Advance Mathematics

Old hens can be bought at Rs. 2 each and young ones at Rs. 5 each. The old hens lays 3 eggs per week and the young one lay 5 eggs per week, each egg being worth 30 paise. A hen costs Rs. 1 per week to feed. I have only Rs. 80 to spend for hens. How many of each kind should I buy to give a profit of more than Rs. 6 per week, assuming that I cannot house more than 20 hens. Solve Graphically.

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Use dynamic programming to solve L.P.P.

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Determine early start (T_E) and late start (T_L) in respect of all nodes and identify critical path in respect of the following network.

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Solve the following game by matrix method

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		B
	I	II	III
I	7	1	7	(1)
A II	9	-1	1	(-1) Row min.
III	5	7	6	(5)
	(9)	(7)	(7)
	Column max.

Use the Graphical method to determine an optimal strategy for player I in the game defined by the following table

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Player II
	B1	B2	B3
Player I	A1	2	-3	-4
	A2	-6	-1	1

In a referendum submitted to the student body at a university, 850 men and 560 women voted. Out of these 500 men and 320 women voted "Yes". Does this indicate a significant difference of opinion between men and women on the matter at 1% level of significance.

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Out of 800 families with 5 children each, how many families would be expected to have

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1. Three boys and 2 girls
2. Two boys and 3 girls
3. One girl
4. At the most two girls

Under the assumption that probabilities for boys and girls are equal.

Field data from a tire company shows that 90% of tires on passenger car fail to pass inspection between 22 to 30 k-miles. The data also shows that the time-to-failure probability of the tires can be described by a normal distribution.

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1. What is the failure rate when a tire has 20 k-miles on it?
2. What is the failure rate when a tire has 25 k-miles on it?

Prove that reliability function R(t) = e^{-∫₀t λ(u)du}

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Use the simplex method to solve the problem :

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Solve the following transportation problem :

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	D1	D2	D3	Supply
O1	50	30	220	1
Source O2	90	45	170	3
O3	250	200	50	4
Demand	4	2	2	8

A truck can carry a total of 10 tons of product. Three types of product are available for shipment. Their weights and value are tabulated. Assuming that at least one of each type must be shipped, determine the loading which will maximize the total value.

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Type	Weight (Ton)	Value (Rs.)
A	1	20
B	2	50
C	2	60

A project has the following time schedule :

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Activity	Time in weeks	Activity	Time in weeks
1-2	4	5-6	4
1-3	1	5-7	8
2-4	1	6-8	1
3-4	1	7-8	2
3-5	6	8-9	1
4-9	5	8-10	8
9-10	7

Construct PERT network and compute :

1. T_E and T_L for each event
2. Float for each activity
3. Critical path and its duration

The marks obtained in Mathematics by the students in a class are approximately normally distributed with mean 62 and variance 36. If 3 students are selected at random, find the probability that at least 1 of them would score more than 80 marks.

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Two players A and B without showing each other, put on a table a coin, with head or tail up. A wins Rs. 8 when both the coins show head and Rs. 1 when both are tails. B wins Rs. 3 when the coins do not match. Given the choice of being matching player (A) or non-matching player (B), which one would you choose and what would be your strategy?

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In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming that the inter-arrival time follows an exponential distribution and the service time distribution is also exponential with an average 36 minutes. Calculate the following :

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1. Then mean queue size.
2. The probability that the queue size exceeds 10.
3. Expected waiting time in queue.

If the input of trains increases to an average 33 per day, what will be change in (i), (ii) and (iii)?