The answers are given already but can u pls give explanation on how they got each answer?

PART A Fill in the blank. Select “A” for Always, “B” for Sometimes, and “C” for Never.

1. A set of 3 vectors from R3 _S(B)_ forms a basis for R3.

2. A set of 2 vectors from R3 _N(C)_ spans R3.

3. A set of 4 vectors from R3 is _N(C)_ linearly independent.

4. A set of 2 vectors from R3 is _S(B)_ linearly independent.

5. If a set of vectors spans a vector space V then it is _S(B)_ a basis for V.

6. If B1 and B2 are two different sets of vectors and each forms a basis for the same vector space, then B1 and B2 _A_ have the same number of vectors.

7. If A is a set consisting of only the zero vector and V is a vector space, then A is _A_ a subspace for V.

8. If S is a linearly independent subset of a vector space V, then a given vector in Span(S) can _A_ be

   expressed uniquely as a linear combination of vectors in S.

Explain or use evidence why each is true or false

If C is any smooth, closed curve then Z C e^{x^2} dx = 0

Let F(x, y) = <P(x, y), Q(x, y)> be a vector field on R 2 and let C be a closed curve formed by the unit circle. If ∂P/∂y = ∂Q/∂x , then R C F · dr = 0.

The coordinate transformation x = u² − v² , y = 2uv maps the quarter-disk S = {(u, v) : u² + v² ≤ 1, v ≥ 0, u ≥ 0} onto the half-disk R = {(x, y) : x² + y² ≤ 1, v ≥ 0}.

If x = g(u, v) and y = h(u, v) is a transformation whose Jacobian is a constant 3, then it must map the square [0, 1] × [0, 1] to a region whose area is 1/3.

The equation in φ = π/12 in spherical coordinates describes a cone in R 3 .

A buck converter is operated from the rectified 230 V ac mains, such that the converter de input voltage
IS
V8 = 325 v ± 20%
A control circuit automatically adjusts the converter duty cycleD, to maintain a constant de output voltage
of V = 240 V de. The de load current I can vary over a 10: 1 range:
lOA::;J:s;lA
The MOSFET has an on-resistance ofO.S Q. The diode conduction loss can be modeled by a 0.7 V
source in series with a 0.2 Q resistor. All other losses can be neglected.
(a) Derive an equivalent circuit that models the converter input and output ports, as well as the loss
elements described above.
(b) Given the range of variation of V8 and I described above, over what range will the duty cycle
vary?
(c) At what operating point (i.e., at what value of V8 and I) is the converter power loss the largest?
What is the value of the efficiency at this operating point?

1. For this question, we define the following vectors: u = (1, 2), v = (−2, 3).

(a) Sketch following vectors on the same set of axes. Make sure to label your axes with a scale. i. 2u ii. −v iii. u + 2v iv. A unit vector which is parallel to v

(b) Let w be the vector satisfying u + v + w = 0 (0 is the zero vector). Draw a diagram showing the geometric relationship between the three vectors u, v and w.

2. Let P 1 and P2 be planes with general equations P1 : −2x + y − 4z = 2, P2 : x + 2y = 7.

(a) Let P3 be a plane which is orthogonal to both P1 and P2. If such a plane P3 exists, give a possible general equation for it. Otherwise, explain why it is not possible to find such a plane. (b) Let ` be a line which is orthogonal to both P1 and P2. If such a line ` exists, give a possible vector equation for it. Otherwise, explain why it is not possible to find such a line.

A survey of Ohio University students was conducted to determine if there was a particular ‘Green’ that was desired by students to live on. A sample of 120 students responses are reproduced below. Do students prefer a particular ‘Green’? Use critical value = 5.99. West Green South Green East Green 40 20 60 Q1: What are the expected values? Q2: What is the calculated chi-squared value? Q3: Was there a significant preference for where students live? A. Yes B. No

5. In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students from the university was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. Calculate the value for the test statistic and chi-square and briefly discuss the results.

The students scored an average final exam score of 83 with a standard deviation of 2. Assume that the scores are approximated by a normal distribution.

• What percent of students scored higher than an 86 on the final exam?
• What percent of students scored less than a 79 on the final exam?
• What percent of students scored between 79 and 86?
• What happens when you try to find the percent of students that scored less than 60?

1. We are interested in estimating the proportion of students at a university who smoke. Out of a random sample of 200 students from this university, 40 students smoke.

(1) Calculate a 95% confidence interval for the proportion of students at this university who smoke and interpret this interval in context.

(2) If we wanted the margin of error to be no larger than 2% at a 95% confidence level for the proportion of students who smoke, how big of a sample would we need?