dx. (Correct Answer: D) du u 2 C: 1 2 C: 2 3

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1 Mathematics 00 Solutions for Final Eam. If f = + 2 +, find f. Hint: logarithmic differentiation Correct Answer: B A: 22 ln2 + B: 42 ln2 + C: 24 ln2 + D: 44 ln2 + E: 42 ln If f = 2 e +, find an antiderivative F of f such that F0 = 4.Correct Answer: A. Find 4. Find A: e B: e C: 2 e D: e E: e + d. Correct Answer: D A: 0 ln2 + 8 B: 0 ln2 8 C: 0 ln2 8 D: 0 ln E: 0 ln2 e ln d. Correct Answer: E A: e B: 4 2 e C: e D: 4 4 e E: 4 2 e 5. The substitution u = tan2t transforms the integral π 6 π 8 sec 2 2t tan 2 dt to Correct Answer: C 2t A: u 2 B: / 2 u 2 C: 2 u 2 D: 2 u E: 2 π 6 π 8 u 2 6. Find the area between the curves y = 2 and y = 2. Correct Answer: A A: 8 B: 4 C: 2 D: 6 7. Which of the following integrals represents the volume of the solid formed by by revolving the region bounded by the two curves y = 2 2 and y = 2 about the -ais? Correct Answer: C A: D: π [ ]d B: π d E: π [ ]d C: π [ ]dy 8. Let f, y = 2y ln 2 + y. Find f y,. Correct Answer: D E: 4 [ ]d A: 0 B: 4 C: 2 D: E: 2 For questions 9 and 0 use the following : We want to find the minimum value of f, y, z = 2 + y 2 + z 2 subject to the constraint g, y, z = 2 + y + z 4 = 0, using Lagrange s method. 9. What is the system of equations that we must solve?correct Answer: A

2 2 A: D: 2 + 2λ = 0 2y + λ = 0 2z + λ = y + z 4 = y 2 + z 2 = y + z 4 = 0 B: E: 2 + 2λ = 0 + 2λy = 0 + 2λz = y 2 + z 2 = λ = 0 2y + λ = 0 2z + λ = 0 C: 2 + 2λ = 0 2y + λ = 0 2z + λ = y 2 + z 2 = 0 0. What is the minimum value of f, y, z = 2 +y 2 +z 2 subject to the constraint g, y, z = 2 + y + z 4 = 0?Correct Answer: C A: 4 B: 0 C: 4 D: 24 E: 48. Let f = sintan 2. Find f.correct Answer: E A: costan 2 B: 2 costan 2 C: 2 costan 2 D: costan 2 sec 2 2 E: 2 costan 2 sec Solve for y the differential equation dy d =, if y0 = 0.Correct Answer: B ey A: y = ln B: y = ln2 2 + C: y = 2 + D: y = ln 2 E: y = ln Find the mean µ and variance σ 2 for the random variable X with probability density function f = 8 2 Correct Answer: B on [0,2]. A: µ = 2, σ2 = 20 B: µ = 2, σ2 = 20 C: µ = 4, σ2 = 20 D: µ = 2, σ2 = 20 E: µ = 2, σ2 = The daily number of pieces of mail handled by a certain post office is normally distributed with µ = 2, 500 and σ = 500. Find the probability of needing to handle more than,200 pieces on a randomly selected day.correct Answer: B A: B: C: D: E: Find the value of k so that f = k on [0, 4] and f = 0 elsewhere defines a probability density function. Correct Answer: C A: 6 B: 8 C: 6 D: 4 E: 2 6. Which of the following equations are linear in, y and z i π sin + y + z = 2 ii sin π + y + z = 2 iii = sin 2 yz Correct Answer: D A: i and ii B: ii and iii C: i and iii D: ii only E: iii only 7. Solve for, y, z the linear system: Correct Answer: C 2 y + z = + 2z = 0 y z =

3 A: s, 5s D:, s 2s, s, s B: s, 5s, 2s E: 2s, 5s 5 8. Find the 2,4-entry of the row-reced echelon form of Answer: D, s C: 2s, 5s, s A: 0 B: C: 2 D: 2 E: 2 For questions 9, 20, 2 let A =, B = Which one of the following matri procts is not defined? Correct Answer: C A: A 4 B: AB C: BA 2 D: B T A T E: A 2 B 20. Find the,-entry of A.Correct Answer: E A: 6 B: C: 2 D: E: 6 2. Find the 2,-entry of B T A if it eists.correct Answer: A.Correct A: 4 B: 0 C: D: 4 E: Let A = 0 2. It is given that A = 0. Then the value of in the solution to the linear system is Correct Answer: A + y + z = 0 2y + z = + z = 2 A: B: 2 C: 0 D: 2 E: 5 [ ] If A =, and B is the row-reced echelon form of A, the first row of B is:correct Answer: D A:, 0, 0, 0 B:,, 0, 0 C:,, /2, 0 D:, 0, /2, 0 E:, /2, 0, If ranka=4 and rank[a b] = 5 then the linear system A = b has i no solution. ii eactly one solution. iii eactly four or five solutions.

