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

True

False

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

]]>True

False

(latest quiz included and FAQs and Glossary is also included in this file)

Plz must learn the FAQs and Glossary for Mcqs and short question…

It is very helpful By MC110402046

Tayyaba Rehmna MCS 4th semester

File made on 25-july-2013 ]]>

Mth603 - Midterm Solved Mcqs and Quizes by moaaz.pdf

Mth603 - Midterm Solved Mcqs and Quizes by moaaz.pdf ]]>

While solving a system of linear equations, which of the following approach is economical for the computer memory?

Select correct option:

Direct

Analytical

Graphical

Question # 2 of 10 (Total Marks: 1)

The basic idea of relaxation method is to reduce the largest residual to …

Select correct option:

One

Two

**Zero (Page 83)**

None of the given choices

Question # 3 of 10 (Total Marks: 1)

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its

________.

Select correct option:

**main diagonal (Page 104)**

last column

last row

first row

Question # 4 of 10 (Total Marks: 1)

If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix ,

the eigenvalues of A are the diagonal entries of A.

Select correct option:

**TRUE Click here for detail**

FALSE

Question # 5 of 10 (Total Marks: 1)

A 3 x 3 identity matrix have three and different eigen values.

Select correct option:

**TRUE Click here for detail**

FALSE

Question # 6 of 10 (Total Marks: 1)

Which of the following is a reason due to which the LU decomposition of the system of linear equations;

x+y = 1, x+y =2 is not possible?

Select correct option:

Associated coefficient matrix is singular

All values of l’s and u’s can’t be evaluated

Determinant of coefficient matrix is zero

**All are equivalent**

Question # 7 of 10 (Total Marks: 1)

Gauss - Jordan Method is similar to …

Select correct option:

Gauss–Seidel method

Iteration’s method

Relaxation Method

**Gaussian elimination method (Page 95)**

Question # 8 of 10 (Total Marks: 1)

While using Relaxation method, which of the following is the largest Residual for 1st iteration on the

system; 2x+3y = 1, 3x +2y = - 4 ?

Select correct option:

-4

**3**

2

1

Question # 9 of 10 (Total Marks: 1)

Gauss–Seidel method is also known as method of …

Select correct option:

**Successive displacement (Page 263)**

Iterations

False position

None of the given choices

Question # 10 of 10 (Total Marks: 1)

Jacobi’s Method is a/an…

Select correct option:

**Iterative method (Page 69)**

Direct method

QUIZ.NO.1(2)

Question # 1 of 10 (Total Marks: 1)

The characteristics polynomial of a 3x 3 identity matrix is __________, if x is the eigen values of the

given 3 x 3 identity matrix. where symbol ^ shows power.

Select correct option:

(x-1)^3

(x+1)^3

**x^3-1**

x^3+1

Question # 2 of 10 (Total Marks: 1)

The can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue

the dominant eigenvalue of A.

Select correct option:

**TRUE**

FALSE

Question # 3 of 10 (Total Marks: 1)

In … method, a system is reduced to an equivalent diagonal form using elementary

transformations.

Select correct option:

Jacobi’s

Gauss-Seidel

Relaxation

**Gaussian elimination (Page 262)**

Question # 4 of 10 (Total Marks: 1)

The linear equation: 2x+0y-2=0 has -------- solution/solutions.

Select correct option:

**Unique (Page 48)**

no solution

infinite many

finite many

Question # 5 of 10 (Total Marks: 1)

Under elimination methods, we consider, Gaussian elimination and …methods.

Select correct option:

Gauss-Seidel

Jacobi

**Gauss-Jordan elimination (Page 48)**

None of the given choices

Question # 6 of 10 (Total Marks: 1)

Which of the following method is not an iterative method?

Select correct option:

Jacobi’s method

Gauss-Seidel method

Relaxation methods

**Gauss-Jordan elimination method**

Question # 7 of 10 (Total Marks: 1)

An eigenvector V is said to be normalized if the coordinate of largest magnitude is equal

to zero.

Select correct option:

TRUE

**FALSE (Page 97)**

Question # 8 of 10 (Total Marks: 1)

Exact solution of 2/3 is not exists.

