IGNOU BMTE-144 Solved Assignment 2023 | B.Sc (G) CBCS Abstract Classes ®

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

₹101.00

Get Instant Access

Share with your Friends

IGNOU BMTE-144 Assignment Question Paper 2023

Course Code: BMTE-144

Assignment Code: BMTE-144/TMA/2023

Maximum Marks: 100

PART – A (40 marks)

1. a) Find the approximate root of the equation \(2 x^3=3 x+6\) using Newton-Raphson method. Perform only 3 iterations with \(\mathrm{x}_0=2\).

b) The roots of the quadratic equation \(x^2+a x+b=0\) are given by \(\alpha\) and \(\beta\). Show that the iteration \(\mathrm{x}_{\mathrm{k}+1}=\frac{-\left(\mathrm{ax} \mathrm{x}_{\mathrm{k}}+\mathrm{b}\right)}{\mathrm{x}_{\mathrm{k}}}\) will converge near \(\mathrm{x}=\alpha\) when \(|\alpha|>|\beta|\).

c) If \(\delta^2 \mathrm{f}\left(\mathrm{x}_0\right)=\mathrm{C}_1 \mathrm{~h}^2 \mathrm{f}^{\prime \prime}\left(\mathrm{x}_0\right)+\mathrm{C}_2 \mathrm{~h}^4 \mathrm{f}^{(4)}\left(\mathrm{x}_0\right)+\cdots\), find the values of \(\mathrm{C}_1\) and \(\mathrm{C}_2\).

2. a) The Gauss-Seidel method is used to solve the system of equations
\[
\left[\begin{array}{ccc}
4 & 0 & 2 \\
0 & 5 & 2 \\
5 & 4 & 10
\end{array}\right]\left[\begin{array}{l}
x_1 \\
x_2 \\
x_3
\end{array}\right]=\left[\begin{array}{c}
4 \\
-3 \\
2
\end{array}\right]
\]
Determine the rate of convergence of the method.

b) Find the interpolating polynomial by Newton’s divided difference formula for the following data:
\begin{tabular}{|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0 & 1 & 2 & 4 \\
\hline \(\mathrm{y}\) & 1 & 1 & 2 & 5 \\
\hline
\end{tabular}

c) Using synthetic division method, show that 2 is a simple root of the equation
\[
p(x)=x^4-2 x^3+x^2-x-2=0 .
\]

3. a) Using Gauss-Jordan method, find the inverse of the matrix
\[
\left[\begin{array}{ccc}
1 & 1 & 3 \\
1 & 3 & -3 \\
-2 & -4 & -4
\end{array}\right] .
\]

b) Find the largest step length that can be used for constructing a table of values for the function
\[
f(x)=\frac{4}{3} x^3+5 \ln x, 10 \leq x \leq 20,
\]
so that a quadratic interpolation can be used with an accuracy of \(5 \times 10^{-6}\).

4. a) Find the missing values in the following table:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \(\mathrm{y}\) & 0 & 2 & – & 18 & – & 90 \\
\hline
\end{tabular}

b) Using Classical Runge-Kutta fourth order method, find an approximate value of \(\mathrm{y}(1.2)\) for the IVP \(\frac{d y}{d x}=x y, y(1)=2\) with \(h=0.2\).

PART – B (40 marks)

5. a) For the following data, use Gauss backward difference method to obtain the interpolating polynomial \(\mathrm{f}(\mathrm{x})\) :
\begin{tabular}{|c|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\
\hline \(\mathrm{f}(\mathrm{x})\) & 1.40 & 1.56 & 1.76 & 2.00 & 2.28 \\
\hline
\end{tabular}
Hence, find the value of \(\mathrm{f}(0.45)\).

b) The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson’s \(\frac{1}{3}\) rule, find the distance travelled by the vehicle in this hour:
\begin{tabular}{|c|l|l|l|l|l|l|}
\hline \(\mathrm{t}=\) time in \(\mathrm{min}\). & 10 & 20 & 30 & 40 & 50 & 60 \\
\hline \(\mathrm{v}=\) velocity in \(\mathrm{km} / \mathrm{hr}\). & 80 & 60 & 70 & 75 & 70 & 80 \\
\hline
\end{tabular}

6. a) Evaluate \(\int_0^1 \frac{1}{1+\mathrm{x}^2} \mathrm{dx}\) by using trapezoidal rule with \(\mathrm{h}=0.5\) and \(\mathrm{h}=0.25\). Use Romber’s method to find the best value of \(\pi\).

b) Estimate the eigenvalues of the matrix
\[
\left[\begin{array}{ccc}
1 & -1 & 2 \\
-1 & 1 & 2 \\
2 & 2 & -2
\end{array}\right]
\]
using the Gerschgorin bounds.

