Q1 /10 Q2 /10 Q3 /10 Q4 /10 Q5 /10 Q6 /10 Q7 /10 Q8 /10 Q9 /10 Q10 /10 Total /100
|
|
- Morris Butler
- 5 years ago
- Views:
Transcription
1 Midterm Maths Calculus III July 23, 2012 Name: Solutions Instructions You have the entire period (1PM-3:10PM) to complete the test. You can use one half-page for formulas, but no electronic devices may be used. In particular, you may not use calculators or cell phones. Cell phones should be on silent and out of sight. Except for the true/false question, you must show all of your work. Partial credit will be awarded, but correct answers without supporting work will not get credit. Make sure that your final answer is clearly labelled as such. If you need more space, you may continue on the other side of the page, or use the last page of the exam. Be sure to indicate the number of the problem if you do so. For grading purposes only (do not fill in) Q1 /10 Q2 /10 Q3 /10 Q4 /10 Q5 /10 Q6 /10 Q7 /10 Q8 /10 Q9 /10 Q10 /10 Total /100
2 Q1: Find the determinant of the following matrix A = Perform row operations to transform A into an upper triangular matrix: det A = (R 1 R 2 ) = (R 3 3R 1 + R 3, R 4 2R 1 + R 4 ) = (R 3 R 2 + R 3, R 4 3R 2 + 2R 4 ) = (R 3 R 4 ) = 1 (12) = 6. 2
3 Q2: Find the inverse of the following matrix A = (R 1 R 2 ) /2 1 1/2 3/ /2 0 1/2 5/ R 3 4R 4 + R 3 R 2 R 4 + R 2 R 1 R 4 + R 1 (R 2 R 3 + R 2 ) (R 1 R 2 + 2R 1 ) ( R1 1 2 R 1 R R 2 ). So A 1 = 1/2 1 1/2 3/2 1/2 0 1/2 5/
4 Q3: Suppose A is a 2 2 matrix such that A ( ) 4 = 5 ( ) 4 and A 5 ( ) 5 = 2 6 ( ) 5. 6 What is A? ( ) ( ) 4 5 and are eigenvectors of A corresponding to the eigenvalues 1 and respectively. Therefore A = ( ) ( ) ( ) = ( ) ( ) 6 5 = 5 4 ( )
5 Q4: The set of solutions to the following system of equations is a subspace of R 4 (you don t need to prove this): Find a basis for the set of solutions. 2w + x y 2z = 0 w + x + y z = 0 3w 2x + 3z = 0 Row reduce the augmented matrix to reduced row-echelon form: (R 1 R 2 ) ( ) R2 2R 1 + R 2 R 3 3R 1 + R 3 ) R 3 R 2 + R 3 R 1 R 2 + R 1 R 2 R 2 So the free variables are y and z. Let y = s, z = t. Then w 2y z = 0 and x + 3y = 0 give w = 2y + z = 2s + t and x = 3y = 3s, so w 2s + t 2 1 x y = 3s s = s t 0 0. z t 0 1 Therefore a basis is ,
6 Q5: Circle either true or false (you do not need to give reasons) (a) T/F : The set V = {f(x) C(R) : f(1) = 1} is a subspace of C(R), the vector space of continuous functions f : R R. (b) T/F : If 0 is an eigenvalue of an n n matrix A, then rank(a) < n. (c) T/F : If A is a 5 5 matrix and rank(a) = 3, then the system Ax = b always has infinitely many solutions. (d) T/F : If a square matrix has an eigenvalue with multiplicity greater than 1, then it is not diagonalizable. (e) T/F : If A and B are square matrices of the same size with B symmetric, then ABA is symmetric. (f) T/F : If a diagonalizable matrix A satisfies det(a λi) = λ 3 λ 2 + λ + 1, then A 2 = I. ( ) ( ) (g) T/F 1 1 : If and are both eigenvectors of matrices A and B, then 2 1 AB = BA. (h) T/F : The differential equation y sin(x)y = cos(2x) has a unique solution with initial conditions y(0) = 2, y (0) = 1. (i) T/F : If y p is a particular solution to a second order nonhomogenous linear DE and y p (0) = y p(0) = 0, then y p (x) = 0 for all x. (j) T/F : If y 1 and y 2 are solutions of a nonhomogenous linear DE, then 2y 1 y 2 is also a solution.