Select correct option:

**TRUE**

FALSE

Question # 9 of 10 (Total Marks: 1)

When the condition of diagonal dominance becomes true in Jacobi’s Method.Then its means that the

method is …

Select correct option:

Stable

Unstable

**Convergent (Page 70)**

Divergent

Question # 10 of 10 (Total Marks: 1)

Gauss–Seidel method is similar to …

Select correct option:

Iteration’s method

Regula-Falsi method

Jacobi’s method

**None of the given choices (Page 263)**

QUIZ.NO.1(3)

Question # 1 of 10 (Total Marks: 1)

Sparse matrices arise in computing the numerical solution of …

Select correct option:

Ordinary differential equations

**Partial differential equations (Page 69)**

Linear differential equations

Non-linear differential equations

Question # 2 of 10 (Total Marks: 1)

While solving by Gauss-Seidel method, which of the following is the first Iterative solution for the

system; x-2y =1, x+4y=4 ?

Select correct option:

**(1, 0.75)**

(0,0)

(1,0)

(0,1)

Question # 3 of 10 (Total Marks: 1)

While solving a system of linear equations by Gauss Jordon Method, after all the elementary row

operations if there lefts also zeros on the main diagonal then which of the is true about the system?

Select correct option:

System may have unique solutions

System has no solution

System may have multiple numbers of finite solutions

**System may have infinite many solutions**

Question # 4 of 10 (Total Marks: 1)

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

… methods

Select correct option:

Indirect

**Iterative (Page 48)**

Jacobi

None of the given choices

Question # 5 of 10 (Total Marks: 1)

If the Relaxation method is applied on the system; 2x+3y = 1, 3x +2y = - 4, then largest residual in 1st

iteration will reduce to -------.

Select correct option:

**zero**

4

-1

-1

Question # 6 of 10 (Total Marks: 1)

While using Relaxation method, which of the following is the Residuals for 1st iteration on the system;

2x+3y = 1, 3x +2y =4 ?

Select correct option:

(2,3)

(3,-2)

(-2,3)

**(1,4)**

0= 1-2x-3y

0= 4-3x -2y

Question # 7 of 10 (Total Marks: 1)

If the order of coefficient matrix corresponding to system of linear equations is 3*3 then which of the
following will be the orders of its decomposed matrices; ‘L’ and ‘U’?
Select correct option:
Order of ‘L’ = 3*1, Order of ‘U’ = 1

Order of ‘L’ = 3

**Order of ‘L’ = 3 3, Order of ‘U’ = 33**

Order of ‘L’ = 3

Question # 8 of 10 (Total Marks: 1)

While solving the system; x–2y = 1, x+4y = 4 by Gauss-Seidel method, which of the following ordering

is feasible to have good approximate solution?

Select correct option:

x+4y = 1, x-2y = 4

x+2y = 1, x- 4y =4

x+4y = 4, x–2y = 1

**no need to reordering**

Question # 9 of 10 (Total Marks: 1)

Full pivoting, in fact, is more …than the partial pivoting.

Select correct option:

Easiest

**Complicated (Page 51)**

Question # 10 of 10 (Total Marks: 1)

For the equation

the root of the equation lies in the interval…

► (1, 3)

► (1, 2)

**► (0, 1)**

► (1, 2)

QUIZ.NO.1(4)

Question # 1 of 10 (Total Marks: 1)

…lies in the category of iterative method.

► Bisection Method

► Regula Falsi Method

► Secant Method

**► all of the given choices (Page 8)**

Question # 2 of 10 (Total Marks: 1)

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

multiplicities).

True

**False**

Question # 3 of 10 (Total Marks: 1)

The Jacobi’s method is a method of solving a matrix equation on a matrix that has ____zeros along its

main diagonal.

**No Click here for detail**

At least one

Question # 4 of 10 (Total Marks: 1)

If the root of the given equation lies between a and b, then the first approximation to the root of the equation by bisection method is …

Question # 5 of 10 (Total Marks: 1)

To apply Simpson’s 3/8 rule, the number of intervals in the following must be

► 10

► 11

**► 12**

► 13

Question # 6 of 10 (Total Marks: 1)

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric________ definite

matrices A.

Select correct option:

**Positive Click here for detail**

negative

Question # 7 of 10 (Total Marks: 1)

Differences methods find the ________ solution of the system.

