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MTH603 Mid Term Past and Current Solved Paper Discussion

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:
If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).
True
FalseSince 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 nonzero since P is invertible) is an eigenvector for B with eigenvalue λ.

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



Question # 1 of 10 (Total Marks: 1)
While solving a system of linear equations, which of the following approach is economical for the computer memory?
Select correct option:
Direct
Iterative (Page 69)
Analytical
GraphicalQuestion # 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 choicesQuestion # 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 rowQuestion # 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
FALSEQuestion # 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
FALSEQuestion # 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 equivalentQuestion # 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
1Question # 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 choicesQuestion # 10 of 10 (Total Marks: 1)
Jacobi’s Method is a/an…
Select correct option:
Iterative method (Page 69)
Direct methodQUIZ.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:
(x1)^3
(x+1)^3
x^31
x^3+1Question # 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
FALSEQuestion # 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
GaussSeidel
Relaxation
Gaussian elimination (Page 262)Question # 4 of 10 (Total Marks: 1)
The linear equation: 2x+0y2=0 has  solution/solutions.
Select correct option:
Unique (Page 48)
no solution
infinite many
finite manyQuestion # 5 of 10 (Total Marks: 1)
Under elimination methods, we consider, Gaussian elimination and …methods.
Select correct option:
GaussSeidel
Jacobi
GaussJordan elimination (Page 48)
None of the given choicesQuestion # 6 of 10 (Total Marks: 1)
Which of the following method is not an iterative method?
Select correct option:
Jacobi’s method
GaussSeidel method
Relaxation methods
GaussJordan elimination methodQuestion # 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
FALSEQuestion # 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)
DivergentQuestion # 10 of 10 (Total Marks: 1)
Gauss–Seidel method is similar to …
Select correct option:
Iteration’s method
RegulaFalsi 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
Nonlinear differential equationsQuestion # 2 of 10 (Total Marks: 1)
While solving by GaussSeidel method, which of the following is the first Iterative solution for the
system; x2y =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 solutionsQuestion # 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 choicesQuestion # 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
1Question # 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= 12x3y
0= 43x 2yQuestion # 7 of 10 (Total Marks: 1)
If the order of coefficient matrix corresponding to system of linear equations is 33 then which of the
following will be the orders of its decomposed matrices; ‘L’ and ‘U’?
Select correct option:
Order of ‘L’ = 31, Order of ‘U’ = 13
Order of ‘L’ = 32, Order of ‘U’ = 2*3Order of ‘L’ = 33, Order of ‘U’ = 33
Order of ‘L’ = 34, Order of ‘U’ = 43Question # 8 of 10 (Total Marks: 1)
While solving the system; x–2y = 1, x+4y = 4 by GaussSeidel method, which of the following ordering
is feasible to have good approximate solution?
Select correct option:
x+4y = 1, x2y = 4
x+2y = 1, x 4y =4
x+4y = 4, x–2y = 1
no need to reorderingQuestion # 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
FalseQuestion # 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 oneQuestion # 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
► 13Question # 6 of 10 (Total Marks: 1)
The GaussSeidel method is applicable to strictly diagonally dominant or symmetric________ definite
matrices A.
Select correct option:
Positive Click here for detail
negativeQuestion # 7 of 10 (Total Marks: 1)
Differences methods find the ________ solution of the system.
Select correct option:
numerical
AnalyticalQuestion # 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
► 7Question # 9 of 10 (Total Marks: 1)
Bisection and false position methods are also known as bracketing method and are always
Divergent
Convergent (Page 26)
.67Question # 10 of 10 (Total Marks: 1)
The Inverse of a matrix can only be found if the matrix is
Singular
None Singular: Every square nonsingular matrix will have an inverse.
Scalar
DiagonalQUIZ.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 differenceQuestion # 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
differentQuestion # 5 of 10 (Total Marks: 1)
Eigenvalues of a symmetric matrix are all _______ .
Real (page 104)
complex
zero
positiveQuestion # 6 of 10 (Total Marks: 1)
The Jacobi iteration converges, if A is strictly diagonally dominant.
TRUE (Page 69)
FALSEQuestion # 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 methodQuestion # 8 of 10 (Total Marks: 1)
Two matrices with the same characteristic polynomial need not be similar.
TRUE
FALSEQuestion # 9 of 10 (Total Marks: 1) No.69
The determinant of a diagonal matrix is the product of the diagonal elements.
True
FalseQuestion # 10 of 10 (Total Marks: 1)
The GaussSeidel method is applicable to strictly diagonally dominant or symmetric positive definite matrices A.
True
FalseQUIZ.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
ScalarQuestion # 2 of 10 (Total Marks: 1)
For differences methods we require the set of values.
True
FalseQuestion # 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
FalseQuestion # 4 of 10 (Total Marks: 1)
Central difference method seems to be giving a better approximation, however it requires more computations.
True
FalseQuestion # 5 of 10 (Total Marks: 1)
Iterative algorithms can be more rapid than direct methods.
True
FalseQuestion # 6 of 10 (Total Marks: 1)
Central Difference method is the finite difference method.
True
FalseQuestion # 1 of 10 (Total Marks: 1)
Back substitution procedure is used in …
Select correct option:
Gaussian Elimination Method
Jacobi’s method
GaussSeidel method
None of the given choicesQuestion # 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
FalseQuestion # 8 of 10 (Total Marks: 1)
Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.
True (Page102)
FalseQuestion # 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 distinctQuestion # 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
QuarticQUIZ.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
FalseQuestion # 2 of 10 (Total Marks: 1)
The predictorcorrector method an implicit method. (multistep methods)
True (Page 212)
FalseQuestion # 3 of 10 (Total Marks: 1)
Generally, Adams methods are superior if output at many points is needed.
True
FalseQuestion # 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
FunctionQuestion # 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
ConvergeQuestion # 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
ZeroQUIZ.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 of13
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 nonzero
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−Ε1
δ = Ε+Ε1
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
GaussSeidel
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 solutions15
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 oneMTH603 – 2
ND QUIZ FILE (24112011)
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; x3y+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 pivot16
Question # 4 of 10 ( Start time: 11:53:06 PM ) Total Marks: 1
Differences methods are iterative methods.
Select correct option:
TRUE
FALSEQuestion # 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:
GaussSeidel method
GaussJordon 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 choicesQUIZ.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
NoneQuestion # 3 of 10 ( Start time: 11:36:46 PM ) Total Marks: 1
The eigenvectors of a square matrix are the nonzero vectors that, after being multiplied by the
matrix, remain … to the original vector.
Select correct option:
Perpendicular
Parallel
Diagonal18
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 offdiagonal 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 