For what value of k are the following vectors linearly independent?

Check whether the following set is convex:

Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

Find all the basic feasible solutions of the following system of linear equations:

Check if any of them is degenerate solution. Justify your answer.

Two breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.

Find the initial basic feasible solution of the following transportation problem using North-West Corner method. 

A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on Aand B are ₹ 4 and ₹ 3 per belt. The production of each type Arequires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt Arequire a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.

Obtain the dual of the following primal LP problem:

Maximize z = x₁ - 2x₂ + 3x₃

Subject to -2x₁ + x₂ + 3x₃ = 2

                   2x + 3x₂ + 4x₃ =1

                   X₁,X₂X₂ ≥ 0

Reduce the following two person zero sum game to 2× 2 game using principle of dominance. And hence solve the game.

State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.

a) The intersection of finite number of convex sets is not convex.

b) If value of the 2× 2 matrix game  is 4, then p ≥ 4 .

c) If 10 is added to each of the entries of the cost matrix of a 3× 3 assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.

d) For maximization LP model, the simplex method is terminated when all values c_j − z_j ≥ .0

e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

हो जहां 

फलन f(x) = (2+x)⁴, 1 ≤ x ≤ 2 के समदूरी मानों की तालिका से एक ऐसा अंतर hज्ञात कीजिए जिससे कि इस तालिका में द्वितीय घात अंतर्वेशन | त्रुटि | ≤10^-6 को संतुष्ट करता हो।

पुनरावृत्ति विधि

जहां Nएक धन अचर है, एक परिमाण की ओर अभिसरित होती है। यह परिमाण ज्ञात कीजिए। इस विधि की अभिसरण दर भी ज्ञात कीजिए।

अंतर समीकरण y_k+2 - 4y_k+1 + 4y_k = 0, k=0,1,... का हल ज्ञात कीजिए। y_o = 1 और y₁ = 6 के लिए विशेष हल भी ज्ञात कीजिए। (2)

h=0.1 के लिए x = 1, y = 0 से आरंभ करके x = 1.5 तक समीकरण y' = x + y का हल कोटि चार की रूंगे-कुट्टा विधि द्वारा प्राप्त कीजिए।

मान लीजिए f_n, t = t_n पर f (t) के मान को निरूपित करता है। यदि f(t) = t³ हो तो  का मान प्राप्त कीजिए। 

f(x) = xe^x के मानों की निम्नलिखित तालिका से h = 0.1 और h = 0.2 पर 0(h²) का केन्द्रीय अंतर सूत्र लागू करके f" (2.0) ज्ञात कीजिए। रूंडन त्रुटि और वास्तविक त्रुटि परिकलित कीजिए।

इकाई वृद्धि करके x = 300 से x = 310 तक आधार 10 पर 10 लघुगणक लीजिए। log₁₀ x का प्रथम अवकलज परिकलित कीजिए जबकि x = 310 हो।