Select correct option:

**numerical**

Analytical

Question # 8 of 10 (Total Marks: 1)

To apply Simpson’s 1/3 rule, the number of intervals in the following must be

**► 2 (Simpson’‘s 1/3 rule must use an even number of elements’)**

► 3

► 5

► 7

Question # 9 of 10 (Total Marks: 1)

Bisection and false position methods are also known as bracketing method and are always

Divergent

**Convergent (Page 26)**

.67

Question # 10 of 10 (Total Marks: 1)

The Inverse of a matrix can only be found if the matrix is

Singular

**None Singular: Every square non-singular matrix will have an inverse.**

Scalar

Diagonal

QUIZ.NO.1(5)

Question # 1 of 10 (Total Marks: 1)

In interpolation is used to represent the δ

Forward difference Δ

**Central difference (Page 117)**

Backward difference

Question # 2 of 10 (Total Marks: 1)

The base of the decimal system is _______

**10**

0

2

8

None of the above.

Question # 3 of 10 (Total Marks: 1)

Bisection method is … method

► Open Method

**► Bracketing Method (page 26)**

Question # 4 of 10 (Total Marks: 1)

A 3 x 3 identity matrix have three and __________eigen values.

**same**

different

Question # 5 of 10 (Total Marks: 1)

Eigenvalues of a symmetric matrix are all _______ .

**Real (page 104)**

complex

zero

positive

Question # 6 of 10 (Total Marks: 1)

The Jacobi iteration converges, if A is strictly diagonally dominant.

**TRUE (Page 69)**

FALSE

Question # 7 of 10 (Total Marks: 1)

Below are all the finite difference methods EXCEPT _________.

jacobi’s method

**newton’s backward difference method**

Stirlling formula

Forward difference method

Question # 8 of 10 (Total Marks: 1)

Two matrices with the same characteristic polynomial need not be similar.

**TRUE**

FALSE

Question # 9 of 10 (Total Marks: 1) No.69

The determinant of a diagonal matrix is the product of the diagonal elements.

**True**

False

Question # 10 of 10 (Total Marks: 1)

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite matrices A.

**True**

False

QUIZ.NO.1(6)

Question # 1 of 10 (Total Marks: 1)

The determinant of a _______ matrix is the product of the diagonal elements.

**Diagonal Page No.70**

Upper triangular

Lower triangular

Scalar

Question # 2 of 10 (Total Marks: 1)

For differences methods we require the set of values.

**True**

False

Question # 3 of 10 (Total Marks: 1)

If x is an eigen value corresponding to eigen value of V of a matrix A. If a is any constant, then x – a is an eigen value corresponding to eigen vector V is an of the matrix A - a I.

**True**

False

Question # 4 of 10 (Total Marks: 1)

Central difference method seems to be giving a better approximation, however it requires more computations.

True

**False**

Question # 5 of 10 (Total Marks: 1)

Iterative algorithms can be more rapid than direct methods.

**True**

False

Question # 6 of 10 (Total Marks: 1)

Central Difference method is the finite difference method.

**True**

False

Question # 1 of 10 (Total Marks: 1)

Back substitution procedure is used in …

Select correct option:

**Gaussian Elimination Method**

Jacobi’s method

Gauss-Seidel method

None of the given choices

Question # 7 of 10 (Total Marks: 1)

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal.

**True**

False

Question # 8 of 10 (Total Marks: 1)

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

True (Page102)

False

Question # 9 of 10 (Total Marks: 1)

Power method is applicable if the eigen values are ______________.

**realanddistinct (Page102)**

real and equal

positive and distinct

negative and distinct

Question # 10 of 10 (Total Marks: 1)

Simpson’s rule is a numerical method that approximates the value of a definite integral by using polynomials.

**Quadratic (Page174)**

Linear

Cubic

Quartic

QUIZ.NO.1(7)

Question # 1 of 10 (Total Marks: 1)

.In Simpson’s Rule, we use parabolas to approximating each part of the curve. This proves to be very efficient as compared to Trapezoidal rule.

**True Click here for detail**

False

Question # 2 of 10 (Total Marks: 1)

The predictor-corrector method an implicit method. (multi-step methods)

**True (Page 212)**

False

Question # 3 of 10 (Total Marks: 1)

Generally, Adams methods are superior if output at many points is needed.

**True**

False

Question # 4 of 10 (Total Marks: 1)

The Trapezoidal rule is a numerical method that approximates the value of a.______________.

Indefinite integral

**Definiteintegral (Page176)**

Improper integral

Function

Question # 5 of 10 (Total Marks: 1)

The need of numerical integration arises for evaluating the definite integral of a function that has no

explicit ____________ or whose anti derivative is not easy to obtain.

**Antiderivative**

Derivatives.

Question # 6 of 10 (Total Marks: 1)

An indefinite integral may _________ in the sense that the limit defining it may not exist.

