# IGNOU BMTE-144 Solved Assignment 2023 | B.Sc (G) CBCS

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

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Details For BMTE-144 Solved Assignment

## 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 $$\mathrm{h}=\mathrm{x}_1-\mathrm{x}_0, \mathrm{x} \in\left[\mathrm{x}_0, \mathrm{x}_1\right]$$.

$$sin\left(\theta +\phi \right)=sin\:\theta \:cos\:\phi +cos\:\theta \:sin\:\phi$$

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