7. a) Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix \(\left(\begin{array}{lll}1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\) using the power method. Take \((1,0,0)^{\mathrm{T}}\) as the initial approximation and perform 4 iterations.

b) The method
\[
\mathrm{x}_{\mathrm{n}+1}=\frac{1}{9}\left[5 \mathrm{x}_{\mathrm{n}}+\frac{5 \mathrm{~N}}{\mathrm{x}_{\mathrm{n}}^2}-\frac{\mathrm{N}^2}{\mathrm{x}_{\mathrm{n}}^5}\right], \mathrm{n}=0,1,2, \ldots
\]
where \(\mathrm{N}\) is a positive constant, converges to \(\mathrm{N}^{1 / 3}\). Find the rate of convergence of the method.

8. a) Find the inverse of the matrix \(\left[\begin{array}{lll}2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2\end{array}\right]\) using Gauss-Jordan method.

b) Divide the polynomial
\[
x^5-6 x^4+8 x^3+8 x^2+4 x-40
\]
by \((x-3)\) by the synthetic division method and find the remainder.

c) Determine a unique polynomial \(\mathrm{f}(\mathrm{x})\) of degree \(\leq 3\) such that \(\mathrm{f}\left(\mathrm{x}_0\right)=1, \mathrm{f}^{\prime}\left(\mathrm{x}_0\right)=2\), \(\mathrm{f}\left(\mathrm{x}_1\right)=2, \mathrm{f}^{\prime}\left(\mathrm{x}_1\right)=3\), where \(\mathrm{x}_1-\mathrm{x}_0=\mathrm{h}\).

PART – C (20 marks)

9. a) Obtain the interpolating polynomial in simplest form which fits the following data:
\begin{tabular}{|c|c|c|c|c|}
\hline \(\mathrm{x}\) & -1 & 0 & 1 & 2 \\
\hline \(\mathrm{f}(\mathrm{x})\) & 3 & -4 & 5 & -6 \\
\hline
\end{tabular}

b) Prove that \(\mu^2=1+\frac{\delta^2}{4}\).

c) Determine the order of convergence of the iterative method
\[
x_{n+1}=\frac{x_{n-1} f\left(x_n\right)-x_n f\left(x_{n-1}\right)}{f\left(x_n\right)-f\left(x_{n-1}\right)}
\]
for finding a simple root of the equation \(f(x)=0\).

10. a) Solve the initial value problem using Euler method
\[
y^{\prime}=\frac{1}{x^2-3 y}, y(3)=2 .
\]
Find \(\mathrm{y}(3.1)\) taking \(\mathrm{h}=0.1\).

b) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations
\[
\left[\begin{array}{ccc}
1 & 1 & 1 \\
4 & 3 & -1 \\
3 & 5 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
6 \\
4
\end{array}\right] .
\]
Show that the method is convergent and hence find its rate of convergence.

c) Write the error in linear interpolation. Hence, show that
\[
\mid \text { error }\left|\leq \frac{h^2}{8} \max \right| f^{\prime \prime}(x) \mid
\]
where \(\mathrm{h}=\mathrm{x}_1-\mathrm{x}_0, \mathrm{x} \in\left[\mathrm{x}_0, \mathrm{x}_1\right]\).

BMTE-144 Sample Solution 2023

Frequently Asked Questions (FAQs)

How do I access the Complete Solution?

You can access the Complete Solution through our app, which can be downloaded using this link:

App Link

Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Is this Complete Solution for IGNOU Assignments?

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Is the Complete Solution aligned with the IGNOU requirements?

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Is the Complete Solution guaranteed error-free?

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

Can I access the Complete Solution anytime?

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

What if I need help with a specific assignment question?

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

Can I access the educational materials on multiple devices?

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

How do I make a payment for accessing the solutions?

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

What are the instructions for formatting my assignments?

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.

Terms and Conditions

The educational materials provided in the app are the sole property of the app owner and are protected by copyright laws.
Reproduction, distribution, or sale of the educational materials without prior written consent from the app owner is strictly prohibited and may result in legal consequences.
Any attempt to modify, alter, or use the educational materials for commercial purposes is strictly prohibited.
The app owner reserves the right to revoke access to the educational materials at any time without notice for any violation of these terms and conditions.
The app owner is not responsible for any damages or losses resulting from the use of the educational materials.
The app owner reserves the right to modify these terms and conditions at any time without notice.
By accessing and using the app, you agree to abide by these terms and conditions.
Access to the educational materials is limited to one device only. Logging in to the app on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Our educational materials are solely available on our website and application only. Users and students can report the dealing or selling of the copied version of our educational materials by any third party at our email address (abstract4math@gmail.com) or mobile no. (+91-9958288900).

In return, such users/students can expect free our educational materials/assignments and other benefits as a bonafide gesture which will be completely dependent upon our discretion.