=0⋅. find the value of 3 1 ( 19 ) correct to four decimal points by newton-raphson method. 6. find...

4051 [ Turn over

Name : …………………………………………….………………

Roll No. : …………………………………………...……………..

Invigilator’s Signature : ………………………………………..

CS/B.TECH(NEW)(APM/CSE/IT/AUE/CHE/BT/ME

/PE/CE/CT/LT/TT/FT/SEM-4/M(CS)-401/2012

2012

NUMERICAL METHODS Time Allotted : 3 Hours Full Marks : 70

The 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 × 1 = 10

i) If 35 is approximated to 1·6667, then absolute error is

a) 0·000033 b) 0·000043

c) 0·000034 d) none of these.

ii) If 21)(x

xf = then the divided difference ),( baf is

a) 2)()(

abba + b) 2)(

)(ab

ba −

c) 2211

ba− d) 22

1ba −

.



4051 2

iii) The value of ) ( 32

xEΔ is

a) x b) 6x

c) 3x d) 2x .

iv) If yxxy

+=dd and y ( 1 ) = 0, then y ( 1. – 1 ) according to

Euler's method is, 10 ⋅=h ( say ),

a) 0·1 b) 0·3

c) 0·5 d) 0·9.

v) If 20 =y , 41 =y , 82 =y , 324 =y then 3y is equal to

a) 5 b) 15

c) 6 d) 16·5.

vi) The order of h in the error expression of Trapezoidal rule

is

a) 1 b) 2

c) 3 d) 4.

vii) Regula-Falsi method is

a) conditionally convergent

b) linearly convergent

c) divergent

d) none of these.



4051 3 [ Turn over

viii) Pivoting is very much essential because

a) determinant of the coefficient matrix should be

grater than zero

b) pivot element should not have very large value

compared to the elements of the matrix

c) it reduces the possibility of division by zero

d) change of convergence is higher.

ix) Which of the following is true ?

a) ! ) 1 ( +=Δ nxnn b) ! nxnn =Δ

c) 0 =Δ nnx d) nxnn =Δ .

x) An nn × matrix A is said to be diagonally dominant if

a) ∑≠=

≥n

jij

ijii aa1

b) ∑≠=

≤n

jij

ijii aa1

c) ∑≠=

>n

jij

ijii aa1

d) ∑≠=

<n

jij

ijii aa1

.



4051 4

xi) The condition of convergence of Newton-Raphson's

method is

a) 2} )({ |)().(| xfxfxf ′′<′

b) 2} )({ |)().(| xfxfxf ′<′′

c) 2} )({ |)().(| xfxfxf ′′>′

d) 2} )({ |)().(| xfxfxf ′>′′ .

xii) For xyxy=

dd and 2) 0 ( =y , the value of 2k according to

Runge-Kutta method of 2nd order is ( h = 0·2 )

a) 0·1 b) 0·01

c) 0·4 d) 0·04. GROUP – B

( Short Answer Type Questions ) Answer any three of the following. 3 × 5 = 15

2. Given 360 =+ uu , 551 =+ uu , 742 =+ uu . Find 3u , where xv

is a function of x.

3. Using the following table find xy

dd at x = 0 & 1·5.

x : 0 1 2 3

y : 1 2 11 34



4051 5 [ Turn over

4. Solve the following system of equations using Gaussian

elimination method :

4054313432

9

=++=+−

=++

zyxzyx

zyx

5. Find the value of 31) 19 ( correct to four decimal points by

Newton-Raphson method.

6. Find the cubic polynomial by Lagrange's interpolation

formula which takes the following value :

x : 0 4 5 8

f ( x ) : 1 2 1 10

GROUP – C ( Long Answer Type Questions )

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

7. a) Find a root of the equation 0104 =−− xx that lies

between 1 & 2 using Newton-Raphson method correct to

3 places of decimal.

b) Solve the system of equations

722156

8562711054

=++=−+

=++

zyxzyx

zyx

by Gauss-Seidel method. 7 + 8



4051 6

8. a) Solve the following system of equations by

LU-factorization method :

7435324723

=++=++=++

zyxzyx

zyx

b) Using Runge-Kutta method of order 4, final y ( 0·2 )

given that yexy x 23

dd

+= , y ( 0 ) = 0,taking h = 0·1.

7 + 8

9. a) Find the root of the equation 01cos3 =−− xx by

Regula-falsi method, correct to three decimal places.

b) Evaluate ∫π2

0

d cos xx by using (i) Trapezoidal and

(ii) Simpson's 31 rd rule, where h = 15°. 7 + 8

10. a) Compute ) 41 ( ⋅=y by Milne's predictor & corrector's

method from ) (21

dd yx

xy

+= where 5953) 1 ( ⋅=y ,

8333) 11 ( ⋅=⋅y , 0884) 21 ( ⋅=⋅y , 3624) 31 ( ⋅=⋅y .

b) Derive Newton's divided difference formula.



4051 7 [ Turn over

c) Given that ) (logdd

10 yxxy

+= with the initial condition

that y = 1 when x = 0. Find y for x = 0·2 and x = 0·5

using Euler's modified formula. 5 + 5 + 5

11. a) If )(xfy = is a polynomial degree 5 with

0)0(0 == fy , 3)1(1 == fy , 14)2(2 == fy ,

45)3(3 == fy , 84)4(4 == fy , 170)5(5 == fy ,

258)6(6 == fy . It is found that there is one error in

the value of 3y . Find the correct value of 3y .

b) Why implicit method is preferred over explicit method

though it requires more computations ?

c) Show that the rate of convergence in Newton-Raphson

method is quadratic. 8 + 3 + 4

=============

4002 [ Turn over

Name : …………………………………………….………………

Roll No. : …………………………………………...……………..

