BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2015

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1 No. of Printed Pages : 12 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2015 MTE-12 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 50 (Weightage : 70%) Note : Question no. 1 is compulsory. Answer any four questions out of questions no. 2 to 7. Use of calculators is not allowed. 1. Which, of the following statements are true and which are false? Give a short proof or a counter-example in support of your answer. 10 (a) In a two-dimensional LPP solution, the objective ftinction can assume the same value at two distinct extreme points. (b) Both the primal and dual of an LPP can be infeasible. (c) An unrestricted primal variable converts into an equality dual constraint. (d) In a two-person zero-sum game, if the optimal solution requires one player to use a pure strategy, the other player must do the same. MTE-12 1 P.T.O.

2 (e) If 10 is added to each entry of a row in the cost matrix of an assignment problem, then the total cost of an optimal assignment for the changed cost matrix will also increase by (a) Solve the following linear programming problem using simplex method : Maximize z = 3x 1 + 5x 2 + 4x 3 subject to 2x 1 + 3x 2 < 8 2x 2 + 5x 3 - < 10 3x 1 + 2x 2 + 4x 3 < 15 x1, x2 x3 > 0 1, 2 3 (b) Using the principle of dominance, reduce the size of the following game : Hence solve the game (a) Find all basic feasible solutions for the following set of equations : 2x1 + 6x2 + 2x3 + x4 = 3 6x 1 + 4x 2 + 4x 3 + 6x4 = 2 x1, x2, x3, x4 >0 1' 2' 3' 4 MTE-12 2

3 (b) Examine convexity of the following sets : 4 (i) S1 = ((xi, X2) E R2 14x1 3X2 < 6, xi + x2 > 1} (ii) S2 = {(x, E R2 I x2 + y2 1} 4. (a) Solve the following linear programming problem by graphical method : 5 Maximize z = 5x 1 + 7x 2 subject tox1 x2 4 3x 1 + 8x 2 < x1 + 7x2 35 x x > 0 1, 2 (b) Find the dual of the following LPP : Minimize z = x1 + x2 + subject to x 1 3x 2 + 4x 3 = 5 2x2 3 2x 2 x 3 >- 4 x1, x2 0 and x3 is unrestricted in sign. MTE-12 3 P.T.O.

4 5. (a) Find the initial basic feasible solution of the following transportation problem using matrix-minima method : Destinations I II III Supply A Sources B C D Demand Also, find the optimal solution. 5 (b) Solve the following game graphically : 5 Player 'B' B i B 2 I 2 7 Player 'A' II 3 5 III 11 2 MTE-12 4

5 6. (a) A firm manufactures two types of products, A and B, and sells them at a profit of 2 on type A and 3 on type B. Each product is processed on two machines M1 and M2. Type A requires one minute of processing time on M1 and two minutes on M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M 2 is available for 10 hours during any working day. Formulate the problem as LPP. (b) Solve the following assignment problem : A (a) The following table is obtained in the intermediate stage while solving an LPP by simplex method : C.' s B CB X1 x..2 x.3 S1 S2 R.H.S. S 1 X /2 9/2 5/ /2 1/4 Check whether an optimal solution of the LPP will exist or not. 31/2 MTE-12 5 P.T.O. 7/4

6 (b) Write the LPP model of the following transportation problem : (c) Find the range of values of p and q which will render the entry (2, 2), a saddle point for the following game : Player B Player A MTE-12 6

7 RT121 : 2 iu2 37-1W-d-AT : 50 (25ci : 70%) Vie TIR W. 1 ow 31-4Tel / YRV ri edi WNW W47 / cle-35de<7 TrenTT 1,4 4? arigia * I 1. PHRiRsid w?..tri 4 Vlq-14 w2r4 TIM t arrm? 3q Tifk-Tqf 344A ZIT 51c-gcWul TKT *If7 10 LPP * ti,(1 1 ITR t atwr-ati 1,11-94 Trgrrq t (ii) 1i LPP * atru 3 acfl aritru t t (Tr) i atsif-difira atru -RT 4 Ago 7FaT t I (174) it-cer6 -RFT! (.1 4, zrr ITZ W-41arm w cnoi t, 141 ail cntil -rr- MTE-12 7 P.T.O.

