MTH168 Mathematics II
Mathematics II: most repeated questions
Questions that have come up in more than one paper, matched by what they ask rather than exact wording. Most repeated first.
- 40×Asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075Find LU factorization of A= bmatrix3&2\\ 2&3 bmatrix .Answer coming20
See the wording in each paper
- Find LU factorization of A= bmatrix3&2\\ 2&3 bmatrix .TU Board 2082
- Find the eigenvalues for the given matrix A= bmatrix4&0&1\\ 1& 6& 2\\5&0&0 bmatrix .TU Board 2082
- Let A= bmatrix1&2\\3&0 bmatrix . Find A^3 .TU Board 2082
- Let A= bmatrix3&1\\4&2 bmatrix . Find 5 A 2 I . Is (5A) equal to 5 A ? Justify.TU Board 2082
- Let T be defined by T(x)=Ax , where A= bmatrix1& 5& 7\\ 3&7&5 bmatrix . Find a vector x whose image under T is b , where b= bmatrix 2\\ 2 bmatrix , and determine whether x is unique or not.TU Board 2082
- Define the inverse of a matrix. Find the inverse of matrix A = bmatrix 1 & 0 \\ 3 & 4 bmatrix if it exists.TU Board 2081
- Determine the least squares error in the least square solution of Ax=b, where A = bmatrix 4 & 0 \\ 0 & 2 \\ 1 & 1 bmatrix, x = bmatrix 1 \\ 2 bmatrix, b = bmatrix 2 \\ 0 \\ 1 bmatrixTU Board 2081
- Find LU factorization of A = bmatrix 2 & 5 \\ 6 & 7 bmatrixTU Board 2081
- Let A = bmatrix 1 & 6 \\ 5 & 2 bmatrix Find the eigenvalues and eigenvectors of AA.TU Board 2081
- Let A = bmatrix 2 & 0 \\ 0 & 2 bmatrix and Define T : R^2 R^2 by T(x) = A x . Find the image under T ( u = bmatrix 1 \\ 3 bmatrix, \, v = bmatrix u \\ u bmatrix ) b) Prove that a map T : R^2 R^2…TU Board 2081
- Define linear transformation with an example. Let A = [figure in the original paper] , v = [figure in the original paper] , b= [figure in the original paper] , x = [figure in the original paper] ,…TU Board 2080
- Find LU Factorization. Given the matrix: bmatrix 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 bmatrixTU Board 2080 (new course)
- Find the basis and dimension of Nul A where A = bmatrix 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 bmatrix.TU Board 2080 (new course)
- Find the least square solution of Ax=b where A= bmatrix 1 & 3 & 3\\ 1 & 5 & 1\\ 1 & 7 & 2\\ bmatrix, b= bmatrix 5\\ 3\\ 5\\ bmatrix and compute the associated least square error.TU Board 2080 (new course)
- Find the least square solution of Ax=c where A= [figure in the original paper] , c= [figure in the original paper] and compute the associated least square error.TU Board 2080
- Is bmatrix3\\2\\ bmatrix an eigen vector of bmatrix5& 3\\ 4&9\\ bmatrix ? If so, find eigenvalue.TU Board 2080 (new course)
- Let A = bmatrix 1 & 5 & 7 \\ 3 & 7 & 5 \\ bmatrix , u = bmatrix 1 \\ 2\\ 3\\ bmatrix, b= bmatrix 2\\ 2\\ bmatrix, and define a transformation T : R^3 → R² by T(x) = Ax then a) find T(u). b) Find x ∈…TU Board 2080 (new course)
- Define linear transformation with an example. Let A = bmatrix1 & 3\\ 3 & 5\\ 1 & 7 bmatrix , v = bmatrix2\\ 1 bmatrix , b = bmatrix3\\ 2\\ 4 bmatrix , x = bmatrixx 1\\ x 2 bmatrix and define a…TU Board 2079
- Define null space of a matrix A. If A = bmatrix 1 & 3 & 2\\ 5 & 9 & 1 bmatrix , and v = bmatrix5\\ 3\\ 2 bmatrix Then show that v is null of A.TU Board 2079
- Evaluate the determinant of the matrix bmatrix5 & 7 & 2 & 2\\ 0 & 3 & 0 & 4\\ 5 & 8 & 0 & 3\\ 0 & 5 & 0 & 6 bmatrixTU Board 2079
- Find the eigenvalues of the matrix pmatrix6 & 3 & 8\\ 0 & 2 & 0\\ 1 & 0 & 3 pmatrixTU Board 2079
- Define linear transformation with an example. Let A = bmatrix1 & 3\\ 3 & 5\\ 1 & 7 bmatrix , v = bmatrix2\\ 1 bmatrix , b = bmatrix3\\ 2\\ 4 bmatrix , x = bmatrixx 1\\ x 2 bmatrix and define a…TU Board 2078
- Define null space of a matrix A. Let A = bmatrix 1 & 3 & 2\\ 5 & 9 & 1 bmatrix , and v = bmatrix5\\ 3\\ 2 bmatrix Then show that v is in the null ATU Board 2078
