BMTE-144 English Medium Solved Assignment 2026

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BMTE-144 English Medium Solved Assignment 2026 Available

1. State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.

a) The equation x ^ 3 – 4x – 16 = 0 has not root in the interval [3,5].

b) Delta*E = E*Delta where Eis the shift operator and A is the forward difference operator.

c) Every 3 * 3 system of linear equations can be solved using the LU decomposition method.

d) For the data (2,4), (1,5), (3,6) the Newton’s divided difference f[x_{0}, x_{1}, x_{2}] * is * 3/2

e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.

2 a) Find the missing values in the following table:

x012345

y02-18 90

b) Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP y(1) = 2 with h = 0.2 (5) d/dx (y) = xy

3. a) Find the approximate root of the equation 2x ^ 3 = 3x + 6 using Newton-Raphson method. Perform only 3 iterations with x_{0} = 2 (3)

b) The roots of the quadratic equation x ^ 2 + ax + b = 0 are given by a and B. Show that the iteration will converge near when laι I >1 beta1 x = alpha (4) x k + 1 = -(ax k +b) x k

c) If delta^ 2 f(x 0 )=C 2 h^ 2 f^ prime prime (x_{0}) + C_{2} * h ^ 4 * f ^ (4) * (x_{0}) +*, find the values of C, and (3) c_{2}

The Gauss-Seidel method is used to solve the system of equations Determine the rate of convergence of the method. [[4, 0, 2], [0, 5, 2], [5, 4, 10]] * [[x_{1}], [x_{2}], [x_{2}]] = [[4], [- 3], [2]]

b) Find the interpolating polynomial by Newton’s divided difference formula for the following data: (3) 0 y1125

c) Using synthetic division method, show that 2 is a simple root of the equation

p(x) = x ^ 4 – 2x ^ 3 + x ^ 2 – x – 2 = 0

5. a) Obtain the interpolating polynomial in simplest form which fits the following data:

X -1012 f(x) 3-45-6

b) Prove that mu ^ 2 = 1 + (delta ^ 2)/4

c) Determine the order of convergence of the iterative method for finding a simple root of the equation f(x) = 0 x n = t = x n-t f(x n )-x n f(x n-1 ) f(x n )-f(x n-1 )

6. a) Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the (161 matrix 120 using the power method. Take (1, 0, o )^ T as the initial approximation 003 and perform 4 iterations. (5)

b) The method where converges to N ^ (1/3) Find the rate of convergence of the method. x n + 1 = 1 9 [5x n + 5N x n ^ 2 – N^ 2 x n ^ 3 ]. a = 0, 1, 2

7. a) Evaluate by using trapezoidal rule with h = 0.5 and h = 0.25 Use Romber’s method to find the best value of π. integrate 1/(1 + x ^ 2) dx from 0 to 1

b) Estimate the eigenvalues of the matrix using the Gerschgorin bounds. [[1, – 1, 2], [- 1, 1, 2], [2, 2, – 2]]

8.a) Solve the initial value problem using Euler method y’ = 1/(x ^ 2 – 3y), y(3) = 2

Find taking h = 0.1 y(3.1)

b) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations