IGNOU MTM 10 SOLVED ASSIGNMENT

Included:

MTE 10 (January 2025 - July 2025) - ENGLISH

Assignment

Course Code: MTE-10

Assignment Code: MTE-10/TMA/2025

Maximum Marks: 100

1. a) The equation   has a positive root in the interval ]1, 2[. Write a fixed point iteration method and show that it converges. Starting with initial approximation x = 1.5 find the root of the equation correct to three decimal places.

b) Find an appropriate root of   with 10^-5 accuracy by

i) Newton Raphson Method

ii) Secant Method

What conclusions can you draw from here about the two methods?

2.a) Using Maclaurin’s expansion for sin x , find the approximate value of 4 sin  with the error bound 5 10 ^-5

b) Find an approximate value of the positive real root of xe ^x= 1 using graphical method. Use this value to find the positive real root of xe^x = 1 correct to three decimal places by fixed point iteration method.

c) Using x _o= 0 find an approximation to one of the zeros of x³-4x+1=0 by using Birge-Vieta Method. Perform two iterations.

3.a) Solve the system of equations

using Gauss elimination method with pivoting.

b) Find the inverse of the matrix   using Gauss Jordan Method.

c) Solve the following linear system Ax = b of equations with partial pivoting

Store the multipliers and also write the pivoting vectors.

4.a) Solve the system of equations

 by using the Gauss Jacboi and Gauss Seidel method. The exact solution of the system is   Perform the required number of iterations so that the same accuracy is obtained by both the methods. What conclusions can you draw from the results obtained?

b) Starting with  find the dominant eigenvalue and corresponding eigenvector for

the matrix using the power method.

5. a) The solution of the system of equations is attempted by the Gauss

Jacobi and Gauss Seidel iteration schemes. Set up the two schemes in matrix form. Will the iteration schemes converge? Justify your answer.

b) Obtain an approximate value of  using composite Simpson’s rule with h = 0.25 and

h = 0.125  Find also the improved value using Romberg integration.

c) Find the minimum number of intervals required to evaluate   with an accuracy of

by using the Trapezoidal rule.

a) From the following table, find the number of students who obtained less than 45 marks.

Marks	No. of Students
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

b) Calculate the third-degree Taylor polynomial about 

c) Use the polynomial in part (a) to approximate and find a bound for the error involved. 

d) Use the polynomial in part (a) to approximate 

a) Using sin( 0.1) = 0.09983 and sin( 0.2) = 0.19867 , find an approximate value of sin( 0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

b) Consider the following data

x	1.0	1.3	1.6	1.9	2.2
f(x	0.7651977	0.6200860	0.4554022	0.2818186	0.110362

Use Stirling’s formula to approximate  

c) Solve the  using R-K method of 0(h⁴ )  with

h = 0.1 and obtain the value of y(0.2) . Also find the error at t = 0.2 , if the exact solution  is   

a) The position f(x) of a particle moving in a line at various times xk is given in the following

table. Estimate the velocity and acceleration of the particle at  x =1.2

b) A solid of revolution is formed by rotating about the x-axis the area bounded between   the curve given by the table   

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule    ii) Simpson’s rule

c) Take 10 figure logarithm to base 10 from by unit increment. Calculate the first derivative of log

 

9. a) For the table of values of x f(x) = xe ^xgiven by

x	1.8	1.9	2.0	2.1	2.2
f(x	10.8894	12.7032	14.7781	17.1489	19.8550

  Find f"(2.0) using the central difference formula of 0(h²) for h = 0.1 and h = 0.2 . Calculate T.E. and actual error

Calculate T.E. and actual error

b) Suppose n f denotes the value of  then find the value of  

c) Use Runge-Kutta method of order four to solve  carry to  x =1.5 with h = 0.1.

d) Find the solution of the difference equation   the particular solution when  

10. a) The iteration method

where N is positive constant, converges to some quantity. Determine this quantity. Also find the rate of convergence of this method.

b) Determine the spacing h in a table of equally spaced values for the function

 so that the quadratic interpolation in this table satisfies

c) Determine a unique polynomial f(x) of degree  3 such that

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