- Find a least square solution of the inconsistent system Ax = b for A = bmatrix 1 & 2\\ 2 & 3\\ 1 & 3 bmatrix , b = bmatrix4\\ 2\\ 1 bmatrixTU Board 2078
- Find the eigen value of bmatrix3 & 6 & 8\\ 0 & 0 & 6\\ 0 & 0 & 2 bmatrixTU Board 2078
- Find the LU factorization of bmatrix2 & 4 & 1 & 5 & 2\\ 4 & 5 & 3 & 8 & 1\\ 2 & 5 & 4 & 1 & 8\\ 6 & 0 & 7 & 3 & 1 bmatrixTU Board 2078
- If A = bmatrix7 & 2\\ 4 & 1 bmatrix . find a formula for A^n, where A = PDP^ 1 P = bmatrix1 & 1\\ 1 & 2 bmatrix and D = bmatrix5 & 0\\ 0 & 3 bmatrixTU Board 2078
- Let A = bmatrix0 & 1\\ 1 & 0 bmatrix and define T:R^2 → R^2 by T(x) = Ax, find the image under T of u = bmatrix1\\ 3 bmatrix and v = bmatrix1\\ 5 bmatrixTU Board 2078
- Define null space . Find the basis for the null space of the matrix A = bmatrix1 & 2 & 3\\ 2 & 3 & 4 bmatrixTU Board 2076
- Find the eigen values of the matrix bmatrix6 & 5\\ 8 & 6 bmatrixTU Board 2076
- Find the least square solution of Ax=b for A = bmatrix 1 & 6\\ 1 & 2\\ 1 & 1\\ 1 & 7 bmatrix and b = bmatrix 1 \\ 2\\ 1\\ 6 bmatrixTU Board 2076
- Find the QR factorization of the matrix bmatrix2 & 1\\ 3 & 1 bmatrixTU Board 2076
- For what value of h will y be in span v 1 , v 2, v 3 if v 1 = bmatrix1\\ 1\\ 2 bmatrix , v 2 = bmatrix5\\ 4\\ 7 bmatrix , v 3 = bmatrix 3\\ 1\\ 0 bmatrix and y = bmatrix 4\\ 3\\ h bmatrixTU Board 2076
- Let us define a linear transformation T:R^2 → R^2 by T(x) = bmatrix0 & 1\\ 1 & 0 bmatrix bmatrixx 1\\ x 2 bmatrix = bmatrix x 2\\ x 1 bmatrix . Find the image under T of u = bmatrix4\\ 1 bmatrix , v…TU Board 2076
- What is the condition of a matrix to have an inverse? Find the inverse of the matrix A = bmatrix 5 & 1 & 2\\ 1 & 0 & 3\\ 4 & 3 & 8 bmatrix If it exists.TU Board 2076
- Find LU factorization of the matrix pmatrix2 & 5\\ 6 & 7 pmatrixTU Board 2075
- Find the dimension of the null space and column space of A = bmatrix 3 & 6 & 1 & 1 & 7\\ 1 & 2 & 2 & 3 & 1\\ 2 & 4 & 5 & 8 & 4 bmatrixTU Board 2075
- Find the eigenvalues of the matrix pmatrix6 & 3 & 8\\ 0 & 2 & 0\\ 1 & 0 & 3 pmatrixTU Board 2075
- Find the least square solution of Ax = b for A = bmatrix1 & 3 & 5\\ 1 & 1 & 0\\ 1 & 1 & 2\\ 1 & 3 & 3 bmatrix and b= pmatrix3\\ 5\\ 7\\ 3 pmatrixTU Board 2075
- What is the condition of a matrix to have an inverse? Find the inverse of the matrix A = bmatrix 5 & 1 & 2\\ 1 & 0 & 3\\ 4 & 3 & 8 bmatrix If it exists.TU Board 2075
- 6×Asked in 2080, 2079, 2078, 2075Define group. Show that the set of integers is a group with respect to addition operation.Answer coming5
See the wording in each paper
- Define group. Show that the set of integers is a group with respect to addition operation.TU Board 2080
- Define ring and show that set of positive integers with respect to addition and multiplication operation is not a ring.TU Board 2080
- Define group. Show that the set of integers is not a group with respect to subtraction operation.TU Board 2079
- Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.TU Board 2079
- Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring.TU Board 2078
- Define group. Show that the set of all integers Z forms group under addition operation.TU Board 2075
- 4×Asked in 2080, 2079, 2078Define system of linear equations. When a system of equations is consistent? Make echelon form to solve: 2a – 3b + 4c = 5 b – 2c = 4 a + 3b – c = 2Answer coming10
See the wording in each paper
- Define system of linear equations. When a system of equations is consistent? Make echelon form to solve: 2a – 3b + 4c = 5 b – 2c = 4 a + 3b – c = 2TU Board 2080
- What is a system of linear equations ? When the system is consistent ? Find the condition on g, h, k that makes the system consistent. x 1 – 4x 2 + 7x 3 = g 3x 2 – 5x 3 = h 2x 1 + 5x 2 – 9x 3 = kTU Board 2080 (new course)
- When a linear system of equation is consistent? Find the values of h and k for which the system: 2x 1 – x 2 = h; 6x 1 + 3x 2 = k is consistent?TU Board 2079
