MTH603-Numerical Analysis Quiz MCQS #Objective #Questions #MidTerm

1. If n x n matrices A and B are similar, then they have the ___ eigenvalues (with the same multiplicities)

- same ✔
- different

2. If the product of two matrices is an identity matrices that is AB = I, then which of the following is true?

- A is transpose of B
- A is inverse of B ✔
- A is singular
- B is singular

3. Iterative algorithms can be more rapid than direct methods.

- False
- True ✔

4. While using Relaxation method, which of the following is the largest Residual for 1st iteration on the system; 2x + 3y = 1, 3x + 2y = -4?

- -4 ✔
- 3
- 2
- 1

5. Which of the following systems of linear equations has a strictly diagonally dominant coefficient matrix?

- -2x+7x+2x=5, 6x-2x+3x=1, x+x-5x=-13
- -2x+7x+2x=5, 6x-2x+3x=1, x+x-5x=-13
- x+x-5x=-13, 6x-2x+3x=1, -2x+7x+2x=5
- 6x-2x+3x=1, -2x+7x+2x=5, x+x-5x=-13

6. In the context of Jacobi's method for finding Eigen values and Eigen vectors of a real symmetric matrix of order 2*2, if |-5| be its largest off-diagonal then which of the following will be its corresponding off-diagonal values of Orthogonal Matrix?

- Cos(theta), -Cos(theta)
- Sin(theta), Cos(theta)
- Sin(theta), -Sin(theta)
- -Sin(theta), Cos(theta)

7. While using power method, from the resultant normalize vector

8. Every non-zero vector x is an eigenvector of the identity matrix with Eigen value___

- one
- two
- three
- four

9. Which of the following systems of linear equations has a strictly diagonally coefficient matrix?

- -x+12x+5x=8, 9x+5x-3x=12, 2x-4x+7x=-15
- 9x+5x-3x=12, -x+12x+5x=8, 2x-4x+7x=-15
- 2x-4x+7x=-15, -x+12x+5x=8, 9x+5x-3x=12
- 9x+5x-3x=12, 2x-4x+7x=-15, -x+12x+5x=8

10. Let[A] be a 3 x 3 real symmetric matrix with |a| be numerically the largest off-diagonal element of A, then we can construct orthogonal matrix S1 by Jacobi's method as

11. If the pivot element happens to be zero, then the i-th column elements are searched for the numerically ___ element

- Smallest
- Largest ✔

12. Exact solution of 2/3 is not exists

- True ✔
- False

13. A and its transpose matrix have ___ eigenvalues

- same ✔
- different

14. If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities)

- True ✔
- False

15. Power method is applicable if the eigen values are real and distinct

- True ✔
- False

16. By using determinants, we can easily check that the solution of the given system of linear equation ___ and it is ___

- exists, unique ✔
- exists, consistent
- trivial, unique
- nontrivial, inconsistent

17. Power method is applicable if the eigen vectors corresponding to eigen values are linearly ___

- independent ✔
- dependent

18. When the condition of diagonal dominance becomes true in Jacobi's Method. Then its means that the method is ___

- Stable
- Unstable
- Convergent ✔
- Divergent

19. While using Jacobi method for the matrix

A = [ | 1 | 1/4 | 1/3 | ] |

1/4 | 1/3 | 1/2 | ||

1/3 | 1/2 | 1/5 |

the value of 'theta Θ' can be found as

- tan 2Θ = 2a
_{13}/a_{11}-a_{33}✔

20. While using Jacobi method for the matrix

and 'theta Θ=0.4480' the orthogonal matrix S1 will be given by

A = [ | 1 | 1/4 | 1/2 | ] |

1/4 | 1/3 | 1/4 | ||

1/2 | 1/4 | 1/5 |

S _{1}= [cos 0.4480 0 -sin 0.4480 ] ✔ 0 1 0 sin 0.4480 0 cos 0.4480

21. Full pivoting, in fact, is more ___ than the partial pivoting

- Easiest
- Complicated ✔

22. While using the Gauss-Seidel Method for finding the solution of the system of equation, the following system

x + 2y + 2z = 3

x + 3y + 3z = 2

x + y + 5z = 2

- x = 3 - 2y - 2z, y = 2/3 - x/3 - z, z = 2/5 - x/5 - y/5

23. By using determinants, we can easily check that the solution of the given system of linear equation exists and it is unique