Invigilator’s Signature : ………………………………………..

CS/B.TECH/NEW/APM/CSE/IT/AUE/CHE/BT/ ME/ PE/CE/CT/LT/TT/FT/SEM-4/M(CS)-401/2013

2013

NUMERICAL METHODS Time Allotted : 3 Hours Full Marks : 70

The 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 × 1 = 10

i) The number of significant figures in 0·03409 is

a) five b) six

c) seven d) four.

ii) The kind of error occurs when π approximated by

3·14 is

a) truncation error b) round-off error

c) inherent error d) relative error.


4002 2

iii) If 12)0( =f , 6)3( =f and 8)4( =f , then the

interpolation function )( xf is

a) 1232 +− xx b) xx 52 −

c) xxx 523 −− d) 1252 +− xx .

iv) Newton-Raphson method for solution of the equation

0)( =xf fails when

a) 1)( =′ xf b) 0)( =′ xf

c) 1)( −=′ xf d) none of these.

v) In Gaussian elimination method, the given system of

equation represented by Ax = B is converted to another

system YUx = where U is

a) diagonal matrix

b) null matrix

c) identity matrix

d) upper triangular matrix.

vi) Error in Weddle method of integration is

a) 0 b) 4

4)(

180Mabh

−−

c) 2

2)(

12Mabh

−− d) 6

6)(

840Mabh

−− .


4002 3 [ Turn over

vii) In Trapezoidal rule, the portion of curve is replaced by

a) straight line b) circular path

c) parabolic path d) none of these.

viii) Which of the following is an iterative method ?

a) Gauss Elimination Method

b) Gauss Jordan Method

c) LU decomposition Method

d) Gauss-Seidel Method.

ix) The number 9·6506531 when rounded-off to 4 places of

decimal will give

a) 9·6506 b) 9·6507

c) 9·6505 d) none of these.

x) 03yΔ may be expressed as

a) 0123 33 yyyy −+− b) 012 2 yyy +−

c) 0123 33 yyyy ++− d) none of these.


4002 4

xi) Which of the following statements applies to the bisection method used for finding roots of functions ?

a) Convergence within a few iteration

b) Guaranteed to work for all continuous functions

c) Is faster than the Newton-Raphson method

d) Requires that there be no error in determining the sign of the function.

xii) Runge-Kutta formula has a truncation error, which is of the order

a) 2h b) 4h

c) 5h d) none of these.

xiii) In finite difference method, 2

2

ddxy is replaced by

a) 211

2

2

h

yyy nnn +−−+ b) 2

11 2

h

yyy nnn −++−

c) h

yyy nnn22 11 +−

−+ d) 211

4

2

h

yyy nnn +− −+ .

GROUP – B

( Short Answer Type Questions )

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

2. a) Show that ⎥⎦

⎤⎢⎣

⎡ Δ+=Δ

)()(1log)(log

xfxfxf .

b) Define forward difference operator Δ and shift operator E. Prove that EE .. Δ=Δ .


4002 5 [ Turn over

3. Find the missing terms in the following table :

x 0 5 10 15 20 25 y 6 10 ? 17 ? 31

4. Evaluate ∫ +

1

021

dxx using Simpson's

31 rd rule taking n = 6.

Hence find the value of π .

5. Using Runge-Kutta method of 4th order solve 22

22

dd

xyxy

xy

+−

=

with 1)0( =y at 20 ⋅=x .

6. Solve the following system of linear equations by Gaussian Elimination method :

18543 =++ zyx , 1382 =+− zyx , 20725 =+− zyx .

GROUP – C ( Long Answer Type Questions )

Answer any three of the following. 3 × 15 = 45 7. a) What do you mean by interpolation ? Derive Newton's

backward interpolation formula. Can you apply this

formula for unequispaced interpolating points ? 7

b) Using Trapezoidal and Simpson's 31 rd rule compute

∫⋅25

4

dlog xxe by taking seven ordinates correct up to four

decimal places. 8

8. a) Find the value of 2 from the following table : 7

x 1·9 2·1 2·3 2·5 2·7

xxf =)( 1·3784 1·4491 1·5166 1·5811 1·6432


4002 6

b) Solve the following system of equations by

LU-factorization method : 8

15243 =++ zyx

1825 =++ zyx

10232 =++ zyx

9. a) Find a root of the equation 21log10 ⋅=xx by the

method of false position correct to three decimal places.

7

b) Find the inverse of the matrix ⎥⎥⎦

⎤

⎢⎢⎣

⎡=

221232123

A by using

Gaussian elimination method. 8

10. a) Apply Milne's method to find )80( ⋅y for the equation

2

dd yx

xy

+= , given that 0)0( =y , 020)20( ⋅=⋅y ,

08050)40( ⋅=⋅y , 18390)60( ⋅=⋅y . 8

b) Evaluate ∫⋅

−

60

021

d

x

x , using Weddle's rule taking

12 equal subintervals. 7


4002 7 [ Turn over

11. a) Using Gauss-Seidel method find the solution of the

following system of linear equations correct up to two

decimal places :

1353 =++ zyx

425 =+− zyx

126 −=−+ zyx 7

b) Using finite difference method solve the boundary value

problem :

01dd

2

2=++ y

xy with 0)0( =y , 0)1( =y . 8

=============

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