7 Q6: Find the general solution to y (3) + 3y 4y = 0. Auxilliary equation: 0 = λ 3 + 3λ 2 4 = (λ 1)(λ 2 + 4λ + 4) = (λ 1)(λ + 2) 2, so the roots are λ = 1 (mult 1) and λ = 2 (mult 2). Therefore the general solution is y = µ 1 e x + µ 2 e 2x + µ 3 xe 2x, µ 1, µ 2, µ 3 R.
8 Q7: Find a particular solution to y 3y + 2y = 2xe 2x. Guess: y p = (A + Bx)e 2x. However, 2 is a root (of multiplicity 1) of the auxilliary equation λ 2 3λ + 2 = 0, so replace y p by y p = x(a + Bx)e 2x = (Ax + Bx 2 )e 2x. y p = (A + 2Bx)e 2x + (2Ax + 2Bx 2 )e 2x = (A + (2A + 2B)x + 2Bx 2 )e 2x, y p = (2A+2B+4Bx)e 2x +(2A+(4A+4B)x+4Bx 2 )e 2x = (4A+2B+(4A+8B)x+4Bx 2 )e 2x, so 2xe 2x = y p 3y p + 2y p = (A + 2B + 2Bx)e 2x. Therefore A + 2B = 0, 2B = 2, which implies B = 1, A = 2, and a particular solution is y p = ( 2x + x 2 )e 2x.
9 Q8: Solve the initial value problem on (0, ) x 2 y 3xy + 13y = 0, y(1) = 0, y (1) = 6. This is a Cauchy-Euler problem. Auxilliary equation: 0 = λ(λ 1) 3λ + 13 = λ 2 4λ + 13 = (λ 2) 2 + 9, so λ = 2 ± 3i. Therefore the general solution is Since y = µ 1 x 2 cos(3 ln x) + µ 2 x 2 sin(3 ln x) = x 2 [µ 1 cos(3 ln x) + µ 2 sin(3 ln x)]. y = 2x[µ 1 cos(3 ln x) + µ 2 sin(3 ln x)] + x 2 [ 3µ 1 sin(3 ln x)x 1 + 3µ 2 cos(3 ln x)x 1 ] = x[(2µ 1 + 3µ 2 ) cos(3 ln x) + ( 3µ 1 + 2µ 2 ) sin(3 ln x)], the initial conditions imply µ 1 = 0, 2µ 1 + 3µ 2 = 6. Thus the solution is y = 2x 2 sin(3 ln x).
10 Q9: An experiment was carried out to ascertain the damping constant of high-fructose corn syrup. When a 2kg mass was attached to a spring, and both were submerged in a vat of high-fructose corn syrup, the resulting motion was found to be critically damped. When the mass was released with an upwards velocity of 5m.s 1 from a point 2m below the equilibrium position, it passed through the equilibrium position after 1 second. What is the damping constant of high-fructose corn syrup? [Hint: λ = ρ 2m, where ρ is the damping constant.] Since the motion is criticallly damped, the equation of motion is of the form x(t) = (µ 1 + tµ 2 )e λt. Since x (t) = µ 2 e λt λ(µ 1 + tµ 2 )e λt = (µ 2 λµ 1 tλµ 2 )e λt, the initial conditions imply that µ 1 = 2 and µ 2 λµ 1 = 5. After 1 second the mass was at the equilibrium position, so µ 1 + µ 2 = 0. Therefore µ 2 = µ 1 = 2, so 2 2λ = 5, which implies λ = 3/2. Therefore ρ = 2mλ = 2(2)(3/2) = 6 (in N.s.m 1 ).
