Write the dual of the following LPP

Minimize   Z = 

Subject to the constraints

unrestricted.

Use dual simplex method to solve the following LPP.

Min

Subject to

a) The table below is a Cayley table for the group ({e,a, b,c,d},∗). Fill in the blanks.

b) Let G be a finite group. Show that the number of elements g of G such that g³ = e is odd, where e is the identity of G.

c) Check if is an abelian group with respect to matrix

Based on the previous data, the probabilities of a batsman making various scores in One Day Internationals are given below:

Simulate the runs scored by the batsman in the next five One Day Internationals using the following 25, 39, 65, 76, 12.

A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He finds that when he starts production run, he can produce 25,000 bearings per day. The cost of holding a bearing in stock for one year is Rs. 2 and the set up cost of a production run is Rs. 180. Find the EOQ. How frequently should the production run he made

A department has five employees with five jobs to be performed. The time (in hours) each men will take to perform each job is given in the table below

How should the jobs be assigned, one job per employee, so as to minimize the total man-hours

A department has five employees with five jobs to be performed. The time (in hours) each men will take to perform each job is given in the table below

How should the jobs be assigned, one job per employee, so as to minimize the total man-hours

Use the simplex method to solve the following L.P.P.

Max 

Subject to

Find the critical path for the project whose network diagram is given below

A television repairman finds that the time spent on his jobs has an exponential distribution with a mean of 30 minutes. If he repairs sets in the order in which they come in, and if arrival of sets follows a Poission distribution approximately with an average rate of 10 per 8 hours day, what is the repairman’s expected idle time each day, How many jobs are ahead of the average set just brought in?

(a) A company has three factories 1 2 F ,F and F3 which supply goods to four warehouses 1 2 3 W ,W ,W and . W4 The daily factory capacities of 1 2 F ,F and F3 are, respectively, six units, one unit and ten units. The demand of the warehouses 1 2 3 W ,W ,W and W4 are, respectively, seven, five, three and two units. Unit transportation cost are as follows:

Find an initial basic feasible solution by the Vogel’s approximation method

(b) Three custom officers check the luggage of the passengers of an airport. The passengers are found to arrive at an average rate of 30 per 8 hours a day. The amount of time a custom officer spends with the passenger is found to have an exponential distribution with mean service time 32 minutes.

(i) Find the probability that all the custom officers are idle.

(ii) Find the expected number of passengers in the queues.

(iii) Find the expected waiting time of passenger in the system

Find the sequence of jobs that minimizes the total elapsed time required to complete the following task on two machines.

Also, find the optimal elapsed time.

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)

i) If A and B are two sets such that A ⊆ B, then A × B = B.

ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real numbers lying between 2.5 and 2.55, then T SU is an infinite set.

iii) The set {x x=1 1(mod 30)is a group with respect to multiplication (mod 30).

iv) If G is a group with an abelian quotient group G/N, then N is abelian.

v) There is a group homomorphism f with Ker f and Im f {0}.

vi) There is a 1 – 1 correspondence between the odd permutations of S₃₅ and the even permutations of S₃₅.

vii) If R is a ring such that , a =− a ∀a ∈ R then R is Boolean.

viii) Given any ring R, there is an ideal I of R such that R/1 is commutative.

ix) If S is an ideal of a ring R and f a ring homomorphism from R to a ring R' , then

x) ‘ring’, as we now define it, was first presented to us by Dedekind.

A firm makes two products A and B has a total production capacity of 9 tonnes per day, with A and B utilizing the same production facilities. The firm has a permanent contract to supply at least 2 tonnes of A per day to another company. Each tone of A requires 20 machine hours of production time and each tone of B requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. All the firm’s output can be sold and the profit made is Rs. 80 per tonne of A and Rs. 120 per tonne of B. Formulate the problem of maximising the profit as an LPP and solve it graphically

Which of the following statements are true and which are false? Give a short proof or a counter example in support of your answer.

(a) The optimal solution for the following LPP is z = 30 :

 Subject to 

(b) The optimal solution of an ILLP can be obtained by rounding off the optimal solution of its LP relaxation.

(c) If the availabilities and requirements of a balanced transportation problem are integers, the optimal solution to the problem will have integer values.

(d) The following max /3/4 F / F problem can be reduced to a machine problem

(e) For the mixed generator  (mod 8 ) if  then 3 r is zero.

a) Prove that n 2 4 n > for n ≥ 5.

b) Give an example, with justification, of a function with domain \{ 2,3 }and codomain  Is this function 1 – 1? Is it onto? Give reasons for your answers.

c) Give a set of cardinality 5 which is a subset of 

d) Check whether the relation R ={(x, y)∈ × |xyis the square of an integer} is an equivalence relation on .

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)

i) If A and B are two sets such that A ⊆ B, then A × B = B.

ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real numbers lying between 2.5 and 2.55, then T SU is an infinite set.

iii) The set {x x=1 1(mod 30)is a group with respect to multiplication (mod 30).

iv) If G is a group with an abelian quotient group G/N, then N is abelian.

v) There is a group homomorphism f with Ker f and Im f {0}.

vi) There is a 1 – 1 correspondence between the odd permutations of S₃₅ and the even permutations of S₃₅.

vii) If R is a ring such that , a =− a ∀a ∈ R then R is Boolean.

viii) Given any ring R, there is an ideal I of R such that R/1 is commutative.

ix) If S is an ideal of a ring R and f a ring homomorphism from R to a ring R' , then

x) ‘ring’, as we now define it, was first presented to us by Dedekind.

Which of the following statements are true or false? Give reasons for your answer. 

i) Pre-operational thinking is the characteristic of a two year old child.

ii) Each mathematical problem have a unique solution.

iii) ‘Today is a bright day’ is an unambiguous statement.

iv) The sum of the interior angles of a Pentagon is 450o

. v) If the capacity of a 3D-objects increases, then the volume also increases

i) What is an equation? Does all the equation involve a variable. Give an example of an equation with a variable in it and which does not have a variable in it.

ii) Here is a think of a number game: ‘Think of a number, then double it, add six to the sum, divide the sum by half and then subtract 3 from it the number’. Did you receive the same number you had started with? Why? Justify.

A class 5 child believe that the division always makes a number smaller. Describe an activity that could help her correct her misconceptions. Also describe an activity to assess how far the activity is effective.