- True ✔
- False

24. While using Jacobi method for the matrix

and 'theta Θ= 0.7191' the orthogonal matrix S1 will be given by

A = [ | 1 | 1/4 | 1/3 | ] |

1/4 | 1/3 | 1/2 | ||

1/3 | 1/2 | 1/5 |

S _{1}= [cos 0.7191 0 -sin 0.7191 ] ✔ 0 1 0 sin 0.7191 0 cos 0.7191

25. The linear equation x + y = 1 has ___ solution/solutions

- no solution
- unique ✔
- infinite many
- finite many

26. For a system of linear equations, the corresponding coefficient matrix has the value of determinant; |A|=-3, then which of the following is true?

- The system has unique solution ✔
- The system has finite multiple solutions
- The system has infinite many solutions
- The system has no solution

27. In Gauss-Jacobi's method, the corresponding elements of x

_{i}^{(r+1)}replaces those of x_{i}^{r}as soon as they become available- True
- False ✔

28. An augmented matrix may also be used to find the inverse of a matrix by combining it with the ___ matrix

- Inverse
- Square
- Identity ✔
- None

29. Power method is applicable it the eigen vectors corresponding to eigen values are linearly independent

- True ✔
- False

30. While using the relaxation method for finding the solution of the below given system, which of the following increment will be introduced?

6x

_{1}- 2x_{2}+ 3x_{3}= 1-2x

_{1}+ 7x_{2}+ 2x_{3}= 5x

_{1}+ x_{2}- 5x_{3}= -13- dx
_{3}= R_{3}/a_{33}✔

31. Let |A| be a 3 x 3 real symmetric matrix with |a

_{23}| be the numerically largest off-diagonal element then using Jacobi's method the value of theta can be found by- tan 2 Θ = 2a
_{23}/a_{22}-a_{33 }✔

32. The linear equation; 0x+0y=2 has ___ solution/solutions

- unique
- no solution ✔
- infinite many
- finite many

33. The root of the equation xe

^{x}-5=0 is bounded in the interval- [-2, 1]
- [-1, 1]
- [0, 1] ✔
- [1, 2]

34. Which of the following is a forward difference table for the given values of x and y?

x 0.1 0.5 0.9

y 0.003 0.148 0.370

x y Δy Δ ^{2}y0.1 0.003 0.145 0.077 0.5 0.148 0.222 ✔ 0.9 0.37

35. In ___ method, a system is reduced to an equivalent diagonal form using elementary transformations

- Jacobi's
- Gauss-Seidel
- Relaxation
- Gaussian elimination ✔

36. If the determinant of a matrix A is not equal to zero then the system of equations will have ___

- a unique solution ✔
- many solutions
- infinite many solutions
- None

37. A 3 x 3 identity matrix have three and ___ eigen values

- same ✔
- different

38. Numerical methods for finding the solution of the system of equations are classified as direct and ___ methods

- Indirect
- Iterative ✔
- Jacobi
- None

39. Which of the following is a forward difference table for the given values of x and y?

x 0.1 0.7 1.3

y 0.003 0.248 0.697

x y Δy Δ ^{2}y0.1 0.003 0.245 0.204 0.7 0.248 0.449 ✔ 1.3 0.697

40. While using Relaxation method, which of the following is the Residuals for 1st iteration on the system; 2x + 3y = 1, 3x + 2y = 4?

- (2, 3)
- (3, -2)
- (-2, 3)
- (1, 4) ✔