1. The questions below relate to the following case study: System: Test Score System As a Systems Analyst, you are appointed to document requirements of a Test Score System. You are also required to study agile project management approach and apply it in the project. As part of the business process, the lecturer marks all the student papers, aggregates the scores and captures them. The Head of Department oversees the work of the lecturer. His duties in this regard include moderation of papers, amendment and approval of the scores, and the submission of the scores to the admin office. The test scores are captured per course, a course belongs to a department and a department belongs to a faculty. Students can enrol for more than one course and lecturers are allowed to facilitate many courses. The Test Score System should also have a functionality for students to check their scores, lodge a dispute on scores they have a problem with, rate the course and chat with lecturers. Q.2.1 Develop a simple UML activity diagram to model the activities between the lecturer and the Head of Department. (11) Q.2.2 Identify six entities from the case study and draw an Entity Relationship Diagram (ERD) using the correct cardinality symbols to present the relationships amongst the entities. You do not have to include attributes for any of the entities.

2. The questions below are still based on the case study in Question 2. Q.3.1 Since you will be applying agile project management, identify and discuss a process you will put in place to control the scope of the project. (6) Q.3.2 The case study has numerous use cases and detailed information about use cases is described with a use case description. List any three aspects of a use case covered in a use case description. (3) Q.3.3 Identify use cases in relation to the student’s interaction with the Test Scores System and develop a use case diagram. (9) Q.3.4 Distinguish between a User Story and a Use Case and provide an example for each, (8) Q.3.5 Designing the user interface for smartphones has several items affecting layout and formatting. List any four of the items.

Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈ Z,b ∈ N} (recall that [(a,b)] = {(c,d) ∈Z×N : (a,b) ∼ (c,d)}).

We define an addition and a multiplication on X as follows: [(a,b)] + [(c,d)] = [(ad + bc,bd)] and [(a,b)]·[(c,d)] = [(ac,bd)]

Prove that this addition and multiplication is well-defined on X.

Suppose 400 balls are distributed into 200 boxes in such a way that no box contains more than 200 balls and each box has at least one ball. Then there are some boxes that together contain exactly 200 balls.

7. Prove that congruence modulo 10 is an equivalence relation on the set of integers. What do the equivalence classes look like?

Show that the set of all real numbers of the form a+b sqrt(2)+c sqrt(3)+d sqrt(6),where a,b,c,d ∈Q, forms a subfield of R

Find the eigenvalues and eigenvectors of the given matrix.

((0.6 0.1 0.2),(0.4 0.1 0.4), (0 0.8 0.4))

What is Determinant Search Method? Explain by using an example.

On January 1, 2017, Smith deposits 1,000 into an account earning nominal annual interest rate of i(4) = 0.04 compounded quarterly with inter- est credited on the last day of March, June, September, and December. If Smith closes the account during the year, simple interest of 4% is paid on the balance from the most recent interest credit date.

(a) What is Smith’s close-out balance on September 23, 2017?

(b) Suppose all four quarters in the year are considered equal, and time is measured in years. Derive expressions for Smith’s accumulated amount func- tion A(t), the close-out balance at time t. Consider separately the four inter- vals 0 ≤ t ≤ 0.25, 0.25 ≤ t ≤ 0.50, 0.50 ≤ t ≤ 0.75 and 0.75 ≤ t ≤ 1 and draw the time diagram for each of these cases.

(c) Using part (b), show that if 0 ≤ t ≤ 0.25, then it follows that δt = δt+0.25 = δt+0.50 = δt+0.75.

Terri Vogel, an amateur motorcycle racer, averages 129.71 seconds per 2.5 mile lap (in a seven-lap race) with a standard deviation of 2.28 seconds. The distribution of her race times is normally distributed. We are interested in one of her randomly selected laps.

A)Find the percent of her laps that are completed in less than 131 seconds. (Round your answer to two decimal places.)

B) The fastest 4% of her laps are under how many seconds? (Round your answer to two decimal places.)

C)

The middle 80% of her lap times are from ___  seconds to ___

For the following functions f and g

: f(x, y) = e ax − (1 − a)lny a > 0 g(x, y, z) = −3x 2 − 3y 2 − 3z 2 + 2xy + 2xz + 2yz

1.

Calculate the Hessian matrices of f and g noted Hf (x, y) and Hg(x, y, z)

2. Show that Hg(x, y, z) is define negativly for all (x, y, z) ∈ Dg

3. For what value o a is , Hf (x, y) define positivly for any (x, y) ∈ Df ?

A typical long-playing phonograph record (once known as an LP) plays for about 2424 minutes at 33133313 revolutions per minute while a needle traces the long groove that spirals slowly in towards the center. The needle starts 5.75.7 inches from the center and finishes 2.52.5 inches from the center. Estimate the length of the groove.

A liquid culture of infectious bacteria was accidentally dumped into a river 900 m upstream of the intake pipe for the municipal water plant; 10^(11) organisms are introduced into the river which has a cross-section of 0.6 m^2 .

(a) If the river velocity is 40 cm/s and the longitudinal dispersion coefficient is 0.1 m^2 /s, what is the maximum concentration expected at the plant?

(b) How spread out (in the direction of flow) is the cloud of organisms at the time the peak concentration occurs at the intake pipe? The “spread” can be described by the standard deviation (σ) on either side of the peak.

(c) How does the answer to (a) change if the organisms have a half-life of 10 hours in the river?

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal basis {v1,v2.....vk-1}. Work out a orthonormal basis of W in terms of that of V and the orthogonal complement of V in W.

Provide detailed reasoning.