8 (;) qr< -ft-zr-dq TITR7zrr tft at 'l 311 Giccs) w4-31-rro 14-4-d.iT Sri1: r4 10 Aim t 2. () rhrr,rnd isict, rr z = 3x + 5x + 1 4x3 2 ~Ff atrwalft-kui *1177 cf) 2x1 + 3x2 8 2x 2 + 5x 3 < 10 3x 1 + 2x 2 + 4x 3 < 15 x1, x2, x > 0 1' 2' 3 (13.) 3piu-d-r wr,14,t rii-irirsid wr 31TITT7 ITUF : sicnik () tsid atftlift Tirru trr-a-r : 2x 1 + 6x 2 + 2x 3 + x 4 3 6X1 + 4X2 + 4x3 + 63(4 = 2 x1, x2, x3, x>0 1, 2' 3' x4 MTE-12 8

9 T11:1-11. ardr54at q.1-4 Q.N7 : 4 (i) S1 = ((xi, x2) ER2 14x1 + 3x2 6, xi + x2 1) (ii) S2 = ((x, y) ER2 x2 + y2 1) 4. () PHICiRgo trqff xi 91H4 TFRAT t mrati fart z = 5x1 + 7x2 falftr* difr+7,4 trft7 (t) xi +x2 5_4 3x1 + 8x xi + 7x2 35 xi, x2 >_0 LPP Ti c tt-4r : z = xl + x2 + x3 -Tic! chtul ch x1 3x2 + 4x3 = 5 xi- 2x x 2 x 3 > 4 xi, x2 >_ o at x 3 RIG ated-ifiral I MTE-12 9 P.T.O.

10 5. () att&z-f-d1:1 f r 1 I.t)( (go trit-4-f Tri:R:Err Trffr ritt amft t-471ff Tff 4=tṟ4R : 41d I II III 1:0 A ti i B C D T *acii4 Tff trr ftitṟe%rff fart a. *=tr-a-r : 5 %fall `B' B 1 B2 I 2 7 ft-drgt 'A' II 3 5 III 11 2 MTE-12 10

11 6. ()R- TF4A*Bt3r*IK* 3Nic 44Tet t, mcnik A TR 2 * sichit B 3 * t I saeb dm< t M1 AK m2 Tu etit t I A mcnik *.30-11c Wqr4 31 M2 * 2 gme. r1jil 4I B mcbit * drii< t 61-0 * r1 4.1d AK M2 wr RA WfẔi Raki Hk1.1 M H aftr iii Tel.t.1616 H411 1 M2 10 rm -61 ucrorig t I TFR;ErT LPP * 4tr7R (1:4) PiHR1RSICI PticH err W-A-R : A B C D E () TTT fart. 1T1 LPP cht itteiadt air > srm -rra-wr 4 6 Ci'S B CB X1 X2 X3 S1 S2 R.H.S. S 1 X /2 9/2 5/ /2 1/4 31/2 7/4 W-4R i LPP * atr -( t zrr Ter l 3 MTE P.T.O.

12 (3) rhrmrno tift4f TPTFET LPP W:t17 : (T) p aftt qukftgrr Tff *1-F-A7 ci i 31c1 (2, 2) tirtut : 4 R 31-T11 B r1 A p 6 MTE ,500

1. Which of the following statements are true and which are false? Give reasons for your answers. 10 (a)

1. Which of the following statements are true and which are false? Give reasons for your answers. 10 (a) No. of Printed Pages : 12 01097 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2016 ELECTIVE COURSE : MATHEMATICS MTE-1 2 : LINEAR PROGRAMMING MTE-12 Time : 2 hours Maximum Marks : 50 (Weightage