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D 1226 (Pages: 3) Name .

Reg. No .

SEVENTH SEMESTER B.TECH. (ENGINEERING) DEGREE EXAMINATION.,DECEMBER 2009

EE 04 705 (B)-NUMERICAL ANALYSIS AND OPTIMIZATION TECHNIQUES

(2004 admissions)

Time: Three Hours Maximum : 100 Marks

Answer all questions.

Section I

1. Find a root of the equation: x) - 4x - 9 =0 • using Bisection method in four stages.

2. Using Newton's forward formula. find the value off(1.6) ifx: 1 1...' 1.8 2'.2

fUc.) : 3.49 4.82 5.96 6.5

:3, Given y I = X - f- and y(O) =O. Determine the value of y{I) by Milne's predictorcorrectormethod

4. Find the real roo t of the equation x loglll x =1.2 by Newton-Raphson method correct

to 3 decimal,places,

5. Whatis a dual problem of an LPP? Write the dual of the fuUowing LPP:

Maximize z =lOx. + J3?'l +19x) subject to the constraints

6x . +5xl +Jx) s 26, 4x. +2x1 +5x) 5" 7 and x" xl'x) ~ 0

6, What is an artificial variable? What is the use of it in LPP?

7. Explain,the.difference between transportation problem and assignment problem.

8. What are the basic features of a dy;iamic'programming problem?

. [ 8 x 5 = 40 m ks ]

Section II

l. (a) Use Lagrange's interpolation formula to find the value ofy when x=lO from thefollowing data.

(7 marks) -Turn over

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(8 marks)

2

ib ) By Relaxation method, solve the system of equations:·.)x--2y+z=50, x +5y -3z=18. -2x+2y +7z=19 .

Or

2. (a) Find the real root of the equation Xl - 2x - 5 =0 that lies between 2 and 3 by

Regula-falsi method correct to 3 decimal places. (8 marks)

D 1226

(b) Solve the following system of equations by Crout's method:2x-3y+IOz=3, - x + 4y + 2z = 20, 5x+2y+z=-12. (7 marks)

3. From the following table, find the value of x for which f(x) is maximum in the

given range of x. Also find the maximum value of «X>.

x 9 10 11 12 13 14-

{(x) 1330 -1340 1320 1250 1120 930

(15 marks)

Or

4. a) Empioy~ method to obtain correct to three decimal places, solution forthe diffe~p.tialequation y' = y2+X2 for x = 0.4 given that y(O)=O. (7 marks)

b) Find the value ofy(O.2> using Runge-Kutta method of fourth order given thaty' = Y- x d yeO) =2 taking h=O.l. (8 marks)

5. A firm manufactures three types of products, A, Band C. The profits are Rs.3,

Rs.2 and Re.4 respectively. The firm has two machines Ma and M! and below is the

required processing time in minutes fOr each machine for each product.

Product

A B c

Machine

Machines M. and Ms have 2000 and 2500 machine-minutes respectively. The firm

must manufacture 100 units of product A, 200 units of product Band 50 units ofproduct C but Dot more than 150 units of A. Formulate an LPP to maximize theprofit. (15 marks)

Or



(). Solve the LPP by simplex method:

Maximize z =6x, +8x 2 subject to the constraints

5X,+IOX1~60, 4x 1 4X1~30 and XI'X!~O

i. olve the following transportation problem:

Destination

(Is mark)

n 1226

Source IIIII

A2117326

B

16

18

27

10

c251418

12

D

13234115

11

13

19

43

Availability

Requirement (15 marks)

Or

8. Four jobs are to be done on' four different machines. The cost Un rupees) of

performing i-th job on the j -th machine is given in the table below. Assign the jobs

to different machines so as to minimize the total cost.

Machine

J s ·

IS 11 13 15

17 12 12 13

14 15 10 14

16 13 11 17

(15 marks)

( 4 )( 15 =60 marks )



C 56404 (Pages: 3) Name .

Reg. No .

