# WBUT Previous Years Question Papers CS

## Numerical Methods And Programming B Tech Third Sem Dec 2008

Time : 3 Hours ]

I Full Marks : 70′

GROUP – A ( Multiple Choice Type Questions)

1. Choose the correct alternatives for any ten of the following :              10 x 1 = 10

i)           Which of the following relations is true ?

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

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

l

f dxii)  By evaluating 2 by a numerical integration method, we can obtain an

J I + X

0

approximate value of

a) loge 2                 b) n

c) e                      d) log102.

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

a) a                   b, J-5L=JLL

^ e                     a

0             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= i

i56 J

fl

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

J= 1 < *J

ij •

J= 1 i*J ■

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.

 /
 IWHlI
 C8/B.TECH (ECE/IT/EE (0)/EEE/ICE)/SEM-3/M(CS)-312/08/(09) 5 vlll) Lagrange’s interpolation formula is used for a) equispaced arguments only
 ixj If /( 3 ) = 5 and /( 5 ) = 3, then the linear interpolation function f ( x) is
 x)

 b) d)
 a) c)
 abc
 a+b-c’

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

 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) | ’ ( x) | < 1 c) | ‘( x) | = 1
 b) [ '(x) | > 1 d) |

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

a) 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   b) – 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.

 x : 0 1 2 3 4 5 fix): 0 — 8 15 — 35
 4. a) Evaluate the missing terms In the following table :

 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.                       3 x 15 = 45

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

 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

5 + 5 + 5

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 numbe

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

o

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. 5 + 5 + 5 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 :

4                                                                                                                                                                    * sin ( x) = e x by Bisection method.                                                                                                                             5 + 5 + 5

Write a C program to interpolate a given function at a specified argument by Lagrange’s interpolation formula.

l

Find the value of log 2 1/3 from

—3 dx using Simpson’s 4

j 1 + x

>3

n = 4.

Calculate the approximate value of J sin x dx by Composite Trapezoidal Rule

0

by using 11 ordinates. Also compare it with the actual value of the integral.

5  + 5 + 5

END