**diverge**

Converge

Question # 7 of 10 (Total Marks: 1)

An improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or -∞ or, in some cases, as both endpoints approach limits.

**TRUE Click here for detail**

FALSE

Question # 8 of 10 (Total Marks: 1)

Euler’s Method numerically computes the approximate derivative of a function.

TRUE

FALSE

Question # 9 of 10 (Total Marks: 1)

Euler’s Method numerically computes the approximate ________ of a function.

Antiderivative

Derivative

Error

Value

Question # 10 of 10 (Total Marks: 1)

If we wanted to find the value of a definite integral with an infinite limit, we can instead replace the

infinite limit with a variable, and then take the limit as this variable goes to _________.

Chose the correct option :

Constant

Finite

Infinity Click here for detail

Zero

QUIZ.NO.1(8)

Question # 1 of 10 (Total Marks: 1)

The Jacobi iteration ______, if A is strictly diagonally dominant.

converges

Diverges

Question # 2 of 10 (Total Marks: 1)

By using determinants, we can easily check that the solution of the given system of linear equation exits

and it is unique.

TRUE

FALSE

Question # 3 of 10 (Total Marks: 1)

The absolute value of a determinant (|detA|) is the product of the absolute values of the eigenvalues of

13

matrix A

TRUE

FALSE

Question # 4 of 10 (Total Marks: 1)

Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.

TRUE

FALSE

Question # 5 of 10 (Total Marks: 1)

Let A be an n ×n matrix. The number x is an eigenvalue of A if there exists a non-zero

vector v such that _______.

Av = xv

Ax=xv

Av + xv=0

Av = Ax1

Av = λv

Question # 6 of 10 (Total Marks: 1)

In Jacobi’s Method, the rate of convergence is quite ______ compared with other methods.

Slow Click here for detail

Fast

Question # 7 of 10 (Total Marks: 1)

Numerical solution of 2/3 up to four decimal places is ________.

0.667

0.6666

0.6667

0.666671.

Question # 8 of 10 (Total Marks: 1)

Symbol used for forward differences is

∆ (Page 12)

δ

μ

Question # 9 of 10 (Total Marks: 1)

The relationship between central difference operator and the shift operator is given by

δ =Ε−Ε-1

δ = Ε+Ε-1

δ = Ε1/2+Ε1/2

δ = E1/2 −Ε1/2

1 1

E E 2 2

(Page 152)

Question # 10 of 10 (Total Marks: 1)

14

Muller’s method requires --------starting points

1

2

3 (Page 41)

QUIZ.NO.1(9)

Question # 1 of 10 (Total Marks: 1)

By using determinants, we can easily check that the solution of the given system of linear equation

______ and it is ______.

Select correct option:

exits, unique

exists, consistent

trivial, unique

nontrivial, inconsistent

Question # 2 of 10 (Total Marks: 1)

Two matrices with the _______ characteristic polynomial need not be similar.

Select correct option:

same

different

Question # 3 of 10 (Total Marks: 1)

In … method, the elements above and below the diagonal are simultaneously made zero.

Select correct option:

Jacobi’s

Gauss-Seidel

Gauss–Jordon Elimination (Page 59)

Relaxation

Question # 4 of 10 (Total Marks: 1)

Which of the following is equivalent form of the system of equations in matrix form; AX=B ?

Select correct option:

XA = B

X = B(Inverse of A)

X =(Inverse of A)B

BX = A

Question # 5 of 10 (Total Marks: 1)

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

Select correct option:

a unique solution

many solutions

infinite many solutions

15

None of the given choices

Question # 6 of 10 (Total Marks: 1)

Sparse matrix is a matrix with …

Select correct option:

Some elements are zero

Many elements are zero (page 69)

Some elements are one

Many elements are one

MTH603 – 2

ND QUIZ FILE (24-11-2011)

EDIT BY MUHAMMAD AWAIS AND MUHAMMAD MOAAZ SIDDIQ

QUIZ.NO.2(1)

Question # 1 of 10 ( Start time: 11:18:07 PM ) Total Marks: 1

How many Eigen vectors will exist corresponding to the function; Exp(ax) = e^ax, when the matrix

operator is of differentiation?

Select correct option:

Infinite many

Finite Multiple

None

Question # 2 of 10 ( Start time: 11:19:00 PM ) Total Marks: 1

Which of the following rearrangement make strictly diagonal dominant, the system of linear

equations; x-3y+z= –2, –6x+4y+11z=1, 5x–2y–2z=9?

