dy/dx - = 1 - y,y(0) = 0 is an example of

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

Iterative method

The Relaxation Method is an iterative method used to solve systems of linear equations. It is an iterative process that starts with an initial guess and repeatedly updates the solution until convergence is achieved. The method is based on the idea of relaxing the values of the variables in a way that minimizes the residual error.

In contrast, direct methods are algorithms that solve systems of linear equations in a single step, without the need for iteration. Examples of direct methods include Gaussian elimination and LU decomposition.

So, the Relaxation Method is an iterative method, not a direct method.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

No Solutions

An inconsistent system of equations is a system that has no solution. This means that there is no set of values for the variables that can satisfy all the equations in the system. In other words, the equations in the system are contradictory, and it is impossible to find a solution that satisfies all of them.

Inconsistent systems are characterized by having no solution, which is why “No Solutions” is the correct answer.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 8 of 10 (Total Marks: 1)
Euler’s Method numerically computes the approximate derivative of a function.
TRUE
FALSE

The correct answer is:

FALSE

Euler’s Method is a numerical method used to approximate the solution of an initial value problem for a differential equation. It is used to find the approximate value of a function at a given point, not to compute the derivative of a function.

Euler’s Method is a way to approximate the solution of a differential equation by iteratively approximating the value of the function at small increments of the independent variable. It is not a method for computing derivatives.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 9 of 10 (Total Marks: 1)
Euler’s Method numerically computes the approximate ________ of a function.
Antiderivative
Derivative
Error
Value

The correct answer is:

Value

Euler’s Method is a numerical method used to approximate the value of a function at a given point, not to compute the derivative or antiderivative of a function. It is used to find the approximate value of a function at a specific point, usually by iteratively approximating the value of the function at small increments of the independent variable.

So, the correct answer is “Value”.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

Infinity

When dealing with a definite integral with an infinite limit, we can replace the infinite limit with a variable, say “b”, and then take the limit as “b” approaches infinity. This is a common technique used in calculus to evaluate improper integrals.

For example, if we want to evaluate the definite integral ∫[0,∞) f(x) dx, we can replace the infinite limit with a variable “b” and write it as ∫[0,b] f(x) dx. Then, we can take the limit as b approaches infinity, i.e., lim (b→∞) ∫[0,b] f(x) dx.

So, the correct answer is “Infinity”.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 1 of 10 (Total Marks: 1)
The Jacobi iteration ______, if A is strictly diagonally dominant.
converges
Diverges

The correct answer is:

converges

The Jacobi iteration is a method for solving systems of linear equations. If the matrix A is strictly diagonally dominant, then the Jacobi iteration is guaranteed to converge. This means that the sequence of approximate solutions generated by the Jacobi iteration will converge to the exact solution of the system.

In particular, a matrix A is said to be strictly diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the non-diagonal elements in the same row. When A is strictly diagonally dominant, the Jacobi iteration is convergent, meaning that it will produce a sequence of approximate solutions that get arbitrarily close to the exact solution as the number of iterations increases.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

TRUE

By using determinants, specifically the determinant of the coefficient matrix, we can easily check if the solution of a system of linear equations exists and is unique.

If the determinant is non-zero, then the system has a unique solution.
If the determinant is zero, then the system either has no solution or infinitely many solutions.
This is a fundamental property of linear algebra, and it’s a powerful tool for determining the solvability and uniqueness of solutions to systems of linear equations.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

TRUE

The absolute value of the determinant of a matrix A, denoted by |det(A)|, is indeed equal to the product of the absolute values of the eigenvalues of matrix A.

This is a well-known result in linear algebra, and it can be proven using various methods, including the fact that the determinant of a matrix is equal to the product of its eigenvalues.

So, if λ1, λ2, …, λn are the eigenvalues of matrix A, then:

|det(A)| = |λ1| × |λ2| × … × |λn|

This result has important implications in many areas of mathematics, science, and engineering, including stability analysis, signal processing, and machine learning.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 4 of 10 (Total Marks: 1)
Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.
TRUE
FALSE

The correct answer is:

TRUE

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

To be more precise, if λ1 and λ2 are distinct eigenvalues of a symmetric matrix A, and v1 and v2 are their corresponding eigenvectors, then:

v1 · v2 = 0

This means that v1 and v2 are orthogonal vectors.

However, if λ1 = λ2, then the eigenvectors v1 and v2 are not necessarily orthogonal. In fact, they can be any linearly independent vectors in the eigenspace corresponding to the eigenvalue λ1.

So, the orthogonality of eigenvectors of a symmetric matrix only holds for distinct eigenvalues.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

Av = xv

This is the definition of an eigenvalue and eigenvector. Specifically, if A is an n × n matrix, then x is an eigenvalue of A if there exists a non-zero vector v such that:

Av = xv

This means that when the matrix A is multiplied by the vector v, the result is a scaled version of v, where the scaling factor is x. The vector v is called an eigenvector of A corresponding to the eigenvalue x.

The other options are not correct:

Ax = xv is not a correct statement, as the matrix A is multiplied by the vector v, not the scalar x.
Av + xv = 0 is not a correct statement, as it does not define an eigenvalue or eigenvector.
Av = Ax1 is not a correct statement, as it does not define an eigenvalue or eigenvector.
Av = λv is close, but λ is not necessarily equal to x. The correct statement is Av = xv, where x is the eigenvalue.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 6 of 10 (Total Marks: 1)
In Jacobi’s Method, the rate of convergence is quite ______ compared with other methods.
Slow
Fast

The correct answer is:

Slow

Jacobi’s method is an iterative method for finding the eigenvalues and eigenvectors of a symmetric matrix. While it is a simple and intuitive method, its rate of convergence is relatively slow compared to other methods, such as the QR algorithm or the power iteration method.

In particular, Jacobi’s method has a linear rate of convergence, which means that the error in the eigenvalue estimates decreases linearly with each iteration. This can lead to slow convergence, especially for large matrices or matrices with closely spaced eigenvalues.

In contrast, other methods like the QR algorithm or the power iteration method have a quadratic or cubic rate of convergence, which means that the error in the eigenvalue estimates decreases much more rapidly with each iteration. This can lead to much faster convergence and more accurate results.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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.

The correct answer is:

0.6667

The numerical solution of 2/3 up to four decimal places is indeed 0.6667.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 8 of 10 (Total Marks: 1)
Symbol used for forward differences is
∆ (Page 12)
δ
μ

The correct answer is:

∆

In numerical analysis, the symbol ∆ (delta) is commonly used to represent forward differences. Forward differences are used to approximate derivatives and are defined as the difference between a function value at a point and the function value at a nearby point. The symbol ∆ is used to denote this difference.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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 

 

The correct answer is:

δ = (E1/2 - E-1/2)

The central difference operator δ is related to the shift operator E by the formula:

δ = (E1/2 - E-1/2)

This formula expresses the central difference operator in terms of the shift operator E, which is a fundamental concept in numerical analysis and difference equations.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Question # 10 of 10 (Total Marks: 1)

14

Muller’s method requires --------starting points
1
2
3 (Page 41)

The correct answer is:

3

Muller’s method is a numerical root-finding method that requires three starting points to converge to a root of a function. It is an iterative method that uses a quadratic interpolation to approximate the root, and it requires three initial guesses to start the iteration process.

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

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

The correct answer is:

exists, unique

By using determinants, we can easily check that the solution of a given system of linear equations exists and is unique, provided the determinant of the coefficient matrix is non-zero. This is a fundamental result in linear algebra, and it provides a convenient way to determine the solvability and uniqueness of a system of linear equations.