11 Q10: Find the general solution to the linear system This is X = AX, where x 1 = 2x 2 x 3, x 2 = x 2, x 3 = 2x 1 + 4x 2 + 3x A = Eigenvalues of A: λ = 0 1 λ λ = (1 λ) λ λ = (1 λ)[λ2 3λ+2] = (λ 1) 2 (λ 2). So eigenvalues are 1 (mult. 2) and 2 (mult. 1). Eigenspace for λ = 1: (A I 0) = so there are two free variables, k 2 = s and k 3 = t, so k 1 = 2k 2 k 3 = 2s t, hence 2s t 2 1 K = s = s 1 + t 0. t Thus 1 and 0 are linearly independent eigenvectors corresponding to 0 1 λ = 1. Eigenspace for λ = 2: (A 2I 0) = / so there is one free variable, k 3 = s, and k 1 = 1/2k 3 = 1/2s, k 2 = 0, hence 1/2s 1/2 K = 0 s = s 0 1.,,
12 Extra page 1 Thus 0 is an eigenvector corresponding to λ = 2. 2 Therefore the general solution is x x 2 = µ 1 1 e t + µ 2 0 e t + µ 3 0 e 2t, µ 1, µ 2, µ 3 R. x
Midterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10
Midterm 1 NAME: RULES: You will be given the entire period (1PM-3:10PM) to complete the test. You can use one 3x5 notecard for formulas. There are no calculators nor those fancy cellular phones nor groupwork
More informationJune 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations
June 20 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations The topics covered in this exam can be found in An introduction to differential equations
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationMA 262 Spring 1993 FINAL EXAM INSTRUCTIONS. 1. You must use a #2 pencil on the mark sense sheet (answer sheet).
MA 6 Spring 993 FINAL EXAM INSTRUCTIONS NAME. You must use a # pencil on the mark sense sheet (answer sheet).. On the mark sense sheet, fill in the instructor s name and the course number. 3. Fill in your
More information1. In this problem, if the statement is always true, circle T; otherwise, circle F.
Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation
More informationFinal. for Math 308, Winter This exam contains 7 questions for a total of 100 points in 15 pages.
Final for Math 308, Winter 208 NAME (last - first): Do not open this exam until you are told to begin. You will have 0 minutes for the exam. This exam contains 7 questions for a total of 00 points in 5
More informationMATH 223 FINAL EXAM APRIL, 2005
MATH 223 FINAL EXAM APRIL, 2005 Instructions: (a) There are 10 problems in this exam. Each problem is worth five points, divided equally among parts. (b) Full credit is given to complete work only. Simply
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationMATH 1553 SAMPLE FINAL EXAM, SPRING 2018
MATH 1553 SAMPLE FINAL EXAM, SPRING 2018 Name Circle the name of your instructor below: Fathi Jankowski Kordek Strenner Yan Please read all instructions carefully before beginning Each problem is worth
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationName: Final Exam MATH 3320
Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following
More informationMATH 1553, C. JANKOWSKI MIDTERM 3
MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationMath 21b. Review for Final Exam
Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a
More informationMA 262, Fall 2017, Final Version 01(Green)
INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT 0A: Mathematical Methods II Instructor: Hadi Salmasian Final Exam Solutions April 08 Surname First Name Student # Seat # Instructions:
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION A)
MATH 1553 PRACTICE MIDTERM 3 (VERSION A) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationMath 323 Exam 2 Sample Problems Solution Guide October 31, 2013
Math Exam Sample Problems Solution Guide October, Note that the following provides a guide to the solutions on the sample problems, but in some cases the complete solution would require more work or justification
More informationMath 250B Final Exam Review Session Spring 2015 SOLUTIONS
Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT B: Mathematical Methods II Instructor: Hadi Salmasian Final Exam Solutions April 7 Surname First Name Student # Seat # Instructions: (a)
More informationLinear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions
Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with
More informationNo books, notes, any calculator, or electronic devices are allowed on this exam. Show all of your steps in each answer to receive a full credit.
MTH 309-001 Fall 2016 Exam 1 10/05/16 Name (Print): PID: READ CAREFULLY THE FOLLOWING INSTRUCTION Do not open your exam until told to do so. This exam contains 7 pages (including this cover page) and 7
More informationMath 20F Practice Final Solutions. Jor-el Briones
Math 2F Practice Final Solutions Jor-el Briones Jor-el Briones / Math 2F Practice Problems for Final Page 2 of 6 NOTE: For the solutions to these problems, I skip all the row reduction calculations. Please
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
More informationAPPM 2360: Midterm exam 3 April 19, 2017
APPM 36: Midterm exam 3 April 19, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your lecture section number and (4) a grading table. Text books, class notes, cell
More informationDON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end.
