# WBUT Question Papers EC 3rd Semester Numerical Methods And Programming 2008

WBUT Question Papers Electronics Communication

B Tech 3rd Semester 2008

Numerical Methods And Programming

GROUP – A ( Multiple Choice Type Questions)

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

i) Which of the following relations is true ?

a) E = 1 + A                                                                b) E = 1 – A

c) E = 1/A                                                                   d) None of these.

ii)————————– By evaluating 2 by a numerical integration method, we can obtain an approximate value of

a) loge 2                                                                      b) n

c) e                                                                               d) log102.

iii) if a be the actual value and e be its estimated value, the formula for relative error is

a) a                                                                         b) J-5L=JLL

c) ea                                                                       d) -L^-

iv) in Trapezoidal rule, the portion of curve is replaced by

a) straight line                                                           b) circular path

c) parabolic path                                                      d) none of these.

v) The error Involved In 4111 order R-k method is given by

a) O (h2)                                                                     b) 0(h4)

c) O (h3)                                                                      d) 0(h5).

vi) An n x n matrix A is said to be diagonally dominant if

a)J= 1

b) J= ii56 J fl

C) | a(t | > X| a ij |

J= 1 < *J

d) ij J= 1

vii)               Find the output of the following program main() { char a, b , a = ‘b’ ; b = a ; printf( “b = %c\n”, b ) ;

a) a

c) garbage value

b) b

d) none of these.

Vlll) Lagrange’s interpolation formula is used for

a) equispaced arguments only    b) unequispaced arguments only

c) both equispaced and unequispaced arguments

d) none of these.

ix) j If /( 3 ) = 5 and /( 5 ) = 3, then the linear interpolation function f ( x) is

xa) /( x) = 8 – x c) /(x)=x2 If / ( x) = ^ , the divided difference [ a, b, c ] is

b)f(x) = 8 + x /(x)=x + x2 + 8.

a)

c)

1

a + b + c 1

a2+b2

b)

d)

abc

a+b-c’

If ^ = x + y and y ( 1 ) = 0, then y ( 1.1) according to Euler’s method is | h = 0-1 ]

a) 01 c) 0-5

b) 0-3 d) 0-9.

xii) Which one of the following results is correct ?

a] Axn – rxx

c) Anex = ex

d) A cos x = – sin x.

xiii) In the method of iteration the function ( x) must satisfy

a) | <t>’ ( x) | < 1 c) | <t> ‘( x) | = 1

b) [ <t>'(x) | > 1 d) | <fr'( x) | = 2.

xiv) The inherent error for Simpson’s ^ rd rule of integration is as (the notations have their usual meanings )

31 – iso-HM                                                                    b) ” T5o(*o)

c) – ~j2~ f” ( x o )                                                        d) none of these.

xv) ( A – V ) x2 is equal to (the notations have their usual meanings

a) h2                                                                               2h2

c) 2 h2                                                                            d) none of these.

GROUP -B ( Short Answer Type Questions )

Answer any three of the following.                          3×5= 15

2. From the following table And the values of / { 12 ) by Newton’s divided difference interpolation formula :

 x : 11 13 14 18 19 21 f(x): 1342 2210 2758 5850 6878 9282

3. Solve the following system by Matrix Inversion Method : 2x + y + z = 10 3x + 2y + 3z = 18 x + 4y + 9z = 16.

4. a) Evaluate the missing terms In the following table :

 x : 0 1 2 3 4 5 fix): 0 — 8 15 — 35

5. What is ternary operator ? Give an example.

Solve by Taylor’s series method = 2x + 3y 2 given y = 0 when x = 0 at x = 02.

Using Euler’s method obtain the solution of ^ = x – y. with y ( 0 ) = 1 and

h = 0-2 at x = 0-4.

6. Find the first approximation of the root lying between 0 and 1 of the equation x3 + 3x – 1 = Oby Newton:Raphson formula.

 x : 0 1 2 3 4 fix): 1 1 15 40 85

GROUP -C ( Long Answer Type Questions )

Answer any three of the following questions.

8. a) From the following table, estimate the number of students who obtained marks

between 40 and 45 :

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

10. a) b)

 X : 4 5 7 10 11 13 f(X): 48 100 294 900 1210 2028

Find the positive real root of x3 = 18 using the bisection method of 4 iterations. Find the root of the equation x3 + x2 + x+ 7 = 0 using Regula Falsi method.

A curve passes through the points as given in the following table. Find the area

 X l 2 3 4 5 6 7 8 9 y 0-2 0-7 1 1-3 15 1-7 1-9 21 2-3

Write a program in C to solve the equation x3-3x-5 = 0 within ( 1, 2 ) by Bisection method correct up to 3 places of decimal.

Write a program in C using recursive function to calculate the sum of all digits of

8 + 7

any number.

11. a) Evaluate J* xex dx by using Trapezoidal rule taking n = 6.

b) Use Lagrange’s interpolation formula to find the value of /( x ) for x – 0. given the following :

 x : – 1 – 2 2 4 f(x): – 1 – 9 11 69

Prove that Newton-Raphson method has a quadratic convergence. Solve the following system of equations by L-U Factorization Method :

X1+X2~X3 = 2

2x t + 3x2 + 5x3 = – 3

3x , + 2x 2 – 3x 3 = 6.                                                                    ’

Solve the following set of equations by Gauss-Seidel method correct to 2 places of decimal :

9x – 2y + z = 50

x + 5y – 3z = 18

– 2x + 2y + 7z = 19.

Write a C program to approximate a real root of the following equation 4Bisection method.Write a C program to interpolate a given function at a specified argument by Lagrange’s interpolation formula.

Find the value of log 2 1/3 from—3 dx using Simpson’s 41 + x >3 c) n = 4. R Calculate the approximate value of J sin x dx by Composite Trapezoidal Rule by using 11 ordinates. Also compare it with the actual value of the integral.