Obtain all the basic solutions to the following system of linear equations:

Which of the solutions are feasible? Justify.

At present a company is purchasing an item ‘x’ from outside suppliers. The assumption of unit is 10000 units/year. The cost of the item is ₹ 5 per unit and the ordering cost is estimated to be ₹ 100 per order. The cost of carrying inventory is 25%. If the consumption rate is uniform, determine the economic order quantity.

Use branch and bound method to solve the following LPP:

Minimize:

Z + 4x₁ + 3x₂

Subject to the constraints:

x₁,x₂ ≥ and are integers.

A road transport company has one reservation clerk on duty at a time. He handles information of bus schedules and make reservations. Consumers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour.

i) What is the average number of customers waiting for the service of the clerk?

ii) What is the average time a customer has to wait before getting service?

A project schedule has the following characterizing:

i) Construct PERT network.

ii) Find the critical path.

On an average 96 patients per 24-hour a day require the services of an emergency clinic. Also on an average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic ₹ 100 per patient treated to obtain an average servicing time of 10 minutes, and that each minute of decrease in this average time would cost ₹ 10 per patient treated. How much would have to budgetted by the clinic to decrease the average size of the queue from  patients to  a patient?

A manufacturing company needs 2500 units of a particular item every year. The company buys it at the rate of ₹ 30 per unit. The order processing cost for this item is estimated at ₹ 15 and the cost of carrying a item in stock comes to about ₹ 4 per year. The company can manufacture this item internally. In that case it saves 20% of the price of the product. However, it estimates a set-up cost of ₹ 250 per production run. The annual production rate would be 4800 units. However, the inventory holding costs remain unchanged.

i) Determine the EOQ and the optimal number of orders placed in a year.

ii) Determine the optimum production lot size and the average duration of the production run.

The time taken (in hours) by five different machines for completing five different jobs is given below:

Find the optimal assignment.

In a factory, there are six jobs to perform each of which should go through two machines A and B in the order A B. The processing timings (in hours) for the jobs are given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T, what is the value of T?

Given x₁₃ = 50units, x₁₄ = 20 units, x₂₁ = 55units, x₃₁ = 30units, x₂ = 35units and x₃₄ = 25 units. Is it an optimal solution to the transportation problem: 

Use dual simplex method to solve the following LPP:

Minimize:

z = 3x₁ + x₂

Subject to the constraints:

Use two-phase method to solve the following LPP: 

Maximize:

z = 3x₁ + 2x₂

Subject to the constraints:

Use simplex method to solve the following LPP:

Maximize:

z = 2x₁ - x₂ + x₃

Subject to the constraints:

x₁ x₂, ≥ 0 and x is unrestricted.

Use the graphical method to solve the following LPP:

Maximize:

z = 2x₁ + 3x₂

Subject to the constraints:

A firm produces there product A, B and C. It uses two types of raw materials I and II of which 5000 and 7500 units respectively are available. The raw material requirements per unit of the products are given below:

The minimum demand of the three products is 600, 650 and 500 units respectively. Assuming the profits per unit of A, B and C as ₹50, ₹50 and 280 respectively. Formulate the problem as LPP model in order to determine the number of units of each product which will maximize the profit.

Which of the following statements are true? Give reasons for your answers.

i) If an item is ordered frequently, then the risk of running out of stock is least.

ii) In deterministic queuing model, arrival rate must not exceed the service rate.

iii) The critical path of a project network represents the minimum time needed to complete the project.

iv) A necessary and sufficient condition for a basic feasible solution for a minimization LPP to be optimum is that all Z_j - C_j  20.

v) If dual has an unbounded solution, primal has a feasible solution.

निम्नलिखित में से कौन से कथन सत्य है और कौन से असत्य? अपने उत्तर के कारण बताइएः

i) 'आज दिन उज्जवल है' एक कथन है।

ii) पेंटागॉन के भीतरी कोणों का योग 450° होता है।

iii) पूर्व-संक्रियात्मक सोच दो वर्ष के उम्र की बच्ची की विशेषता है।

iv) यदि त्रिविमीय वस्तुओं की धारिता बढ़ती है तो उसका आयतन भी बढ़ जाता है।

v) प्रत्येक गणितीय सवाल का एक अद्वितीय हल होता है।

अधिकांश गणित शिक्षण वास्तव में बच्चों को उन पैटर्नो के लिए ज्यादा जागरूक (जानकार) बनाने के लिए प्रोत्साहित करना, जिन्हें वे ढूँढते हैं और अपनी समझ (सोच) में उनका प्रयोग करते हैं। नीचे प्रश्न (i), (ii) और (iii) में दिए गए प्रश्नों को उत्तर देकर आप इस बात को स्पष्ट कर सकते हैं।

"कक्षा 5 की गणित की शिक्षिका ने कक्षा में निम्नलिखित पैटर्न दिखायाः

46 x 44 = 2024

63 × 67 = 4221

71 × 79 = 4909

उसने विद्यार्थियों को पैटर्न की पहचान करने के लिए कहा। थोड़ी देर बाद उसने विद्यार्थियों को "84 × 86" का उत्तर देने के लिए कहा। एक विद्यार्थी ने उत्तर दिया 7224.

i) विद्यार्थी द्वारा प्रयुक्त पैटर्न का पता लगाइए।

ii) बताइए यह क्यों कारगर है?

iii) यह गणितीय सोच को प्रोत्साहित करने में कैसे मदद करता है, वर्णन कीजिए।

जादुई वर्ग क्या है? नीचे दिए वर्ग की प्रविष्टियाँ पूरी कीजिए और उसको जादुई वर्ग बनाइए।

प्रविष्टियों को भरने के लिए इस्तेमाल की गई विधियें/तरीकों को स्पष्ट कीजिए। यह भी वर्णन कीजिए कि वह विधि क्यों कारगर है?

सिद्ध कीजिए कि पहली n सम संख्याओं का योग सम संख्या होता है। क्या इसको सिद्ध करने का तर्क आगमनिक है, निगमनिक है या दोनों? अपने उत्तर की पुष्टि कीजिए।