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

[zaasmi]

Full pivoting, in fact, is more …than the partial pivoting.
Select correct option:
Easiest
Complicated

[zaasmi]

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

Full pivoting, in fact, is more …than the partial pivoting.
Select correct option:
Easiest
Complicated

Full pivoting is generally considered more:

Complicated than partial pivoting.

Explanation:

• Partial Pivoting: This involves only row swaps to ensure that the largest absolute value element is on the diagonal of the current row. It’s simpler and computationally less intensive compared to full pivoting.

• Full Pivoting: This involves both row and column swaps to ensure the largest absolute value element in the entire remaining submatrix is placed on the diagonal. It’s more complex and computationally demanding due to the additional step of considering column swaps.

So the correct option is:

Complicated

[zaasmi]

For the equation
3
x x    3 1 0

, the root of the equation lies in the interval…

► (1, 3)
► (1, 2)
► (0, 1)
► (1, 2)

[zaasmi]

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

For the equation
3
x x    3 1 0

, the root of the equation lies in the interval…

► (1, 3)
► (1, 2)
► (0, 1)
► (1, 2)

To determine the interval in which the root of the equation ( x^3 - x - 1 = 0 ) lies, you can use methods such as evaluating the function at different points to find where the function changes sign (which indicates a root exists in that interval).

Let’s evaluate the function ( f(x) = x^3 - x - 1 ) at various points within the given intervals:

1. Interval (0, 1):

   • ( f(0) = 0^3 - 0 - 1 = -1 )
   • ( f(1) = 1^3 - 1 - 1 = -1 )

   The function does not change sign between 0 and 1.

2. Interval (1, 2):

   • ( f(1) = 1^3 - 1 - 1 = -1 )
   • ( f(2) = 2^3 - 2 - 1 = 5 )

   The function changes sign between 1 and 2, indicating a root lies in this interval.

3. Interval (1, 3):

   • Although this interval includes (1, 2), it is broader. The function already indicates a root in (1, 2), so this interval is valid but not the most specific.

Given the evaluations, the most specific interval where the function changes sign is:

(1, 2)

So, the correct option is:

(1, 2)

[zaasmi]

…lies in the category of iterative method.
► Bisection Method
► Regula Falsi Method
► Secant Method
► all of the given choices

[zaasmi]

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

…lies in the category of iterative method.
► Bisection Method
► Regula Falsi Method
► Secant Method
► all of the given choices

The methods listed—Bisection Method, Regula Falsi Method, and Secant Method—are all iterative methods used to find roots of equations. Each of these methods iteratively approximates the root through successive approximations.

Explanation:

• Bisection Method: Iteratively narrows down the interval where the root lies by halving the interval based on the sign change.

• Regula Falsi Method (False Position Method): Iteratively refines the interval where the root lies by using linear interpolation.

• Secant Method: Uses two initial guesses and iteratively updates them to approximate the root using the secant line.

Therefore, all the methods mentioned fall under the category of iterative methods.

So the correct option is:

all of the given choices

[zaasmi]

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

1. True
2. False

[zaasmi]

@zaasmi 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 different eigenvalues (with the same
multiplicities).

1. True
2. False

The statement is:

False

Explanation:

If ( n \times n ) matrices ( A ) and ( B ) are similar, it means there exists an invertible matrix ( P ) such that ( A = PBP^{-1} ). Similar matrices have the same eigenvalues, including their multiplicities.

So, if ( A ) and ( B ) are similar, they indeed have the same eigenvalues (with the same multiplicities), not different.

Thus, the correct answer is:

False

[zaasmi]

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

No
At least one

[zaasmi]

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

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

No
At least one

Jacobi’s method is a technique for solving a system of linear equations that requires the matrix to be diagonally dominant or, in some cases, can work with a matrix that has at least one non-zero element along its main diagonal. However, for the method to be effective and guarantee convergence, it is typically applied to matrices where the diagonal elements are not zero.

To clarify:

• At least one non-zero element along the main diagonal is necessary to ensure the method’s application because having zero along the main diagonal would prevent the method from proceeding directly.

Therefore, the Jacobi method requires that the matrix has:

At least one non-zero element along its main diagonal

[zaasmi]

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 …

►
( )
2
a b 
Page 9

►
( )
2
a b 

►
( )
2
b a 
► None of the given choices

[zaasmi]

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

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 …

►
( )
2
a b 
Page 9

►
( )
2
a b 

►
( )
2
b a 
► None of the given choices

In the bisection method, the first approximation to the root of the equation, given that the root lies between ( a ) and ( b ), is calculated as the midpoint of the interval ([a, b]). This is done to refine the interval where the root lies.

The formula for the midpoint is:

[ \frac{a + b}{2} ]

So, the correct choice is:

(\frac{a + b}{2})

[zaasmi]

To apply Simpson’s 3/8 rule, the number of intervals in the following must be
► 10
► 11
► 12
► 13

[zaasmi]

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

To apply Simpson’s 3/8 rule, the number of intervals in the following must be
► 10
► 11
► 12
► 13

Simpson’s 3/8 rule requires that the number of intervals ( n ) be a multiple of 3. This is because the 3/8 rule is based on approximating the integral of a function using a cubic polynomial, which necessitates the intervals being divisible by 3 for proper application.

To apply Simpson’s 3/8 rule correctly, the number of intervals ( n ) must satisfy:

[ n = 3k ]

where ( k ) is a positive integer.

Given the options:

• 10 (not a multiple of 3)
• 11 (not a multiple of 3)
• 12 (multiple of 3, since ( 12 = 3 \times 4 ))
• 13 (not a multiple of 3)

The correct choice is:

12

[zaasmi]

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric________ definite
matrices A.
Select correct option:
Positive
negative

[zaasmi]

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

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric________ definite
matrices A.
Select correct option:
Positive
negative

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

Explanation:

• Strictly Diagonally Dominant Matrices: The Gauss-Seidel method converges for strictly diagonally dominant matrices, which ensures that the diagonal elements are sufficiently large compared to the sum of the other elements in the row.

• Symmetric Positive Definite Matrices: For symmetric positive definite matrices, the Gauss-Seidel method is also guaranteed to converge. Positive definiteness ensures that the matrix has a unique solution and the method will converge.

Thus, the correct option is:

Positive