1. What is the definition of an eigenvalue and eigenvector of a matrix?

2. Consider the nonhomogeneous equationy′′(t) +y′(t)−6y(t) = 6e2t.

(a)Find the general solution yh(t)of the corresponding homogeneous problem.

(b)Find any particular solution yp(t)of the nonhomogeneous problem using the method of undetermined Coefficients.

c)Find any particular solution yp(t)of the nonhomogeneous problem using the method of variation of Parameters.

(d) What is the general solution of the differential equation?

3. Consider the nonhomogeneous equationy′′(t) + 9y(t) =9cos(3t).

(a)Findt he general solution yh(t)of the corresponding homogeneous problem.(b)Find any particular solution yp(t)of the nonhomogeneous problem.(c) What is the general solution of the differential equation?

4. Determine whether the following statements are TRUE or FALSE.Note: you must write the entire word TRUE or FALSE. You do not need to show your work for this problem.(

a)yp(t) =Acos(t)+Bsin(t)is a suitable guess for the particular solution ofy′′+y= cos(t).(

b)yp(t) =Atetis a suitable guess for the particular solution ofy′′−y=et.

(c)yp(t) =Ae−t2is a suitable guess for the particular solution ofy′′+y=e−t2.(d) The phase portrait of any solution ofy′′+y′+y= 0is a stable spiral.

5. Consider the matrixA=[−2 0 0,0 0 0,0 0−2].(

a) Find theeigenvaluesofA.

(b) Find theeigenvectorsofA.

(c) Does the set of all the eigenvectorsofAform a basis ofR3?

6. Consider the system of differential equationsx′(t) =−2x+y,y′(t) =−5x+ 4y.

a) Write the system in the form~x′=A~x.

b) Find the eigenvalues of.

c) Find theeigenvectorsofA.

d) Find the general solution of this system.

e) Sketch the phase portrait of the system. Label your graphs.

7. Determine whether the following statements are TRUE or FALSE. You must write the entire word “TRUE” or “FALSE’’. You do not need to show your work for this problem.

a) If|A|6= 0 then A does not have a zero eigenvalue.

(b) IfA=[4 2,0 4]then the solution ofx′=Axhas a generalized eigenvector of A.

(c) LetA=[−1 4 0,0 3 3,1 0−2].The sum of the eigenvalues of A is 18.

(d) Let x′=Ax be a 2x2 system. If one of the eigenvalues of A is negative, the stability structure of the equilibrium solution of this system cannot be a stable spiral.

8. Below (next page) are four matrices corresponding to the 2x2 system of equations x′=Ax,where x= (x1, x2). Match each of the four systems (1)–(4) with its corresponding vector field, one of the four plots (A)–(D), on the next page. You do not need to show your work for this problem.

A=[0 1,1−1]

A=[0−1,1 0]

A=[1 2,−2 1]

A=[−1 0,−1−1]

1. Describe the status of nursing as a profession and as a discipline.

2. The focus of nursing is on the person receiving the care. Explain the aims of nursing as they interrelate to facilitate maximal health and quality of life for patients.

3. define nursing from your own personal perspective and experience. Be sure to include the importance of practicing self-care in relation to the demands of the nursing profession.

4. explore one of the aims of nursing as they interrelate to facilitate maximal health and the quality of life for patients assigned to them by the faculty member. Group members should write examples of the aim in practice from their own clinical experiences and note the appropriate nursing interventions that accompanied each example.

5. Prepare a list of interview questions that will help you learn about these programs and the reasons students chose them.

Let τ ∈ S_n be the cycle (1, 2, . . . , k) ∈ S_n where k ≤ n.

(a) For σ ∈ S_n, prove that στσ^-1 = (σ(1), σ(2), . . . , σ(k)).

(b) Let ρ be any cycle of length k in S_n. Prove that there exists an element σ ∈ S_n so that στσ^-1 = ρ.

describe the process of antigen presentation by MHC-1 and MHC-2

Solve the given equation.

(tan²(θ) − 16)(2 cos(θ) + 1) = 0

θ =

Question 2     

Harry’s business position at 1 July was as follow:

                                                                        RM

Inventory                                                        5,000

Equipment                                                      3,700

Creditor (Maju Ent.)                                       500

Debtor (Abadi Ent.)                                        300

Bank balance                                                  1,200

During July, he:

                                                                        RM

Sold goods for cash – paid to bank                 3,200

Sold goods to Abadi Ent.                                600

Bought goods from Maju Ent.                        3,900

Paid Maju Ent. by cheque                              3,000

Paid general expenses by cheque                   500

Abadi Ent. paid by cheque                             300

Inventory at 31 July was RM6,200

Required:

1. Open ledger accounts (including capital) at 1 July
2. Record all transactions
3. Prepare a trial balance
4. Prepare an income statement for the period
5. Prepare a statement of financial position as at 31 July

what is the difference between type 1 and type 2 diabetes

Consider the function and the value of a.

f(x) =

, a = 9. (a) Use m_tan = lim h→0

to find the slope of the tangent line m_tan = f '(a).

m_tan =  

(b)Find the equation of the tangent line to f at x = a.

(Let x be the independent variable and y be the dependent variable.)

Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of partitions of n + 3 into exactly two distinct parts