Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Credit:Eli TamangEli Tamang

Tribhuwan University

Institute of Science and Technology

2082

Bachelor Level / Second Year / Third Semester / Science

Bachelors in Information Technology (BIT203)

(Numerical Methods)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

Candidates are required to give their answers in their own words as for as practicable.

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.
[2*10=20]
1.
Explain how bisection method differ from secant. Derive the formula for Newton Raphson. Use Newton Raphson method to solve the equation f(x)=x3+2x−2f(x) = x^3 + 2x - 2 correct upto three decimal places. [2+4+4]
2.
Write an algorithm to compute the value of interpolation using Newton’s divided difference method. Write a program to compute the value of interpolation using Newton’s divided difference method. [5+5]
3.
List out any two applications of system of linear equation. Differentiate between Gauss-Seidel and Jacobi iteration method. Solve the following system of equations using Jacobi iteration method: 4x+y+z=74x + y + z = 7, x+5y−2z=3x + 5y - 2z = 3, 3x+2y+6z=143x + 2y + 6z = 14. [2+3+5]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Differentiate between round-off error and truncation error. Explain how they affect numerical computations, with the help of an example. [2+3]
5.
Derive the formula for two points forward difference. Derive the formula for two points backward difference. [2.5+2.5]
6.
Fit the exponential curve y=aebxy = ae^{bx} for (1,15), (2,22), (3,33), (4,48), (5,70) using least square method. [5]
7.
Solve the following system of linear equation using Cholesky decomposition method: 4x+6y−8z=−84x + 6y - 8z = -8, 6x+13y−11z=−16x + 13y - 11z = -1, −8x−11y+29z=57-8x - 11y + 29z = 57. [5]
8.
Find the first and second derivative at x=2.5x = 2.5 of the following data points.
x1.522.533.54f(x)2.3754.57.6251217.87523\begin{array}{|c|cccccc|}\hline x & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\ \hline f(x) & 2.375 & 4.5 & 7.625 & 12 & 17.875 & 23 \\ \hline \end{array}
[5]
9.
Integrate ∫03(2x3+1)dx\int_{0}^{3} (2x^3 + 1) dx using Simpson’s 13\frac{1}{3} rule with n=6n = 6. [5]
10.
Solve dydx=x2+y\frac{dy}{dx} = x^2 + y, with y(0)=1y(0) = 1 for x=1.5x = 1.5, using RK fourth order method. [5]
11.
Explain boundary value problem with example. Describe how higher order differential equation can be solved. [2+3]
12.
Define eigenvalue and eigenvector. Distinguish between regression and interpolation. [2.5+2.5]