262StudyGuide0611262StudyGuideProblems262StudyGuide2000.pdfJune 2011PURDUE UNIVERSITYStudy Guide for the Credit Exam in (MA 262)Linear Algebra and Differential EquationsThe topics covered in this exam can be found in “An introduction to differential equations andlinear algebra” by Stephen Goode, Chapters 1–11. See alsohttp://www.math.purdue.edu/academic/courses for recent course information and materials.The exam consists of 25 multiple choice questions w ith two hour time limit. Nobooks, notes, calculators or any other electronic devices are allowed.IMPORTANT1. Read this material thoroughly if you contemplate trying for advanced placement (and extracredit which counts toward graduation).2. Study all the material listed in the outline.3. Work the practice problems below. The correct answers are at the end.Topics covered in the examLinear Algebra:1. Complex numbers, polar representation, roots of complex numbers.2. Vector spaces: subspace, basis, spanning set, linear combination, linear independence.3. Matrix operations: addition, multiplication, inverse, determinants.4. Row reduction of matrices: row-echelon normal form, rank.5. Systems of linear equations: solve using matrix methods, augmented matrices, Cramer’s rule,solution space.6. Eigenvalues and eigenvectors.Differential Equations:1. First-order differential equations with and without initial conditions. Types include - separablevariables, exact, linear.2. Applications of first-order D.E.’s to mixture problems, growth and decay problems, fallingbodies, electrical circuits, orthogonal trajectories.3. Second and higher-order linear D.E.’s with constant coefficients: Undetermined coefficients,initial value problems. Variation of Parameters.4. Applications of second-order D.E.’s to Newton’s 2nd law, spring mass systems, electrical cir-cuits, etc.Systems of Differential Equations:1. Matrix Formulation.2. Solutions to linear systems with constant coefficient using eigenvalues and eigenvectors.3. Variation of parameters for systems.MA 262 PRACTICE PROBLEMS page 1/8Circle the letter corresponding to your choice of correct answer.1. Find the determinant of A ifA =1112122012300341A. −1B. 0C. 3D. 7E. 112. If A is a 3 × 4matrixandP =p1p2p3p4and Q =q1q2q3q4aresolutions of Axyzw=1213which of the following is alsoa solution?A. P + QB. P − QC. 2P − QD. Q −2PE. all of the above3. Which of the following complex numbers z satisfies z2= riwhere r is some real number?A. i3B. 1 + i√3C.√3+i√3D.√3+iE. none of these4. Which of the following are real vector spaces (in each casemultiplication by a real number and addition have theusual meanings)?(i) The set of all real valued functions f such thatf(x +1)=2f(x) for all x.(ii) The set of all real polynomials p such thatp(1) = p(0) + 1.(iii) The set of all 4 ×4 matrices whose first and last rowsare equal.A. (i) and (iii)B. noneC. (i) and (ii)D. only (iii)E. (ii) and (iii)5. The smallest subspace of R3containing the vectors (1, 2, 1)and (5, 3, 1) is the set of all (x, y, z) satisfying which of thefollowing?A. x, y, z ≥ 0B. x2+ y2+ z2=0C. 3x + y − z =0D. x +5y +7z =0E. x − 4y +7z =06. Which of the following sets of vectors is linearly depen-dent?(i) (1, 0, 1), (1, 1, 0), (0, 1, −1)(ii) (1, 0, 1), (1, 1, 0), (0, 1, 1)(iii) (1, 0, 1), (0, 1, 0), (0, 0, 0)A. noneB. (ii) onlyC. (i) and (ii)D. (i) and (iii)E. (ii) and (iii)7. The characteristic values of A =11− kk +1 1are realifA. k2> 1B. k2≥ 1C. k2=1D. k2≤ 1E. k2< 18. The system of equations126340233xyz=abchas anonzero solutionA. for all nonzero a, b, cB. if a = b + cC. if 2c = a + bD. if b 6= a − cE. if c 6=2a − b9. If the 2 ×2 matrices B =3/21/2xyand A =21−10satisfy B2= A, then the bottom row of B is equal toA. (−1, 0)B.−12,12C.−12, −12D.12, −12E.−12, 110. If A =3142then which of the following is not an entryof A−1?A.23B. −2C. −12D. all are entriesE. none are entries11. The matrixA =211121−2 −2 −1has 1 as a characteristic value. What is the dimensionof the vector space spanned by the characteristic vectorscorresponding to 1?A. 2B. 0C. 1D. 3E. cannot be determined12. If A and B are 4 × 4 matrices and detA =4,detB =6,then det(A−1BT)=A.124B.23C.32D. 24E. Undefined13. If y0+1xy =exexand y(1) = 0, then y =A. xex−1− 1B. (x − 1)exC.1x2(ex−1− 1)D.1x(ex−1− 1)E. (2x − 2)ex14. The solution of (cos2x)y0= y2satisfying y(π/4) = 1 is A. y =2− tan xB. y =2tanx − 1C. y =12 − cot xD. y =1+sinx1+cosxE. y =cos x2cosx − sin x15. The general solution ofdydx=2xy +3x2y2− x2isA.y33+ x2y + x3= cB.y33− x2y − x3= cC. 2xy + y2+3x2= cD. log(x2+ y2)=cE. none of the above16. A tank initially contains 50 gal of water. Alcohol enters atthe rate of 2 gal/min and the mixture leaves the tank atthe same rate. The time t0when there is 25 gal of alcoholin the tank satisfiesA. 25 = et0/2B. 1 = t0/50C. t0=25ln2D.125= et0/25E. t0=2517. The general solution of y(4)+4y = 0 is A. ex(c1+ c2x)+e−x(c3cos x + c4sin x)B. ex(c1cos x+c2sin x)+e−x(c3cos x+c4sin x)C. (c1+ c2x)e−x+(c3+ c4x)exD. e−x(c1+ c2x + c3x2+ c4x3)E. none of the above18. For a particular solution of y00+3y0+2y = xe−xone shouldtry a solution of the formA. Ae−2x+ Be−xB. Ae−x+ BxexC. (A + Bx)e−xD. (Ax + Bx2)e−xE. Ax2e−x19. A body attached to the lower end of a vertical spring hasaccelerationd2xdt2= −16x ft/sec2,wherex = x(t)isthedistance of the body from the equlibrium position at timet. If the body passes through the equilibrium position att = 0 with v = 12 ft/sec, then x =A. 3 cos 4tB. cos 4t +3sin4tC. 3 sin 4tD. 3e4tE. 3 cos 4t − sin 4t20. The substitution t = extransformsy00+(ex− 1)y0+ e2xy = e3xintoA.d2ydt2+dydt+ y = tB. t2d2ydt2+(t − 1)dydt+ t2y = t3C. t2d2ydt2+ tdydt+ t2y = t3D.d2ydt2+(t − 1)dydt+ t2y = t3E. none of the above21. The smallest order of a linear homogeneous differentialequation with constant coefficients which hasy = xe2xcos x as a solution isA. 3B. 4C. 5D. 6E. more than 622.A2× 2 matrix A has 3 as a characteristic value withmultiplicity 2. Then two linearly independent solutions ofx0= Ax are of the formA. k1e3x, (k2+ k1)e3xB. k1e3x, (k2+ tk3)e3xC. k1e3x, k2te2xD. k1e3x, k2e−x23. If a fundamental matrix for x0= Ax isU(t)=e2t00 et, then a particular solution up(t)of x0= Ax +e2tet,