Master Numerical Integration and Differential Equations for MSc Mathematics

Question 1

Which of the following is a method for numerically integrating a function?

Accepted Answer

Trapezoidal rule

Accepted Answer

Simpson's rule

Answer

Runge-Kutta method

Question 2

Which of the following is NOT a stability requirement for an explicit numerical method used to solve differential equations?

Accepted Answer

Self-starting capability

Answer

A sufficiently small step size

Answer

Negative real parts of the eigenvalues of the Jacobian matrix of the right-hand side

Answer

Consistency

Question 3

What is the role of the Jacobian matrix in the context of numerical methods for solving differential equations?

Accepted Answer

Determining stability

Answer

Simplifying the system of equations

Answer

Estimating solution error

Answer

Calculating the next integration step

Question 4

Which numerical method is commonly used to solve first-order ordinary differential equations?

Accepted Answer

Runge-Kutta method

Answer

Gauss-Seidel method

Answer

Power method

Answer

LU decomposition

Question 5

For the Newton-Raphson method to converge for finding roots of nonlinear equations, which condition is necessary?

Accepted Answer

Lipschitz continuity of the derivative

Answer

Continuous first derivative

Answer

Continuous second derivative

Answer

Boundedness of the function

Question 6

Which statement about the error in numerical integration is true?

Accepted Answer

Increasing the number of subintervals generally reduces the error.

Answer

The error is always smaller than the absolute value of the function.

Answer

The error is proportional to the second derivative of the function.

Answer

The error is independent of the numerical integration method used.

Question 7

The order of convergence of a numerical method measures:

Accepted Answer

The rate of error reduction as the step size decreases

Answer

The maximum possible error

Answer

The number of iterations needed for a desired accuracy

Answer

The stability of the method

Question 8

Which of the following is NOT an advantage of using numerical methods to solve mathematical problems?

Accepted Answer

Always producing exact solutions

Answer

Providing approximate solutions when analytical solutions are unavailable

Answer

Handling complex equations that are difficult to solve manually

Answer

Modeling real-world problems

Question 9

Which numerical integration method is most suitable for highly oscillatory functions, exhibiting rapid fluctuations?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Romberg integration

Question 10

Consider the initial value problem y' = y - 1, y(0) = 0. Which numerical method would provide the most accurate solution given a fixed step size constraint?

Accepted Answer

Runge-Kutta method

Answer

Euler's method

Answer

Predictor-corrector method

Answer

Finite difference method

Question 11

What is the order of convergence of the Newton-Raphson method for finding the roots of a nonlinear equation f(x) = 0?

Accepted Answer

Quadratic

Answer

Linear

Answer

Cubic

Answer

Exponential

Question 12

What is the order of accuracy of the trapezoidal rule for numerical integration?

Accepted Answer

O(h^2)

Answer

O(h)

Answer

O(h^3)

Answer

O(h^4)

Question 13

Which of the following is a first-order linear differential equation?

Accepted Answer

y' + y = sin(x)

Answer

y'' + 2y' + y = e^x

Answer

y' = y^2

Answer

y''' - 3y'' + 2y' - y = 0

Question 14

What is the integrating factor for the differential equation y' + 2xy = x?

Accepted Answer

e^(-2x)

Answer

e^(x^2)

Answer

x

Answer

1/x

Question 15

Which of the following is a first-order ordinary differential equation?

Accepted Answer

y' + y = sin(x)

Answer

y'' - 2y' + y = 0

Answer

y''' + 3y'' + 2y' + y = x^2

Answer

y^{(4)} + y = e^x

Question 16

Which method is used to find the solution to a system of linear differential equations?

Accepted Answer

Eigenvalue decomposition

Answer

Gaussian elimination

Answer

Fixed-point iteration

Answer

Monte Carlo simulation

Question 17

Which of the following is an example of a boundary value problem?

Accepted Answer

Find y(x) such that y'(x) = x and y(0) = 1

Answer

Find the critical points of f(x) = x^3 - 2x^2

Answer

Approximate the integral of sin(x) from 0 to π

Answer

Determine the eigenvalues of the matrix A

Question 18

What is the main idea behind the method of weighted residuals?

Accepted Answer

To solve differential equations by minimizing a weighted sum of the residuals

Answer

To approximate integrals using a set of basis functions

Answer

To find zeros of a function using an iterative algorithm

Answer

To solve linear systems using Gaussian elimination

Question 19

Which of the following is a type of finite element method?

Accepted Answer

Galerkin method

Answer

Newton's method

Answer

Simpson's rule

Answer

Runge-Kutta method

Question 20

What is the purpose of boundary conditions in solving differential equations?

Accepted Answer

To specify the values of the solution at specific points or boundaries

Answer

To ensure that the solution is unique

Answer

To determine the order of the differential equation

Answer

To guarantee that the solution is continuous

Question 21

Which of the following is NOT a type of differential equation?

Accepted Answer

Algebraic equation

Answer

Ordinary differential equation

Answer

Partial differential equation

Answer

Discrete differential equation

Question 22

The Euler method is a numerical method for solving differential equations of which order?

Accepted Answer

First

Answer

Second

Answer

Third

Answer

Fourth

Question 23

Which of the following can be used to analyze the stability of a numerical method for solving differential equations?

Accepted Answer

Von Neumann analysis

Answer

Runge-Kutta method

Answer

Taylor series expansion

Answer

Gauss-Seidel method

Question 24

What is the order of convergence of the trapezoidal rule for approximating integrals?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 25

Which of the following is a necessary condition for the existence and uniqueness of solutions to initial value problems for first-order ODEs?

Accepted Answer

Lipschitz continuity of the right-hand side function

Answer

Continuity of the right-hand side function

Answer

Differentiability of the right-hand side function

Answer

Integrability of the right-hand side function

Question 26

What is the numerical stability criterion for the explicit Euler method?

Accepted Answer

h ≤ 1/λ

Answer

h ≤ 2/λ

Answer

h ≤ 1/2λ

Answer

h ≤ λ

Question 27

Which of the following is NOT a method for solving nonlinear differential equations?

Accepted Answer

Method of undetermined coefficients

Answer

Picard iteration

Answer

Newton's method

Answer

Continuation methods

Question 28

What is the purpose of orthogonalisation in numerical linear algebra?

Accepted Answer

To construct an orthogonal basis for a subspace

Answer

To find the eigenvalues and eigenvectors of a matrix

Answer

To solve systems of linear equations

Answer

To find the least squares solution to a linear system

Question 29

Consider the integral ∫[0,1] e^(-x^2) dx. Which of the following numerical integration methods will provide the most accurate approximation?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Monte Carlo integration

Question 30

Which of the following is NOT a commonly used boundary condition for solving partial differential equations?

Accepted Answer

Cauchy boundary condition

Answer

Dirichlet boundary condition

Answer

Neumann boundary condition

Answer

Periodic boundary condition

Question 31

The finite element method is primarily used for solving:

Accepted Answer

Partial differential equations

Answer

Algebraic equations

Answer

Ordinary differential equations

Answer

Integral equations