4 4 iv eactly parameter in the solution. Correct Answer: A A: i B: ii C: iii D: iv E: None of A, B,C, D [ ] c 25. Find the value of c for A = has no inverse.correct Answer: C c A: only B: only C: and D: 9 E: 9 For 26, 27 and 28 use the following: Suppose that the augmented matri of a linear system 0 0 is given by a 2 + a a 26. For what values of a is there no solution? Correct Answer: B A: 0 B: C: D: 0 and E: No value 27. For what values of a are there infinitely many solutions?correct Answer: A A: 0 B: C: D: 0 and E: No value 28. For what values of a is there eactly one solution? Correct Answer: D A: 0 B: all c C: all c 0 D: all c 0 and c E: No value Find the rank of the matri 0.Correct Answer: C 2 2 A: B: 2 C: D: 4 E: Compute det Correct Answer: D A: 0 B: 7 C: D: 666 E: Compute det. Correct Answer: B A: 5 B: 5 C: 0 D: 57 E: Compute y det 2 y y Correct Answer: E

5 5 A: y y B: 2 y 2 C: 2y D: 2yy E: 2y y Use the following for questions, 4 and 5. Let a b c det d e f = 7. g h i. Find 2a b + 2c det c 2d e + 2f f 2g h + 2i i. Correct Answer: D A: 02 B: 7 C: 7 D: 02 E: 0 4. Find a b c det g h i d e f 5. Find Correct Answer: B A: 02 B: 7 C: 7 D: 02 E: Correct Answer: A 2a d g det 2b e h 2c f i A: 02 B: 7 C: 7 D: 02 E: 0 6. Let 2 A = Let p be the 2,-minor of A and q be the,2-cofactor of A. Then, Correct Answer: B 7 A: p = 6, q = B: p = 6, q = C: p = 6, q = D: p = 6, q = E: p = 6, q = 7. Let A and B be 4 4 matrices with det A = and det B = 5. Consider the following statements: i detab = detba. ii detba = 2 iii deta T B 2 = 225 iv det A =. Then Correct Answer: D

6 6 8. Let A = A: only ii and iii are true B: only i and iii are true C: only i, ii and iii are true D: only i and iv are true E: all of them are true It is given that det A = 6. Find the 2,-entry of A.Correct Answer: B A: 20 B: 0 C: 5 D: 0 E: Let 0. Find the inverse of the 2 2 matri [ + + ].. Correct Answer: A 40. It is given that and det det = 0 = 6. Find the value of y in the system of linear equations: 2 + y z = + 4y + 2z = 2 y z = A: [ B: [ [ + C: + ] + + D: [ + ] 4 + E: none of these ] ]

7 7 Correct Answer: B A: 5 B: 5 C: 5 D: 5 E: 6 4. Let u =,, 0, v =,, and w =,,. Let a, b be such that w = au + bv. Which one of the following statements is true? Correct Answer: C A: There is no such a and b. B: a = 2, b =. C: a = 2. D: b =. E: There are infinitely many such a and b. 42. Which of the following vectors is not a linear combination of, 0, and 2, 0,?Correct Answer: A A:, 2, B:, 0, 0 C: 0, 0, D:, 0, 4 E:, 0, 4. Which one of the following sets of vectors spans R? Correct Answer: C 44. Consider the following sets of vectors in R : A: {0, 0, 0} B: {,, 2,, 2, } C: {0,, 0, 0, 0,,, 0, 0,,, 0} D: {,, 2,, 2,, 2, 2, 4} E: {0,, 0, 0, 2, 0, 0,, 0} S = {,, } T = {, 0, 0, 0, 0, 0} U = {, 0, 0, 0,, 0} V = {0, 0,, 0, 0, }. W = {, 0,,,,, 0,,, 0, 0,} Among the sets S, T, U, V, the ones that are linearly independent are Correct Answer: D 45. Consider the subsets S, T, U, V of R given by A: All of them. B: S, U and W only. C: S, T and U only. D: S and U only. E: S, U and V only. S = {a, b, c : a, b, c are integers } T = {a, b, c : ab = 0} U = {a, b, c : b = a + c}

8 8 Then Correct Answer: E V = {a, b, c : a = 0}. A: only S, T, U and V are subspaces of R B: only T, U, and V are subspaces of R C: only S, U, V are subspaces of R D: only S, T, U are subspaces of R E: only U, V are subspaces of R 46. Let S be the subspace of R 4 consisting of all vectors of the form + 2y, 2y,, 2y, where and y are scalars. Which of the following statements is false? Correct Answer: E A: The vector 0, 0, 0, 0 is in S. B: The vector 2, 0,, is in S. C: {,,, 0,,, 0, } is a basis for S D: {2, 0,,,,, 0, } is a basis for S E: The dimension of S is Let A be a 4 4 matri of rank 2. Then the solution of the system A = 0 Correct Answer: C A: is a subspace of R 2 B: is a -dimensional subspace of R 4 C: is a 2-dimensional subspace of R 4 D: is not a subspace. E: is a 2 dimensional subspace of R It is known that the matri has the row-reced echelon form Then a basis for the subspace of R4 generated by {, 0,,, 2, 0, 2, 2,,,,,,,,,,,, } is given by Correct Answer: E A: {, 0,,, 2, 0, 2, 2,,,,,,,, } B: {, 0,,,,,,,,,, } C: { 2, 0, 2, 2,,,,,,,, } D: {, 0,,, 2, 0, 2, 2,,,, } E: {, 0,,,,,,,,,, }

9 9 49. Find the dimension of the subspace of R 4 generated by the set Correct Answer: D {, 0, 0, 0,, 2, 0, 0,, 2,, 0,,2, 0,0}. A: 0 B: C: 2 D: E: A certain homogeneous system of linear equations A = 0 has augmented matri Which one of the following statements is true? Correct Answer: D. A: The set of all solutions is a -dimensional subspace of R 6. B: The set of all solutions is a -dimensional subspace of R 5. C: The set of all solutions is a -dimensional subspace of R 5. D: The set of all solutions is a 2-dimensional subspace of R 5. E: There is a unique solution to this system of equations.