$ runs over all abelian subgroups of $P$ (so, for example, $m(Q_8)=1$ and $m(D_8)=2$). Define $J(P)=\langle A\ | \ A\ \text{is an abelian subgroup of } P \text{ with } d(A)=m(P)$). ($J(P)$ is called the Thompson subgroup of $P$.)

(a) Prove that $J(P)$ is a characteristic subgroup of $P$. (b) For each of the following groups $P$ list all abelian subgroups $A$ of $P$ that satisfy $d(A)=m(P)$: $Q_8, D_8, D_{16}, QD_{16}$ (where $QD_{16}$ is the quasidihedral group of order $16$). [Use the lattices of subgroups for these groups.] (c) Show that $J(Q_8)=Q_8, J(D_8)=D_8, J(D_{16})=D_{16}, J(QD_{16})\cong D_8$. (d) Prove that if $Q\leq P$ and $J(P)$ is a subgroup of $Q$, then $J(P)=J(Q)$. Deduce that if $P$ is a subgroup (not necessarily normal) of the finite group $G$ and $J(P)$ is contained in some subgroup $Q$ of $P$ such that $Q\trianglelefteq G$, then $J(P)\trianglelefteq G$.

Graph the inequality.

$$y \geq 3$$

In a large city school system with 20 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For each, indicate what kind of sampling strategy is involved and what (if any) biases might result. Randomly select 20 parents from each elementary school. Send them a survey, and follow up with a phone call if they do not return the survey within a week.