# Numerical Method | BSc.CSIT (TU) Question Paper 2069 | Third Semester

## Third Semester | Second Year | Tribhuvan University Old Question Collection | Question Bank Numerical Method, Year: 2069 Computer Science and Information Technology (CSc 204) Full Marks: 60 | Pass Marks: 24 | Time: 3 hours

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. Derive a formula to solve nonlinear equation using secant method. Using your formula estimate a real root of following nonlinear equation using secant method correct up to decimal places x2 + ln x = 3. (3+5)
2. Estimate f(3) from the following data using Cubic Spline Interpolation. (8)
 x 1 2.5 4 5.7 f(x) -2 4.2 14.4 31.2

OR
Find the best fitting quadratic polynomial from following data using least square approximation.

 x -2 -1.2 0 1 1.2 2.5 3 4.5 6.3 f(x) 10.39 2.96 -2 -2.63 -2.46 0.83 3.1 12.8 30.4
3. (a) For the function ?(?) = $e^x \sqrt{(sin x + ln x)}$ estimate ?′(6.3) and ?”(6.3) [take ℎ = 0.01] (4)
(b) Evaluate $\int_{1}^{2} (ln x + x^2 sinx)dx$ using Gauss integration 3 point formula. (4)
4. Solve the following system of linear equations using Gauss-elimination or Gauss Jordan method.
3x1 + 5x2 – 3x3 + x4 = 16
2x1 + x2 + x3 + 4x4 = 9
43 – 4x2 – x4 = 1
2x1 + x2 – 3x3 + 9x4 = 5                                                                                                                                       (8)
5. How can you solve higher order differential equation? Explain. Solve the following differential within 0 ≤ ? ≤ 1 using Heun’s method. (3+5)
$\frac{d^2 y}{ dx^2} + 3 \frac{d y}{d x}$ + 2xy = 1 with ?(0)=1 and ?′(0)=1 [?ake ℎ=0.5]
6. (a) How can you obtain numerical solution of a partial differential equation? Explain. (3)
(b) The steady state two dimensional heat-flow in a metal plate is defined by $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}$=0. Given the boundary conditions as sown in figure below, find the temperatures at interior points T1, T2, T3 and T4. (5)
7. Write an algorithm and C-program code to solve non-linear equation using Newton’s method. Your program should read an initial guess from keyboard and display the following if the solution is obtained.
• Estimated root of the equation
• Functional value at calculated root
• Required number of iterations                                                                                                                                   (5+7)
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