What is the rate equation for the elementary termolecular reaction A+2B $\longrightarrow$ products? For 3A $\longrightarrow$ products?

In the following game, decide on the payoff when the following strategy is used.

$$\begin{matrix} \text{ } & \text{ } && \text{ } & \text{B} & \text{ }\\ \text{ } & \text{ } && \text{1} & \text{2} & \text{3}\\ \\ \text{ } & \text{1} && \text{7} & \text{-5} & \text{0}\\ \text{A} & \text{2} && \text{3} & \text{-3} & \text{8}\\ \text{ } & \text{3} && \text{-1} & \text{5} & \text{11}\\ \end{matrix}$$

(2, 2)

Identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate. Tossing two coins

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?

Radioactive phosphorus is used in the study of biochemical reaction mechanisms because phosphorus atoms are components of many biochemical molecules. The location of the phosphorus (and the location of the molecule it is bound in) can be detected from the electrons (beta particles) it produces:

$$\begin{aligned} &{ }^{32} \mathrm{P} \longrightarrow{ }_{15}^{32} \mathrm{~S}+\mathrm{e}^{-} \\ &\text {rate }=4.85 \times 10^{-2} \text { day }^{-1}[32 \mathrm{P}] \end{aligned}$$

What is the instantaneous rate of production of electrons in a sample with a phosphorus concentration of $0.0033$ M?

Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?

The distribution of the ages of the winners of the Tour de France from 1903 to 2016 is approximately bell-shaped. The mean age is 27.9 years, with a standard deviation of 3.3 years. Use the corresponding z-score to determine whether the age is unusual. Explain your reasoning.

$$\begin{matrix} \text{Winner} & \text{Year} & \text{Age}\\ \text{Jan Ullrich} & \text{1997} & \text{24}\\ \end{matrix}$$

Consider the following hypothetical reaction at 298 K: A(aq) + 2B(aq) $\rightleftharpoons$ 3C(aq), K = ? where the initial concentrations are [A] = [B] = [C] = 1.00 M. The reaction is then allowed to proceed until equilibrium is reached, at which time the concentration of C is observed to be 1.96 M. What is the value of K for the above reaction (at 298 K)?

The linear systems in the problem are in echelon form. Solve each by back substitution.

$$\begin{aligned} x_{1}+x_{2}-2 x_{3}+x_{4} &=9 \\ x_{2}-x_{3}+2 x_{4} &=1 \\ x_{3}-3 x_{4} &=5 \end{aligned}$$

Let p be a prime and let $Z=\{z \in \mathbb{C} | z^{p^{n}}=1$ for some $n \in \mathbb{Z}^{+}\}$ (so Z is the multiplicative group of all p-power roots of unity in $\mathbb{C}$). For each $k \in \mathbb{Z}^{+}$ let

$$H_{k}=\left\{z \in Z | z^{p^{k}}=1\right\}$$

(the group of $p^{k}$ th roots of unity). Prove the following: (a) $H_{k} \leq H_{m}$ if and only if $k \leq m$ (b) $H_{k}$ is cyclic for all k (assume that for any

$$n \in \mathbb{Z}^{+},\left\{e^{2 \pi i t / n} | t=0,1, \ldots, n-1\right\}$$

is the set of all $n^{\text {th }}$ roots of 1 in $\mathbb{C}$) (c) every proper subgroup of Z equals $H_{k}$ for some $k \in \mathbb{Z}^{+}$ (in particular, every proper subgroup of Z is finite and cyclic) (d) Z is not finitely generated.

The following problem categorize the eigenvalues and eigenvectors of the coefficient matrix A according to the figure and sketch the phase portrait of the system by hand. Then use a computer system or graphing calculator to check your answer. $x_{1}^{\prime}=2 x_{1}+3 x_{2}, x_{2}^{\prime}=2 x_{1}+x_{2}$

Find the first three powers of each transition matrix. For each transition matrix, find the probability that state 1 changes to state 2 after three repetitions of the experiment.

$$E=\left[\begin{array}{c} 0.8 & 0.1 & 0.1 \\ 0.3 & 0.6 & 0.1 \\ 0 & 1 & 0 \end{array}\right].$$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $f(x)=\frac{1}{x+2}$

Identify the population and the sample. Describe the sample data set. A survey of 1029 U.S. adults found that 23% of those suffering with chronic pain had been diagnosed with a sleep disorder.