Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2080

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.
Write an algorithm and a C-Program to obtain roots of non-linear equation using Newton Raphson Method.[10]
2.
Solve the following ordinary differential equation using shooting method. $y'' + xy' - xy = 2x$ with boundary conditions $y(0)=1$ and $y(2)=10$[10]
3.
Compare and contrast between Jacobi iterative methods and Gauss Seidal method? Solve the following equation using Gauss Seidal method. $x+2y+3z = 5, 2x+8y+22z = 6$ and $3x+22y+82z = -10$[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Use secant method to estimate the root of the equation $x^2-5x+6=0$, with initial estimate $x_1 = 4$ and $x_2 = 2$ (EPS=0.05). [5]
5.
Solve the double integration using Simpson's 1/3 rule. $\int_{2}^{2.6} \int_{4}^{4.4} \frac{dxdy}{xy}$ [5]
6.
What are the sources of errors? Discuss various types of errors encounters in numerical computation. [5]
7.
Fit a second order polynomial to the data in the table below:

$\begin{array}{|c|c|c|c|c|c|}\hline X & 1 & 2 & 3 & 4 & 5 \\ \hline F(x) & 2 & 6 & 12 & 20 & 30 \\ \hline \end{array}$
[5]
8.
Why Numerical Integration is required? Compute the integral: $I = \int_{-1}^{1} e^x dx$ using composite trapezoidal rule for n = 4. [5]
9.
Evaluate $\frac{dy}{dx}$ at x = 5 using Newton's forward interpolation formula using the following table.

$\begin{array}{|c|c|c|c|c|c|}\hline X & 1 & 3 & 5 & 7 & 9 \\ \hline y & -1.20 & 12.80 & 119.60 & 472.80 & 1302.80 \\ \hline \end{array}$
[5]
10.
Find the Eigen values and Eigen vectors of the Matrix: $A=\begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}$ [5]
11.
Solve the Poisson's equation $\partial^2f/\partial x^2+\partial^2f/\partial y^2 = 2x^2y^2$ over the square domain 0<=x<=3 and 0<=y<=3 with f=0 on the boundary and h = 1. [5]
12.
Solve the following differential equation $ \frac{dy}{dx} = 3x + \frac{y}{2} $ with $ y(0) = 1 $ for $ x = 0.2 \quad (h = 0.1) $ using Euler's Method. [5]