# B Tech 3rd Semester 2007

## Numerical Methods And Programming

GROUP-A ( Multiple Choice Type Questions)

1. Choose the correct alternatives for any ten of the following :

1) The no. of significant digits in 1*00234 is

a)4                                                         b) 6

c) 3                                                           d) 5.

2.  IQ Which of the following relations is / are true ?

a) A . V = A – V                                                          b) A . V = A + V

c) A . V = A / V                                                         d) all of these.

3. 1U) The output of the following program will be : #include<stdio.h> main(){int i = 0, x = 0 ; while (i < 0) { if (i%5 = = 0 ) { x + = i;} ++ i; } printft “\nx = %d”, x) ;}

a) 25                                                  b) 30

c)35                                                   d) none of these.

iv) The degree of precision of Trapezoidal rule is

a)1                                                           b) 2

c) 3                                                          d) 4.

v) Which of the following methods is an iterative method ?

a) Gauss Elimination Method

c) Gauss-Jacobi Method

vi) Method of Bisection is

a) conditionally convergent

c) non-conveigent

b) Gauss-Jordan Method

d) Crout’s Method.

b) always Convergent

d) none of these.

vii) Which of the following relations is true ?

a) E – 1 + A                                                      b)

c)E = 1/ A<flE- 1 – A                 d)None of these.

viii) Regula-Falsi Method is used to

a) find the root of a system of linear simultaneous equations

b) differentiate

c) find the root of an algebraic or transcendental equation

d) solve linear differential equations.

ix} The value of a) 3x2 c) 6x2

A2 x is b)d)6×6.

x) The order of h in the error expression of Simpson’s l/3rd rule is

a) 2                                                                   b) 4

c) 3                                                                    d) 5.

xl) When Gauss Elimination method is used to solve AX = B, A is transformed to a

a) null matrix                                               c) identity matrix _

b) upper triangular matrix                       d) diagonally dominant matrix.

Mi) If = x + y and y ( 1 ) = 0, then y ( 1.1) according to Euler’s method is ( h = 0.1 J.

a) 0.1                       c) 0.5

b)0.3                        d)0.9

GROUP -B ( Short Answer Type Questions )

Answer any three of the following. Given the following table, find f(x) and hence find /( 6 )

 x : 0 1 2 3 4 5 f(x): 41 43 47 53 61 71

3. The values of sin x are given below, for different values of x. Form a difference table and from this table find the sin 32°.

 x : 30° 35° O o 45° 50° 55° y m sin x : 0-5000 0-5736 0-6428 0-7071 0-7660 0-8192

7. What are subscripts ? How are they written ? What restrictions apply to the values that can be assigned to subscripts ? Evaluate V12 to three places of decimals by Newton-Raphson method.Find a root of the equation x3 – 3x- 5 = 0 by the method of false position.

Find A ~ 1 , if A =

GROUP -C ( Long Answer Type Questions )

Answer any three of the following questions.

8.a) Find by the method of fixed point iteration the root of x2 – 6x + 2 * 0, which Ilea between 5 and 6 correct upto four significant figures.

b)  Given ^ ^ ~ * with lntial condition y = 1 at x = 0, find y for x = 0-1 by ax y t x

Euler’s method, correct upto 4 decimal places, taking step length h = 0-02.

9. a) Solve the following system of linear equations by Gauss-Jordan elimination

method : – 5x , – x 2 = 9 -xj+5x2-x3 = 4 -x2 + 5x3 = -6 L 1 f x

b) Calculate by Simpson’s ^ rule, the value of the Integral I j + x dx-‘ correct

upto three significant figures by taking six Intervals.                                 10 + 5

10.   a) Solve the following system of equations by LU-factorization method :

8x j – 3x2 + 2x3 = 20 ; 4x j + 1 lx2 – x3 = 33 : 6x l + 3x2 + 12x3 = 36.

b)  Using Gauss-Seidel method, find the solution of the foUowlng system of the linear equations correct upto 2 place of decimal. 3x+y + 5z =13, 5x- 2y + z = 4, x + 6y – 2z = – 1.  8 + 7

11.   a) Find /( 0-9 ) by using Newton divided difference formula. Given

 x ; 0 1 2 4 fix): 5 14 41 98

b) Estimate the missing values from the following table :

 x ; 1 3 5 7 9 11 V : 2 ? 27 64 ? 216

State the necessary assumption.

C)

 x : 10 11 1-2 1-3 1-4 y(x): 7-989 8-403 8-781 9-129 9-451

12.a) Solve the equation ^ = x2 + y 2 : y ( 0 ) = 1. for x = 01 by using Runge- Kutta 4th order method and find the solution correct upto 4 place of dlclmal.

( h = 0 05 )

b)  Find the solution of the following differential equation by Euler s method for x = 1. by taking h = 0-2, ^ = xy, with y = 1 when x = 0.

c)   Using Taylor’s series method solve ^ = 1 + xy with y ( 0 ) = 2. Fin’1 l b’ nlue of y ( 0-2 ). 6 + 5 + 4

13. a) Write a program in C to sove the equation x3-x-4 = 0 within ( 1. 2 ) by Bisection method, correct upto 3 place of decimals.

b)  Solve the equation jjj* = x + y with intial condition y ( 0 ) = 10 and h = 0-1. using predictor-corrector method, to find y ( 0-2 ).

c)   Write a program in C using recursive function to calculate the sum of all digits of any number.