Suppose that k and x are both known to be four-vectors and that in every inertial frame k is a multiple of x. That is, k = λx in frame S, and k' = λ'x' in frame S', and so on. Then the factor λ (the "quotient" of k and x) is in fact a four-scalar with the same value in all frames, λ = λ'. Prove this quotient rule.

Simplify the factorial expression. $\frac { 3 ! } { 5 ! }$

Solve each system of equations. Check your answer.

$$\left\{\begin{array}{l}{x+6 y=1} \\ {2 x-3 y=32}\end{array}\right.$$

Solve. $\frac{x^{2}-6 x-27}{x+3}=3$

Give an example of a multiple equilibria reaction.

Solve the equation.

$$25 a = 10 a ^ { 2 }$$

Suppose a $\in R^{k}, \mathbf{b} \in R^{*} .$ Find $\mathbf{c} \in R^{k}$ and $r>0$ such that

$$|\mathbf{x}-\mathbf{a}|=2|\mathbf{x}-\mathbf{b}|$$

if and only if $|\mathbf{x}-\mathbf{c}|=r.$ (Solution: $3 \mathbf{c}=4 \mathbf{b}-\mathbf{a}, 3 r=2|\mathbf{b}-\mathbf{a}| . )$

To introduce a new beer, a company conducted a survey. It divided the United States into four regions: eastern, northern, southern, and western. The company estimates that 35% of the potential customers for the beer are in the eastern region, 30% are in the northern region, 20% are in the southern region, and 15% are in the western region. The survey indicates that 50% of the potential customers in the eastern region, 40% of the potential customers in the northern region, 36% of the potential customers in the southern region, and 42% of those in the western region will buy the beer. If a potential customer chosen at random indicates that he or she will buy the beer, what is the probability that the customer is from the southern region?

In biological clls, the energy relcased by the oxidation of foods is stored in adenosine triphosphate $(\mathrm{ATP}$ or $\mathrm{ATP}^{4-}).$ The essence of ATP's action is its ability to lose its terminal phosphate group by hydrolysis and to form adenosine diphosphate $(\mathrm{ADP}$ or $\mathrm{ADP}^{3-}).$ $\mathrm{ATP}^{4+}(\mathrm{aq}) + \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightarrow ADP^{5-}(a q) +\mathrm{HPO}_{4}^{2-}(\mathrm{aq}) +\mathrm{H}_3 \mathrm{O}^{-}(\mathrm{aq})$ At pH=70 and $37^{\circ} \mathrm{C}$ (310 K blood temperature) the enthalpy and Gibbs energy of hydrolysis are $\Delta_1 H=-20 \mathrm{kJ} / \mathrm{mol}^{-1}$ and $\Delta_1 G=-31 \mathrm{kJ} / \mathrm{mol}^{-1}$ respectively. Under these conditions, the hydrolysis of 1 mol $ATP^{4+} ( \text{aq})$ results in the extraction of up to 31 kJ of energy that can be used to do non-expansion work, such as the synthesis of proteins from amino acids, muscular contraction, and the activation of neuronal circuits in our brains. (a) Calculate and account for the sign of the entropy of hydrolysis of ATP at pH =7.0 and 310 K. (b) Suppose that the radius of a ypical biological cell is $10 \mu \mathrm{m}$ and that inside it $1 \times 10^{6}$ ATP moleculcs are hydrolysed ech second. What is the power density of the cell in watts per cubic metre $\left(1 W=1 J s^{-1}\right) ?$ A computer battery delivers about 15 W and has a volume of 100 $\mathrm{cm}^{3}.$ Which has the greater power density, the cell or the battery? (c) The formation of glutamine from glutamate and ammonium ions requires 14.2 $\mathrm{kJ} \mathrm{mol}^{-1}$ of energy input. lt is driven by the hydrolysis of ATP to ADP mediated by the enzyme glutamine synthetase. How many moles of ATP must be hydrolysed to form 1 mol glutamine?

Find the best least squares approximating line for the following vectors, and the projections of the vectors onto the one-dimensional subspace: (a)

$$\left[ \begin{array} { l } { 1 } \\ { 4 } \end{array} \right] , \left[ \begin{array} { l } { 1 } \\ { 5 } \end{array} \right] , \left[ \begin{array} { l } { 2 } \\ { 4 } \end{array} \right]$$

, (b)

$$\left[ \begin{array} { l } { 2 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { l } { 4 } \\ { 1 } \end{array} \right] , \left[ \begin{array} { l } { 3 } \\ { 2 } \end{array} \right]$$

, (c)

$$\left[ \begin{array} { l } { 1 } \\ { 2 } \\ { 4 } \end{array} \right] , \left[ \begin{array} { l } { 2 } \\ { 3 } \\ { 5 } \end{array} \right] , \left[ \begin{array} { l } { 2 } \\ { 1 } \\ { 6 } \end{array} \right] , \left[ \begin{array} { l } { 1 } \\ { 1 } \\ { 3 } \end{array} \right]$$

Simplify: 10 cm+100 mm (Write the answer in millimeters.)

(a) Show that if q is time-like, there is a frame S' in which it has the form

$$q' = (0, 0, 0, q'_4).$$

(b) Show that if q is forward time-like in one frame S, then it is forward time-like in all inertial frames.

(a) Show that if a body has speed v < c in one inertial frame, then v < c in all frames. (b) Show similarly that if a signal (such as a pulse of light) has speed c in one frame, its speed is c in all frames.

Prove that if x is time-like and x • y = 0, then y is space-like.