Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2078

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.
Derive the formula for integration using simpsons 3/8 rule. Use Secant Method to estimate the root of equation with initial estimate x₁ = 4 and x₂ = 2, $x^2 - 4x - 10 = 0$.[10]
2.
What do you mean by boundary value problem? Use shootuing method, solve the equation: y'' = 6x², with y(0) = 1 and y(1) = 2 in the interval (0, 1) for y(0.5) taking h = 0.5[10]
3.
Write an algorithm and program to compute the interpolation using Lagrange Interpolation.[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Show that the rate of convergence of Newtons Raphson method is quadratic. [5]
5.
The temperature of a metal strip was measured at various time intervals during heating and the values are given in the table below. If the relation between the time 't' and temperature 'T' is of the form: $T = be^{t/4} + a$ . Estimate the temperature at t = 6 minute.

$\begin{array}{|c|c|c|c|c|}\hline \text{Time (`t' min)} & 1 & 2 & 3 & 4 \\ \hline \text{Temp (`T' $^\circ$C)} & 70 & 83 & 100 & 124 \\ \hline \end{array}$
[5]
6.
Given the following set of data points. Obtain the table of divided difference and use that table to estimate the value of f(1.5).

$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x)=x^3-1 & 0 & 7 & 26 & 63 & 124 \\ \hline \end{array}$
[5]
7.
Solve the following system of linear equation by Gauss Elimination with Pivoting $2x + 2y + z = 6$, $4x + 2y + 3z = 4$,$x - y + 1 = 0$. [5]
8.
Determine the Eigen Values and corresponding Eigen Vectors for the matrix.

$A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
[5]
9.
The table below gives the values of distance travelled by a car at various time intervals during the initial running. Estimate the velocity and acceleration at time t = 7sec.

$\begin{array}{|c|c|c|c|c|c|}\hline \text{Time (`t' sec)} & 5 & 6 & 7 & 8 & 9 \\ \hline \text{Temp (`T' $^\circ$C)} & 10.0 & 14.5 & 19.5 & 25.5 & 32.0 \\ \hline \end{array}$
[5]
10.
Solve the following integral using trapezoidal rule form = 8, $l = \int_{2}^{4} (x^4 + 1)dx$. [5]
11.
Given the equation $y' = 3x^2 + 1$ with y(1) = 2, estimate y(2) by Euler's Method using h = 0.2. [5]
12.
Solve the Poisson's Equation $\nabla^2 f = 2x^2 y^2$ over the square domain $0 \leq x \leq 3$ and $0 \leq y \leq 3$ with $f = 0$ on the boundary and $h = 1$. [5]