Math 307, Midterm 2 Winter 2013 Name: Instructions. DON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end. There
More informationMATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS
MATH Q MIDTERM EXAM I PRACTICE PROBLEMS Date and place: Thursday, November, 8, in-class exam Section : : :5pm at MONT Section : 9: :5pm at MONT 5 Material: Sections,, 7 Lecture 9 8, Quiz, Worksheet 9 8,
More informationMath 250B Midterm III Information Fall 2018 SOLUTIONS TO PRACTICE PROBLEMS
Math 25B Midterm III Information Fall 28 SOLUTIONS TO PRACTICE PROBLEMS Problem Determine whether the following matrix is diagonalizable or not If it is, find an invertible matrix S and a diagonal matrix
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationMATH 1553-C MIDTERM EXAMINATION 3
MATH 553-C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem
More informationMath 308 Practice Final Exam Page and vector y =
Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationJordan Canonical Form Homework Solutions
Jordan Canonical Form Homework Solutions For each of the following, put the matrix in Jordan canonical form and find the matrix S such that S AS = J. [ ]. A = A λi = λ λ = ( λ) = λ λ = λ =, Since we have
More informationPractice Final Exam. Solutions.
MATH Applied Linear Algebra December 6, 8 Practice Final Exam Solutions Find the standard matrix f the linear transfmation T : R R such that T, T, T Solution: Easy to see that the transfmation T can be
More informationTHE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELEC- TRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION.
MATH FINAL EXAM DECEMBER 8, 7 FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number pencil on your answer
More informationNo books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
More informationMath 2114 Common Final Exam May 13, 2015 Form A
Math 4 Common Final Exam May 3, 5 Form A Instructions: Using a # pencil only, write your name and your instructor s name in the blanks provided. Write your student ID number and your CRN in the blanks
More informationMATH 220 FINAL EXAMINATION December 13, Name ID # Section #
MATH 22 FINAL EXAMINATION December 3, 2 Name ID # Section # There are??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is
More informationProblem 1: Solving a linear equation
Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Hadi Salmasian Final Exam April 2016 Surname First Name Seat # Instructions: (a) You have 3
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More information80 min. 65 points in total. The raw score will be normalized according to the course policy to count into the final score.
This is a closed book, closed notes exam You need to justify every one of your answers unless you are asked not to do so Completely correct answers given without justification will receive little credit
More informationMath 21b Final Exam Thursday, May 15, 2003 Solutions
Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationSummer Session Practice Final Exam
Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.
More informationCheat Sheet for MATH461
Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 EXAM I SPRING 2016 FEBRUARY 25, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationSolutions to Exam I MATH 304, section 6
Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationEK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016
EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationIn Class Peer Review Assignment 2
Name: Due Date: Tues. Dec. 5th In Class Peer Review Assignment 2 D.M. 1 : 7 (7pts) Short Answer 8 : 14 (32pts) T/F and Multiple Choice 15 : 30 (15pts) Total out of (54pts) Directions: Put only your answers
More informationTest 3, Linear Algebra
Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all
More informationMATH 251 Examination II July 28, Name: Student Number: Section:
MATH 251 Examination II July 28, 2008 Name: Student Number: Section: This exam has 9 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown.
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationMath 353, Practice Midterm 1
Math 353, Practice Midterm Name: This exam consists of 8 pages including this front page Ground Rules No calculator is allowed 2 Show your work for every problem unless otherwise stated Score 2 2 3 5 4
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationMath Final December 2006 C. Robinson
Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More informationQ1 Q2 Q3 Q4 Tot Letr Xtra
Mathematics 54.1 Final Exam, 12 May 2011 180 minutes, 90 points NAME: ID: GSI: INSTRUCTIONS: You must justify your answers, except when told otherwise. All the work for a question should be on the respective
More informationMath 205B Exam 02 page 1 03/19/2010 Name /7 4/7 1/ /7 1/7 5/ /7 1/7 2/
Math 205B Exam 02 page 1 03/19/2010 Name 3 8 14 1 1. Let A = 1 1 1 3 2 0 4 1 ; then [ A I 4 ] is row-equivalent to 1 2 0 2 Let R = rref(a). 1A. Find a basis for Col(A). 1 0 2 0 0 2/7 4/7 1/7 0 1 1 0 0
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For
More informationCheck that your exam contains 20 multiple-choice questions, numbered sequentially.
MATH 22 MAKEUP EXAMINATION Fall 26 VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More informationThis is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you.
