# WBUT Question Papers

## Numerical Methods EC B Tech Sem Third 2011-2012

Time Allotted : 3 Hours

Marks ‘■ 70

7Tie figures in the margin indicate full marks.

Candidates are required to give their answers in their own words

as far as practicable.

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 is not a computational error ?

■ ‘ f ‘ ■ a) Truncation error     b) Round-off error

c) Inherent error                            d) None of these.

ii)             Newton-Raphson method fails when

a) /'(*)= 1                       b) /’ ( x) = 0

c) f ( x ) = – 1                               d) /” ( x ) = 0.

CS/B.TECH(BME(N)/ECE(N)/EE(N)/EEE(N)/EIE(N)/ICE(N)/PWE(N))/ SEM-3/M(CS)-301 /2011-12                                               ‘

ili) Finite difference method is used to solve

a)           a system of linear simultaneous equations

b)           a system of non-linear simultaneous equations

c)            partial differential equations

d)           non-linear equations.

iv)         Regula-falsi method has a convergence rate of the order of

a) 2 b) 1-62

c) 1                                        d) none of these.

v)           Gauss-Seidel method for solution of a system of linear simultaneous equations converges if

n

a> I “,( I S X I Oy |

j = l j* i

«.      n

b> I a ,i I > X I a ,J I

i J‘,’

f           J*i                                                           .

c)                      I a,i I / I I = i

d)           none of these.

vi)         Modified Euler’s method has a truncation error of the order of

a) h                                        b) h2

c) hA                                      d) h3.

3003 (N)

CS/B.TECH(BME(N)/ECE(N)/EE(N)/EEE(N)/EIE(N)/ICE(N)/PWE(N))/

SEM-3/M(CS)-301 /2011 -12

vii)         Divided difference interpolation formula can be used for

a)         the tabular values with independent variable unequally spaced

b)           inverse interpolation

c)           both (a) and (b)

d)           none of these.                                                     .

viii)      Truncation error in Simpson’s ^ rc* m^e *s £iven by

a) h4/“’U)ai‘-b . b) hS a-^-b

c)                               h4 /'”U) • ci<^<b

d)               ^ n4

ix)         Which of the following relations is true ?

a)           E – 1 – A, A – V = A V

b)           E = 1 – A, A + V = A V

;

Cl E = 1 + A, iA + V = A V

f *

d) E = 1+A, A-V=AV.

x)           Trapezoidal method can be used to integrate numerically a function represented in tabular form

a)           with odd number of intervals only

b)           with even number of intervals only

c)           both (a) and (b)

d)           none of these.

CS/B.TECH(BME(N]/EGE(N)/EE(N)/EEE(N)/EIE(N)/ICE{N]/PWE(N))/ SEM-3/M(CS)~301/2011-12

xl) Condition of convergence for Euler’s method 0> I 1 + kf'(xt. yt) | < 1

b) | 1 + hf'[xt, y() | < 1 0 |1 +                     y () | > 1                             .

d)       I 1 + hf'[xr y .) | > 1.

xii) Milne’s corrector formula is

h 1

a)   =yn + 3(yl-, +4y^ + 4y’<+1)

ynfi =yn-i + §(y’n-, + 4yln + *yln + 1)

i               4h / i              i . i ■*

U n + i =yn+ 3″(yn-i +4yn+4yn+1)

d)          none of these.

GROUP -B

( Short Answer Type Questions )

% . 1 ■ ‘ ‘ . ’ *

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

1. Given t#e function y = ^ , show that the divided difference of nth order

y[xQ, Xx, X2, … , ^n] = (“l)n/(x0x1x2…xr[)

1. Solve the following system of linear equations by Gauss- Seidel iterative method :

9x + 2y + 3z = – 7 , ■ * x ~ 6y + 2z = – 2

CS/B.TECH(BME(N)/ECE(N)/EE(N)/EEE(N)/EIE(N)/ICE(N]/PWE(N))/

SEM-3/M(CvS)-301 /2011 -12 4. Fit a polynomial to the following table of values using Lagrange interpolation formula :

 x : 0 1 3 4 y : – 12 0 6 12

 Find the value of y when

a)          x = 2

b)          x = 3-5.

1. Find the value of using Newton-Raphson method. Result

t

is required to be corrected up to 4 decimal places.

1. Solve the following equation using bisection method :
2. 3x + sin x – e [1] = 0 Take xQ = 1 and a: ] = 0.

Result is required to be corrected up to 2 decimal places.

GROUP -C ( Long Answer Type Questions )

Answer any three of the following. 3 x 15 = 45

1. a) Derive the orde^r of convergence for Newton-Raphson
2. method.                                                                             5

b)          Solve the following initial value problem using Euler’s method :

^ = x2 + y with y ( 0 ) = 1.

Compute the first 5 steps of the solution with h = 01.

Compare the results ( % relative error ) with those obtained from the exact solution

y = 3ex – x2 – 2x – 2.                                               10

1. a) Prove by the method of Induction :

&m y r =vmyr+

m

b) Use Newton’s formula to find the area of a circle of diameter 98 cm.                                              ‘                                      5

 D ( cm ) : 80 85 90 95 100 A ( cm 2 ) 5026 5674 6362 7088 7854

c)

Derive Lagrange interpolation formula.                    ,            5

Derive the expression for total truncation error associated with Simpson’s ^ rd method.                                                                             8

Evaluate the following integral using trapezoidal method :

2

I = J (l/(*2 + 4)Jdx

Take h = 0125. Hence obtain the value of n.                        7

10. a) Solve the following system of equations using LU factorization method.                                                                 8

3x – y + 2z = 12

x + 2y + 3z = 11

2x – 2y – z = 2.

b)   Find the inverse of the following matrix :                                                    7

8-4 0

– 4 8-4 L 0 – 4 8

5

b)                 Derive Newton’s Backward difference interpolation formula.                                                           5

c)                  Derive 4th order Runge-Kutta formula for solution of initial value problem of ordinary differential equation. 5