Select correct option:

5x–2y–2z=9, x–3y+z= –2, –6x+4y+11z=1

–6x+4y+11z=1, x–3y+z= –2, 5x–2y–2z=9

5x–2y–2z=9, –6x+4y+11z=1, x–3y+z= –2

No need to rearrange as system is already in diagonal dominant form.

Question # 3 of 10 ( Start time: 11:22:12 PM ) Total Marks: 1

Which of the following is the meaning of partial pivoting while employing the row transformations?

Select correct option:

Making the largest element as pivot (Page 50)

Making the smallest element as pivot

Making any element as pivot

Making zero elements as pivot

16

Question # 4 of 10 ( Start time: 11:53:06 PM ) Total Marks: 1

Differences methods are iterative methods.

Select correct option:

TRUE

FALSE

Question # 5 of 10 ( Start time: 11:55:28 PM ) Total Marks: 1

Eigenvalues of a _________ matrix are all real.

Select correct option:

symmetric

antisymmetric

rectangular

triangular

Question # 6 of 10 ( Start time: 11:32:38 PM ) Total Marks: 1

If a system of equations has a property that each of the equation possesses one large coefficient and

the larger coefficients in the equations correspond to different unknowns in different equations,

then which of the following iterative method id preferred to apply?

Select correct option:

Gauss-Seidel method

Gauss-Jordon method

Gauss elimination method

Crout’s method

Question # 7 of 10 ( Start time: 11:35:30 PM ) Total Marks: 1

For a system of linear equations, the corresponding coefficient matrix has the value of determinant;

|A| = 0, then which of the following is true?

Select correct option:

The system has unique solution

The system has finite multiple solutions

The system has infinite may solutions

The system has no solution (Page 48)

Question # 8 of 10 ( Start time: 11:36:21 PM ) Total Marks: 1

For the system; 2x+3y = 1, 3x +2y = - 4, if the iterative solution is (0,0) and ‘dxi = 2’ is the increment

in ‘y’ then which of the following will be taken as next iterative solution?

Select correct option:

(2,0)

(0,3)

(0,2)

(1,-4)

17

Question # 9 of 10 ( Start time: 11:37:49 PM ) Total Marks: 1

While using Relaxation method, which of the following is increment ‘dxi’corresponding to the

largest Residual for 1st iteration on the system; 2x+3y = 1, 3x +2y = - 4 ?

Select correct option:

-2

2

3

4

Question # 10 of 10 ( Start time: 11:40:42 PM)Total Marks: 1

If system of equations is inconsistent then its means that it has …

Select correct option:

No Solutions

Many solutions

Infinite Many solutions

None of the given choices

QUIZ.NO.2(2)

Question # 1 of 10 ( Start time: 11:47:15 PM)Total Marks: 1

Relaxation Method is a/an …

Select correct option:

Direct method

Iterative method

Question # 2 of 10 ( Start time: 11:33:36 PM ) Total Marks: 1

How many Eigen values will exist corresponding to the function; Exp(ax) = e^ax, when the matrix

operator is of differentiation?

Select correct option:

Finite Multiple

Infinite many

Unique

None

Question # 3 of 10 ( Start time: 11:36:46 PM ) Total Marks: 1

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the

matrix, remain … to the original vector.

Select correct option:

Perpendicular

Parallel

Diagonal

18

None of the given choices

Question # 4 of 10 ( Start time: 11:38:16 PM ) Total Marks: 1

In Jacobi’s method after finding D1, the sum of the diagonal elements of D1 should be … to

the sum of the diagonal elements of the original matrix A.

Select correct option:

Greater than

Less than

Same (Page 106)

Different

Question # 5 of 10 (Total Marks: 1)

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 and its two equal diagonal values are ‘3’ then

which of the following will be its corresponding argument value ‘theta’ of Orthogonal Matrix?

Select correct option:

Pi/3

Pi/6

Pi/2

Pi/4

Question # 6 of 10 (Total Marks: 1)

If f (x) contains trigonometric, exponential or logarithmic functions then this equation is known as

Transcendental Equation (Page 6)

Algebraic

Polynomial

Linear

Question # 7 of 10 (Total Marks: 1)

In interpolation is used to represent the d

Forward difference ?

Central difference

Backward difference

Question # 8 of 10 (Total Marks: 1)

The Power method can be used only to find the eigen value of A that is largest in absolute value we call

this eigen value the dominant eigen value of A.

True

False

Question # 9 of 10 (Total Marks: 1)

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

Select correct option:

independent

dependent