(a) For a circuit, the parameters are $R_S=2 \mathrm{k} \Omega$ and $R_P=8 \mathrm{k} \Omega$. (i) If the corner frequency is $f_L=50 \mathrm{~Hz}$, determine the value of $C_S$. (ii) Find the magnitude of the transfer function at $f=20 \mathrm{~Hz}, 50 \mathrm{~Hz}$, and $100 \mathrm{~Hz}$. (b) Consider a circuit with parameters $R_S=4.7 \mathrm{k} \Omega$, $R_P=25 \mathrm{k} \Omega$, and $C_P=120 \mathrm{pF}$. (i) Determine the corner frequency $f_H$. (ii) Determine the magnitude of the transfer function at $f=0.2 f_H, f=f_H$, and $f=8 f_H$.

Fill in the blank(s) using elementary row operations to form a row-equivalent matrix.

$$\left[\begin{array}{rrrr}{2} & {4} & {8} & {3} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right] \\ \left[\begin{array}{rrrr}{1} & {-1} & {-3} & {2} \\ {2} & {?} & {?} & {?} \\ {2} & {6} & {4} & {9}\end{array}\right] \\ \left[\begin{array}{cccc}{1} & {-1} & {-3} & {2} \\ {0} & {6} & {?} & {?} \\ {0} & {8} & {?} & {?}\end{array}\right]$$

Choose one of the sequences of transformation you have described in the preceding exercises. Present a different sequence that would also describe the change from the first figure to the second.

Complete each sentence by writing the form of the verb listed in parentheses. Cross out each pronoun that does not agree with its antecedent and write the correct pronoun above it.
The mirror might break if we _____him. (present tense of drop)

Consider the folded-cascode op amp of Fig.earlier when loaded with a $10-\mathrm{pF}$ capacitance. What should the bias current $I$ be to obtain a slew rate of at least 10 $\mathrm{V} / \mu \mathrm{s}$ ? If the input-stage transistors are operated at overdrive voltages of $0.2 \mathrm{~V}$, what is the unity-gain bandwidth realized? If the two nondominant poles have the same frequency of $25 \mathrm{MHz}$, what is the phase margin obtained? If it is required to have a phase margin of $75^{\circ}$, what must $f_t$ be reduced to? By what amount should $C_L$ be increased? What is the new value of $S R$ ?

Discuss the advisability of simply removing the sail in the preceding exercises.

The telescope at the Palomar Observatory in California has a parabolic mirror. The circle at the top of the parabolic mirror has a 200-inch diameter. An equation that approximates the parabolic cross-section of the surface of the mirror is

$$y = \frac { 1 } { 2639 } x ^ { 2 }$$

, where x and y are measured in inches. How far is the focus from the vertex of the mirror? If a point on a parabola whose vertex is at the origin is known, then the equation of the parabola can be found. For instance, if (4, 1) is a point on a parabola with vertex at the origin, then we can find the equation as follows:

$$y = \frac { 1 } { 4 p } x ^ { 2 }$$

\cdot

$$Begin with the general form of the equation of a parabola.$$

1 = $\frac { 1 } { 4 p }$ ( 4 ) ^ { 2 }\cdot

$$The known point is (4, 1). Replace x by 4 and y by 1.$$

1 = $\frac { 4 } { p }$\cdot

$$Solve for p. p=4$$

\cdot

$$p=4 in the equation$$

y = $\frac { 1 } { 4 p }$ x ^ { 2 }

$$. The equation of the parabola is$$

y = $\frac { 1 } { 16 }$ x ^ { 2 }

$$.$$

The 60-Hz line voltage for a 60-kW, 0.76-pf lagging industrial load is 240$\angle 0^{\circ}$ V rms. Find the value of capacitance that when placed in parallel with the load will raise the power factor to 0.9 lagging.

Using your answers to the preceding problems, polynomial division, and the quadratic formula, if necessary, find all of the zeros of the following polynomials.

$$f(x)=x^4-25$$

Answer the following question to test your understanding of the preceding section:
What is an immunocompetent lymphocyte? What does a lymphocyte have to produce in order to become immunocompetent?

Find the partial fraction decomposition of the given rational expression.

$$\frac{x^3+3 x^2-4 x-8}{x^2-4}$$

In a police forensics lab, you examine a package that may contain heroin. However, you find the white powder is not pure heroin but a mixture of heroin $\left(\mathrm{C}_{21} \mathrm{H}_{23} \mathrm{O}_5 \mathrm{~N}\right)$ and lactose $\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)$. To determine the amount of heroin in the mixture, you dissolve $1.00 \mathrm{~g}$ of the white powdery mixture in water in a $100.0-\mathrm{mL}$ volumetric flask. You find that the solution has an osmotic pressure of $539 \mathrm{~mm} \mathrm{Hg}$ at $25^{\circ} \mathrm{C}$. What is the composition of the mixture?

A parallel-plate capacitor has capacitance $C=12.5 \mathrm{pF}$ when the volume between the plates is filled with air. The plates are circular, with radius $3.00 \mathrm{~cm}$. The capacitor is connected to a battery, and a charge of magnitude $25.0 \mathrm{pC}$ goes onto each plate. With the capacitor still connected to the battery. a slab of dielectric is inserted between the plates. completely filling the space between the plates. After the dielectric has been inserted. the charge on each plate has magnitude $45.0 \mathrm{pC}$.(b) What is the potential difference between the plates before and after the dielectric has been inserted?

Suppose you want to use a repulsive electric force to launch a rocket. (a) How much charge do you need to create enough repulsion to bring a 100,000-kg rocket to escape speed (11,200 m/s) while the rocket travels 100 mm? Assume that two identical charged objects, one mounted on the rocket, are separated by an average distance of 1.0 m for the whole launch. (b) If each elementary charge requires adding mass equivalent to the mass of one hydrogen atom to the rocket, how much additional mass would be required?