# Numerical Method | BSc.CSIT (TU) Question Paper 2067 | Third Semester

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## Third Semester | Second Year | Tribhuvan University Old Question Collection | Question Bank Numerical Method, Year: 2067 Computer Science and Information Technology (CSc 204) Full Marks: 60 | Pass Marks: 24 | Time: 3 hours

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Candidates are required to give their answer in their own words as far as practicable.
The figures in the margin indicate full marks.

Attempt all Questions:

1. Discuss methods of Half Interval and Newton’s for solving the nonlinear equation f(x) = 0. Illustrate the methods by figures and compare them stating their advantages and disadvantages. (8)
2. Derive the equation for Lagrange’s interpolating polynomial and find the value of f(x) at x = 1 for the following: (4+4)
 X -1 -2 2 4 F -1 -9 11 69
3. Write Newton-cotes integration formulas in basic form for x = 1, 2, 3 and give their composite rules. Evaluate $\int_{.5}^{1.5} e^{-x^2}$dx using the Gaussian integration three point formula. (4+4)
4. Solve the following algebraic system of linear equations by Gauss-Jordan algorithm. (8)
$\begin{bmatrix} 0 & 2 & 0 & 1 \\ 2 & 2 & 3 & 2 \\ 4 & -3 & 0 & 1 \\ 6 & 1 & -6 & -7 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix} = \begin{bmatrix} 0 \\ -2 \\ -7 \\ 6 \end{bmatrix}$
5. Write an algorithm and program to solve system of linear equations using Gauss-Siedel iterative method. (4+8)
6. Explain the Picard’s proves of successive approximation. Obtain a solution upto the fifth approximation of the equation
$\frac{dy}{dx}$= y + x such that y=1 when x=0
using Picard’s process of successive approximations. (2+6)
7. Define a difference equation to represent a Laplace’s equation. Solve the following Laplace equation.
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ =0 within 0≤?≤3, 0≤?≤3.
For the rectangular plate given as: (3+5)

OR
Derive a difference equation to represent a Poison’s equation. Solve the Poison’s equation ∇2?=2?2?2 over the domain 0≤?≤3, 0≤?≤3 with ?=0 on the boundary and ℎ=1. (3+5)
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