| SESSION | April 2025 |
| PROGRAM | Bachelor of CoMPUTER APPLICATIONS (BCA) |
| SEMESTER | III |
| course CODE & NAME | DCA2101 Computer Oriented Numerical Methods |
Set-I
Q1. Show that
(a)
(b)
Ans1.
(a) δμ = ½(∆ + ∇)
Definitions of Finite Differences:
Let be a function defined at equally spaced points with interval . Then:
- Forward Difference Operator (∆):
- Backward Difference Operator (∇):
- Average
Q2. Solve the system of equations using Gauss Jacobi’s Method:
3x + 20y – z = –18, 2x– 3y + 20z = 25, 20x + y – 2z = 17.
Ans 2.
Given Equations:
3x + 20y – z = –18 2. 2x – 3y + 20z = 25 3. 20x + y – 2z = 17
Step 1: Rearranging equations to isolate each variable
We rewrite each equation to express x, y, z in terms of the other variables:
Equation (1):
Equation (2):
Q3. Fit straight line of the form , to the following data by method of moment
| 2 | 3 | 4 | 5 | |
| 27 | 40 | 55 | 68 |
Ans 3.
To fit a straight line of the form:
using the method of moments, we’ll follow a process that matches the first and second moments of the actual data with the corresponding moments of the fitted line.
Step 1: Given Data
| x | y |
| 2 | 27 |
| 3 | 40 |
| 4 | 55 |
| 5 | 68 |
Let n = 4 (
Set-II
Q4. Apply Gauss forward formula to obtain the value of f(x) at x = 3.5 from the table:
| 1.5 | 2.5 | 3.5 | 4.5 | |
| 8.963 | 24.364 | 66.340 | 180.034 |
Ans 4.
To apply the Gauss Forward Interpolation Formula, we first construct the forward difference table and then apply the formula to find .
Given Table:
| x | f(x) |
| 1.5 | 8.963 |
| 2.5 | 24.364 |
| 3.5 | 66.340 |
| 4.5 | 180.034 |
Let’s denote:
We need to find
Q5. Evaluate using the
- (i) Simpson’s 3/8 Rule
- (ii) Simpson’s 1/3 Rule
- (iii) Trapezoidal Rule
Ans 5.
To evaluate the integral
using Simpson’s 3/8 Rule, Simpson’s 1/3 Rule, and the Trapezoidal Rule, we need to follow numerical integration
Q6. Find the solution for taking interval length 0.1 using Euler’s method to solve: given .
Ans 6.
To solve the differential equation
using Euler’s method with step size and find the solution at , follow the steps below:
Given:
- Differential