~~Rs.~~
Rs. 50

ignou MTE 10 solved assignment 2024

~~Rs.~~
Rs. 50

Last Date of Submission of IGNOU MTE-010 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU BSC MTE 10 Solved Assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 10
Subject Name	Numerical Analysis
Year	2024
Session	-
Language	English Medium
Assignment Code	MTE-010/Assignmentt-1//2024
Product Description	Assignment of BSC (Bachelor in Science) 2024. Latest MTE 010 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-010 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MTE 10/2024

~~Rs.~~
Rs. 50

Questions Included in this Help Book

Ques 1.

a) Find the largest real root $a$ of $f(x)=x^6-x-1=0$ lying between 1 and 2. Perform three iterations by

i) bisection method

ii) secant method $(x_0=2,x_1=1).$

Ques 2.

b) Find the number of positive and negative roots of the polynomial

$p(x)=x^3-3x^3+4x-5.$ Find $p(2)$ and $p'(2)$ using synthetic division method.

Ques 3.

c) Solve $x^3-9x+1=0$ for the root lying between 2 and 4 by the method of false position. Perform two iterations

Ques 4.

a) Using $x_0=-2$ as an initial approximation find an approximation to one of the zeros of $p(x)=2x^4-3x^2+3x-4$

by using Birge-Vieta method. Perform two iterations.

Ques 5.

b) Find by Newton’s method the roots of the following equations correct to three places of decimals

i) $xlog_10x=4.772393$ near $x=6$

ii) $f(x)=x-2sin \,x=2$

Ques 6.

a) The equation $x^2+ax+b=0$ has two real roots p and q such that $|p|<|q|.$ If we use the fixed point iteration $x_{k+1}=\frac{-b}{x_k+a},$ to find a root then to which root does it converge?

Ques 7.

b) Estimate the eigenvalues of the matrix

$\begin{bmatrix} 1 &-2 &3 \\6 &-13 &18 \\4 &-10 &14 \end{bmatrix}$

using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.

Ques 8.

c) Find the inverse of the matrix

$A=\begin{bmatrix} 1 &-1 & 1\\1 &-2 & 4\\1 &2 &2 \end{bmatrix}$

using Gauss Jordan method.

Ques 9.

a) Solve the system of equations

$0.6x+0.8y+0.1z=1$

$1.1x+0.4y+0.3z=0.2 x+y+2z=0.5$

by LU decomposition method and find the inverse of the coefficient matrix

Ques 10.

b) For the linear system of equations $\begin{bmatrix} 1 &2 &-2 \\1 &1 &1 \\2 &2 &1 \end{bmatrix}\begin{bmatrix} x_1\\x_2 \\x_3 \end{bmatrix}=\begin{bmatrix} 1\\3 \\5 \end{bmatrix}$

set up the Gauss-Jacobi and Gauss-Seidal iteration schemes in matrix form. Also check the convergence of the two schemes.

Ques 11.

a) Find the dominant eigenvalue and the corresponding eigenvector for the matrix

$A=\begin{bmatrix} -4 &14 & 0\\-5 &13 &0 \\-1 &0 &2 \end{bmatrix}$

using five iterations of the power method and taking $y^0=[111]^T$ as the initial vector.

Ques 12.

b) Solve the system of equations

$\bg_white 3x+2y+4z=7$

$2x+y+z=7$

$x+3y+5z=2$

with partial pivoting. Store the multipliers and also write the pivoting vectors.

Ques 13.

a) Determine the constants $a,\beta ,y$ in the differentiation formula $y'(x_0)=ay(x_0-h)+\beta y(x_0)+\gamma y(x_0+h)$ so that the method is of the highest possible order. Find the order and the error term of the method.

Ques 14.

b) The function $f(x)=1n(1+x)$ is to be tabulated at equispaced points in the interval [2, 3] using linear interpolation. Find the largest step size h that can be used so that the error $\leq 5\times 10^{-4}$ in magnitude.

Ques 15.

c) Using finite differences, show that the data

x	-3	-2	-1	0	1	2	3
f(x)	13	7	3	1	1	3	7

represents a second degree polynomial. Obtain this polynomial using interpolation and find f (2.5).

Ques 16.

a) Derive a suitable numerical differentiation formula of $0(h^2)$ to find $f''(2.4)$ with $h=0.1$ given the table

x	0.1	1.2	2.4	3.9
f(x)	3.41	2.68	1.37	-1.48

Ques 17.

b) The position $f(x)$ of a particle moving in a line at various times $x_k$ is given in the following table. Estimate the velocity and acceleration of the particle at $x=1.2.$

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 18.

Take 10 figure logarithm to bases 10 from $x=300$ to $x=310$ by unit increment. Calculate the first derivative of $log_{10}x$ when $x=310.$

Ques 19.

a) Show that $\sqrt{1+\mu ^2\delta ^2}=1+\frac{\delta ^2}{2}$ where $\mu$ and $\delta$ are the average and central differences operators, respectively.

Ques 20.

b) A table of values is to be constructed for the function $f(x)$ given by $=\frac{1}{1+x}$ in the interval [1, 4] with equal step length. Determine the spacing h such that quadratic interpolation gives result with accuracy $1\times 10^{-6}.$

Ques 21.

c) Using the classical R-K method of $0(h^4)$ calculate approximate solution of the IVP, $y'=1-x+4y,y(0)=1$ at $x=0.6,$ taking $h=0.1$ and $0.2.$ Use extrapolation technique to improve the accuracy.

Ques 22.

a) Compute the values of

$1=\int _{0}^2{}\frac{dx}{1+x^2}$

by using the trapezoidal rule with $h=0,50.25,0,125.$ Improve this value by using the Romberg’s method. Compare your result with the true value.

Ques 23.

b) Use modified Euler’s method to find the approximate solution of IVP

$y'=2xy,y(1)=1$ at $x=1.5$ with $h=0.1$

If the exact solution is $y(x)=e^{x^{2-1}},$ find the error.

Ques 24.

c) Show that $u_x=c_1e^a^x+c^2e^{-ax}$ is a solution of the difference equation $u_x+1-2u_xcosh\,a+u_{x-1}=0.$

Ques 25.

a) Using the following table of values, find approximately by Simpson’s rule, the arc ength of the graph $y=\frac{1}{x}$ between the points (1, 1) and $\left ( 5,\frac{1}{5} \right )$