SEVENTH SEMESTER B.TECH. (ENGINEERING) DEGREE EXAMINATIONJUNE 2009

.EE 04 705 (B)-NUMERICAL ANALYSIS AND OPrIMIZATION TECHNIQUES

(2004 admissions)

Time: Three Hours

Answer all questions.

Maximum: 100 Marks

Part A

I. (a) Find the real root of the following equations correct to three significant figures by Regula falsi

method xa - x2 - 2 = 0 .

(b) Using Gauss-Seidel method, solve the following systems of equations, tabulating the results

upto the 5th iteration

lOx - 2y + z = 12

x +9y- z =10

2x - y + Llz = 20

, , 2(c) Using the trapezoidal rule, evaluate J y dx from the following table :

0.6

X: 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y: 1.23 1.58 2.03 4.32 6.25 8.36 10.23 12.45

(d) Given dy = x 2 + y, y(O) = 1, determine y(0.02), y(0.04) and y(0.06) using Euler's modifieddx

method.

(e) Use simplex method to

maximize z = 2xI - x2 + xa subject to constraints

3xI + x2 + xa :5 60, 2xI - x2 + 2xa :5 10

Xl + x2 - xa :5 20, and Xl' X2' Xa ~ O.

(f) Use dual simplex method to solve the Linear Programming problem

minimize z = Xl +~ + 3xa subject to the constraints Xl - ~ + xa~ 4 ; Xl + x2 + 2xa :58, x2 - xa ~ 2,

Xl' X2' Xa~ O.

Turnover

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2

(g) Solve the following transportation problem:

To

From A B C Avail

I 50 30 220 I

II 90 45 170 3

III 250 200 50 4

Requirement 4 2 2

(h) Solve the following assignment problems:

M 1 M2 Ma M 4

J [5 8 3

1~]1 10 7 5

J 210 12J 4

a 8 6 9J 4

C 56404

(8 x 5 =40 marks)

Part B

II. (a) Use the method of iteration to solve the equations:

(ii) cosx + 3 =2x.Or

(b) Solve the equations:

13x + 5y - 3z + U = 18

2x + 12y + z - 4u = 13

3x - 4y + 10z + U =29

2x + y - 3z '+ 9u = 31

by (i) Jacobi method proceeding upto 9 iterations; (ii) Gauss-Seidel method proceeding upto6 iterations. -

Takedx2

J XIII. (a) Compute the value of loge 2 using Simpson's rule from the formulae loge

4 equal intervals in (1, 2). Compare the value with that obtained by applying Trapezoidalrule.

Or



3

(b) Using Milne's predictor and corrector formulae, find y(4,4) given:

5xy' + l - 2 =0, y(4) =1, y(4.1) =1.0049, y (4.2) =1.0097 and y(4.3) =1.0143.

C 56404

IV. (a) Use simplex method to solve the linear programming problem. Maximize z = 3x1 + 2x subject

to the constraints:

(1'1') Xl + x2 s 6, 2 6 0Xl + x2 ~ ,Xl' X 2 ~ •

Or

(b) Use dual simplex method to solve the LPP.

Minimize z =Xl + 2x2 + 3xa subject to the constraints

Xl - x2 + xa ~ 4, Xl - x2 + 2xa :5: 8,

X 2 - Xs ~ 2, xl' x2 ' Xs ~ O.

V. (a) Solve the following assignment problems:

A B C D A B C D

(i)I [ 1

4 6

n(ii) lO 25 15 20]II 9 7 10 2 15 30 5 15

III 4 5 11 3 35 20 12 24 .

IV 8 7 8 4 17 25 24 20

Or

(b) Use dynamic programming to fmd the value of:

Maximum Z =Yl' Yz' Ya subject to the constraints Y1 + Y2 + Ya =5, Yl' Yz' Ya ~ 0

Minimize z =Y1 + Yz + Ya + ... + Yn subject to the constraints Yl' Yz ... Yn =d Yj ~ 0

for j =1, 2, ... n.(4 x 15 = 60 marks)

-

h, . a ' ' ~-- ~~---_ ..



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