- Define system of linear equations. When a system of equation is consistent? Determine if the system 2x 1 – 3x 2 + 4x 3 = 5, x 2 – 2x 3 = 4, x 1 + 3x 2 – x 3 = 2 is consistent.TU Board 2078
- 3×Asked in 2080, 2079, 2078Let u = (1, 2, 2, 0). Find a unit vector of v in the same direction of u.Answer coming5
See the wording in each paper
- 3×Asked in 2080, 2079, 2078Determine the column of the matrix A are linearly independent where A = [figure in the original paper]Answer coming5
See the wording in each paper
- Determine the column of the matrix A are linearly independent where A = [figure in the original paper]TU Board 2080
- Determine the column of the matrix A are linearly independent, where A = bmatrix 2 & 8 & 1\\ 0 & 0 & 0\\ 0 & 5 & 3 bmatrixTU Board 2079
- Determine the column of the matrix A are linearly independent, where A = bmatrix0 & 1 & 4\\ 1 & 2 & 1\\ 5 & 8 & 0 bmatrixTU Board 2078
- 2×Asked in 2076, 2075Define determinant. Compute the determinant without expanding bmatrix 2 & 8 & 9\\ 1 & 7 & 0\\ 1 & 4 & 2 bmatrixAnswer coming5
See the wording in each paper
- 2×Asked in 2080, 2079Find the equation y = a 0 + a 1 x of the least squares line that best fits the data points (2,1), (5,2), (7,3), (8,3).Answer coming5
See the wording in each paper
- 2×Asked in 2076, 2075Let A = bmatrix2 & 5\\ 3 & 1 bmatrix and B = bmatrix4 & 5\\ 3 & k bmatrix . Determine the value (s) of k if any will make AB = BA.Answer coming5
See the wording in each paper
- 2×Asked in 2081, 2076Let B=b1,b2, C=c1,c2 be bases for a vector space V, and suppose b 1 = 6c 1 – 2c 2, b 2 = 9c 1 – 4c 2 a) Find the change of coordinates matrix from BB to CC. b) Find [x] C . x = 3b 1 + 2b 2Answer coming5
See the wording in each paper
- Let B=b1,b2, C=c1,c2 be bases for a vector space V, and suppose b 1 = 6c 1 – 2c 2, b 2 = 9c 1 – 4c 2 a) Find the change of coordinates matrix from BB to CC. b) Find [x] C . x = 3b 1 + 2b 2TU Board 2081
- Let B = b 1, b 2 and C = (c 1, c 2) be bases for a vector V, and suppose b 1 = c 1 + 4c 2 and b 2 = 5c 1 – 3c 2. Find the change of coordinate matrix for a vector space and find [x] c for x = 5b 1 +…TU Board 2076
- 2×Asked in 2080, 2078Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from u to v.Answer coming5
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- Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from u to v.TU Board 2080
- Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from u to vTU Board 2078
- 2×Asked in 2080, 2076Show that every field is an integral domain.Answer coming5
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- Show that every field is an integral domain.TU Board 2080 (new course)
- Show that the ring (Z 4, + 4, . 4) is an integral domain.TU Board 2076
- 2×Asked in 2079, 2078Verify that 1^k, ( 2)^k, 3^k are linearly independent signals.Answer coming5
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- 2×Asked in 2076, 2075When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, x + y + 5z = 2, y + z = 0Answer coming10
See the wording in each paper
- When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, x + y + 5z = 2, y + z = 0TU Board 2076
- When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve: x + y + z = 4, x + 2y + 2z = 2, 2x + 2y + z = 5.TU Board 2075
- 2×Asked in 2079, 2078When two column vector in R^2 are equal? Give an example. Computer u + 3v, u – 2v, where u = bmatrix1\\ 3\\ 2 bmatrix , v = bmatrix1\\ 1\\ 3 bmatrixAnswer coming5
See the wording in each paper
- When two column vector in R^2 are equal? Give an example. Computer u + 3v, u – 2v, where u = bmatrix1\\ 3\\ 2 bmatrix , v = bmatrix1\\ 1\\ 3 bmatrixTU Board 2079
- When two column vector in R^2 are equal? Give an example. Computer u + 3v, u – 2v, where u = bmatrix1\\ 3\\ 2 bmatrix , v = bmatrix1\\ 1\\ 3 bmatrixTU Board 2078