### Exam Year

Tribhuvan University

Institute of Science and Technology

Model Set I

Bachelor Level / third-semester / Science

Computer Science and Information Technology( CSC207 )

Numerical Method

Full Marks: 60 + 20 + 20

Pass Marks: 24 + 8 + 8

Time: 3 Hours

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

The figures in the margin indicate full marks.

Group A

Attempts any two questions

1

Derive the formula for Newton Raphson method. Solve the equation x4 – x – 10 = 0 using Newton Raphson method assuming error precision is 0.01. Discuss drawback for this method.

2

How interpolation differs from regression? Write down algorithm and program for Lagrange interpolation.

3

Why partial pivoting is used with Naive Gauss Elimination method? Solve the following system of equations using Gauss Elimination method with partial pivoting? How Gauss Jordan method differs from Gauss elimination method?

2x + 2y – z = 6

4x + 2y + 3z = 4

x + y + z = 0

Group B

Attempts any eight questions

4

Construct Newton’s forward difference table for the given data points and approximate the value of f(x) at x = 12.

 x 10 20 30 40 50 F(x) 0.173 0.342 0.5 0.643 0.766
5

How Simpson’s 1/3 rule differs from Trapezoidal rule? Drive the formula for Simson’s 1/3 rule.

6

Solve $$\frac{dy}{dx} = 3xy$$; y(0) = 1 for y(0.3) taking h = 0.1 by Euler’s method.

7

A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at $$100^{0}c$$, right side at $$200^{0}c$$. Upper part at $$50^{0}c$$, and lower at $$150^{0}c$$. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

8

Fit the curve $$y = ae^{bx}$$ through the following data points.

 x 0 1 3 5 7 9 y 1 0.891 0.708 0.563 0.447 0.355
9

How Doolittle LU decomposition differs from Crout’s Decomposition? Factorize $$\begin{bmatrix}3 & 2 & 1\\ 2 & 3 & 2\\ 1 & 2 & 3\end{bmatrix}$$ by Crout’s method.

10

Solve a boundary value problem by using shooting method $$\frac{d^2y}{dx^2} = 6x$$, y(1) = 2, y(2) = 9

11

Evaluate the following integration using Romberg integration.

$$\int_{0}^{1} \frac{sinx}{x} dx$$

12

Solve the non linear equation logx – cosx = 0 by using false position method correct to four decimal places.