Find a positive real root of x -cos x 0 by bisection method, correct to 3 decimal places between 0 and 1.

Using Newton-Raphson method, find the real root of the equation 3x cos x correct to 4 decimal places.

2. Use Gauss Elimination to solve the following system of equations

2x Y z 10
3x 2y 3z 18
x 4y 9z 16

3. Solve the equations

2x-y 3z 16
3x+ y-z
by the method of LU decomposition.

4. The population of a town (in thousands) was as given below. Estimate the population for the year 1895 using Newton Forward Interpolation Formula.

<img src='./qimages/10298-4.jpg'>

5. Find the cubic Lagrange's interpolating polynomial from the following data:

<img src='./qimages/10298-5.jpg'>

6. Determine the largest eigen value and the corresponding eigen vector of the matrix

<img src='./qimages/10298-6.jpg'>

using Power method.

7. Find the value of y(1·1) using Runge-Kutta method of fourth order given that dy/dx y^2 xy take h 0·05.

8. Evaluate

<img src='./qimages/10298-8.jpg'>

using

Simpson's 1/3rd rule taking h

Simpson's 3/8 rule taking h 1/6

Hence compute an approximate value of n in each case.

Use Euler's method to obtain an approximate value of y (0·4) for the equation dy/dx with h =0·1.

Discuss the salient features of the standard form of a linear programming problem with suitable examples.

10.(a) Explain the following terms:

Fixed point numbers

Floating point numbers

Explain the features of unimodal functions with suitable examples.