Math 54 Fall 2017 Practice Exam 2 Exam date: 10/31/17 Time Limit: 80 Minutes Name: Student ID: GSI or Section: This exam contains 7 pages (including this cover page) and 7 problems. Problems are printed
More informationMATH 1553 PRACTICE FINAL EXAMINATION
MATH 553 PRACTICE FINAL EXAMINATION Name Section 2 3 4 5 6 7 8 9 0 Total Please read all instructions carefully before beginning. The final exam is cumulative, covering all sections and topics on the master
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS Ma322 - Final Exam Spring 2011 May 3,4, 2011 DO NOT TURN THIS PAGE UNTIL YOU ARE INSTRUCTED TO DO SO. Be sure to show all work and justify your answers. There are 8 problems and
More informationThis is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you.
Math 54 Fall 2017 Practice Final Exam Exam date: 12/14/17 Time Limit: 170 Minutes Name: Student ID: GSI or Section: This exam contains 9 pages (including this cover page) and 10 problems. Problems are
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationMath 2030, Matrix Theory and Linear Algebra I, Fall 2011 Final Exam, December 13, 2011 FIRST NAME: LAST NAME: STUDENT ID:
Math 2030, Matrix Theory and Linear Algebra I, Fall 20 Final Exam, December 3, 20 FIRST NAME: LAST NAME: STUDENT ID: SIGNATURE: Part I: True or false questions Decide whether each statement is true or
More informationAPPM 2360 Spring 2012 Exam 2 March 14,
APPM 6 Spring Exam March 4, ON THE FRONT OF YOUR BLUEBOOK write: () your name, () your student ID number, () lecture section (4) your instructor s name, and (5) a grading table. You must work all of the
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More information2018 Fall 2210Q Section 013 Midterm Exam I Solution
8 Fall Q Section 3 Midterm Exam I Solution True or False questions ( points = points) () An example of a linear combination of vectors v, v is the vector v. True. We can write v as v + v. () If two matrices
More informationCheck that your exam contains 30 multiple-choice questions, numbered sequentially.
MATH EXAM SPRING VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these items may result
More informationAPPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014
APPM 2360 Section Exam 3 Wednesday November 9, 7:00pm 8:30pm, 204 ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) lecture section, (4) your instructor s name, and (5)
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationMA 527 first midterm review problems Hopefully final version as of October 2nd
MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes
More informationMath 22 Fall 2018 Midterm 2
Math 22 Fall 218 Midterm 2 October 23, 218 NAME: SECTION (check one box): Section 1 (S. Allen 12:5) Section 2 (A. Babei 2:1) Instructions: 1. Write your name legibly on this page, and indicate your section
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationMath Practice Problems for Test 1
Math 290 - Practice Problems for Test UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT. 5 3 4. Show that w is in the span of v 5 and v 2 2 by writing w as a linear 6 6 6 combination of v and v 2. 2. Find
More informationPractice Final Exam Solutions for Calculus II, Math 1502, December 5, 2013
Practice Final Exam Solutions for Calculus II, Math 5, December 5, 3 Name: Section: Name of TA: This test is to be taken without calculators and notes of any sorts. The allowed time is hours and 5 minutes.
More informationNAME: MA Sample Final Exam. Record all your answers on the answer sheet provided. The answer sheet is the only thing that will be graded.
NAME: MA 300 Sample Final Exam PUID: INSTRUCTIONS There are 5 problems on 4 pages. Record all your answers on the answer sheet provided. The answer sheet is the only thing that will be graded. No books
More informationMath 308 Practice Test for Final Exam Winter 2015
Math 38 Practice Test for Final Exam Winter 25 No books are allowed during the exam. But you are allowed one sheet ( x 8) of handwritten notes (back and front). You may use a calculator. For TRUE/FALSE
More informationMath 51 Midterm 1 July 6, 2016
Math 51 Midterm 1 July 6, 2016 Name: SUID#: Circle your section: Section 01 Section 02 (1:30-2:50PM) (3:00-4:20PM) Complete the following problems. In order to receive full credit, please show all of your
More informationSSEA Math 51 Track Final Exam August 30, Problem Total Points Score
Name: This is the final exam for the Math 5 track at SSEA. Answer as many problems as possible to the best of your ability; do not worry if you are not able to answer all of the problems. Partial credit
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT 30B: Mathematical Methods II Instructor: Alistair Savage Second Midterm Test Solutions White Version 3 March 0 Surname First Name Student
More informationMATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work
More information