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

[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

[zaasmi]

Differences methods find the ________ solution of the system.
Select correct option:
numerical
Analytical

[zaasmi]

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

Differences methods find the ________ solution of the system.
Select correct option:
numerical
Analytical

Difference methods (such as finite difference methods) are used to find the numerical solution of systems of differential equations or partial differential equations.

Explanation:

• Numerical Solution: Difference methods approximate solutions to problems by discretizing the equations and solving them numerically. These methods are used when exact analytical solutions are difficult or impossible to obtain.

• Analytical Solution: Analytical methods seek exact solutions through algebraic manipulations and typically involve solving equations in their exact forms.

Thus, the correct option is:

numerical

[zaasmi]

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

[zaasmi]

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

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

To apply Simpson’s 1/3 rule, the number of intervals ( n ) must be an even number. This is because Simpson’s 1/3 rule requires that the interval be divided into an even number of subintervals for proper application, as the rule is based on approximating the integral of a function using quadratic polynomials over pairs of subintervals.

Explanation:

• Simpson’s 1/3 Rule: This rule applies to cases where the number of intervals is even, ensuring that each pair of intervals is approximated by a quadratic polynomial.

Given the options:

• 2 (even number, so it’s valid for Simpson’s 1/3 rule)
• 3 (odd number, not valid)
• 5 (odd number, not valid)
• 7 (odd number, not valid)

The correct choice is:

2

[zaasmi]

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

[zaasmi]

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

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

Bisection and false position methods (Regula Falsi) are known as bracketing methods. They are convergent methods.

Explanation:

• Bisection Method: This method always converges to a root if the function is continuous and if the initial interval ([a, b]) contains a root (i.e., ( f(a) \cdot f(b) < 0 )). It works by iteratively narrowing down the interval where the root lies.

• False Position Method (Regula Falsi): This method also converges to a root under similar conditions. It improves on the interval using linear interpolation to approximate the root.

Both methods guarantee convergence to a root as long as the initial conditions are satisfied (the function changes sign over the interval).

Thus, the correct option is:

Convergent

[zaasmi]

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