DCA2101 COMPUTER ORIENTED NUMERICAL METHODS APRIL 2025

Description

 

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 (∇):

 

Its Half solved only

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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 (number of observations)

Step 2: Calculate Required Summations

 

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:

 

 

 

Q5. Evaluate  using the

• (i) Simpson’s 3/8 Rule
• (ii) Simpson’s 1/3 Rule
• 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 steps with a chosen number of sub-intervals n.

 

Step 1: Function and Interval

Let:

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 equation:
• Initial condition:
• Step size: