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Question:

Write the dual of the following LP problem:

Min. $Z=3x_1-2x_2+4x_3$

Subject to

$3x_1+5x_2+4x_3\geq 7$

$6x_1+x_2+3x_3=4$

$7x_1-2x_2-x_3\leq 10$

$x_1,x_2,x_3\leq0$

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Question:

Listed in the table below are the activities and sequencing requirements necessary for the completion of a project.

Activity	Predecessor	Duration in weeks
A	__	6
B	A	24
C	A	6
D	A	12
E	A	9
F	C,D,E	18
G	B,F	12
H	G	24

i) Draw a net work diagram for the project.
ii) Find the critical path and the duration for the completion of the project.

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Question:

i) An object is executing simple harmonic motion. Obtain expressions for its kinetic and potential energies.

ii) A spring mass system is characterized by $k=15 Nm^{-1}$ and $m=0.5kg.$ The system is oscillating with amplitude of 0.40 m. Obtain an expression for the velocity of the block as a function of displacement and calculate its value a $x=0.15 m.$

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Question:

The production department for a company requires 3,600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is ₹ 36 and the cost of carrying inventory is 25 percent of the investment in the inventories. The price of the raw material is Image ignouassignments-ignouacademy-com--p-your-96426 10 per kg.

Determine the following:

i) Economic order quantity
ii) Optimal order cycle time and
iii) Minimum yearly inventory cost.

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Question:

) What should be the radius of a space station spinning with an angular speed of 2 rpm such that an astronaut inside the station experiences artificial gravity with

$g=10ms^-2?$

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Question:

A pendulum bob of mass 50 g is suspended on a string from the ceiling of an elevator which is moving downwards with an acceleration 1.5 $ms^{-2}.$ Draw the free body diagram using the non-inertial frame of reference and determine the tension in the string. (Take $g=10$ $(ms^-2)$

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Question:

An investment company wants to study the investment proposals based on the profit factor. While analyzing a new investment proposal, the company estimated the probability distribution for the profit as follows:

Profit (in thousands)	3	5	7	9	10
Probability	0.1	0.2	0.4	0.2	0.1

Using the random numbers:

19, 7, 90, 2, 57, 28

Simulate the profit of the company for six trials.

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Question:

A satellite going around Earth in an elliptic orbit has a speed of 10 km $s^{-1}$ at the perigee which is at a distance of 227 km from the surface of the earth. Calculate the apogee distance and its speed at that point

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Question:

A proton undergoes a head on elastic collision with a particle of unknown mass which is initially at rest and rebounds with 16/25 of its initial kinetic energy. Calculate the ratio of the unknown mass with respect to the mass of the proton

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Question:

. A proton undergoes a head on elastic collision with a particle of unknown mass which is initially at rest and rebounds with 16/25 of its initial kinetic energy. Calculate the ratio of the unknown mass with respect to the mass of the proton.

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Question:

A solid cylinder of mass 3.0 kg and radius 1.0 m is rotating about its axis with a speed of 40 rad $s^{-1}.$ Calculate the torque which must be applied to bring it to rest in 10s. Wha would be the power required?

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Question:

Solve the following cost minimizing assignment problem.

Image ignouassignments-ignouacademy-com--p-doubts-71906

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Question:

The position vector of two particles of mass 4.0 kg and 2.0 kg are, respectively,

$\vec{}r_{1} {}=3t\hat{j}+\hat{j}+2t^{2}\hat{k}$ and $\vec{r}_{2}=3i\hat{}+\left ( t^{2}-1 \right )\hat{j+4t\hat{k}}$ where t is in in metres. Determine the position vector of the centre of mass of the system, the velocity of the cm and the net force acting on the system.

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Question:

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

$Max$ $Z=-2x_1-x_3$

Subject to

$x_1+x_2-x_3\geq 5$

$x_1-2x_2+4x_3\geq 8$

$x_1,x_2,x_3\geq 0.$

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Question:

Solve the following LPP by Simplex method:

$Max.$ $x_1+7x_2$

such that

$3x_1+5x_2\leq 15$

$5x_1+2x_2\leq 10$

$x_1,x_2\geq 0$

From the optimal table of the solution to the problem find the optimal solution of the dual of the problem. Verify complementary slackness property for the primal-dual pair.

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Question:

An insect of mass 20 g crawls from the centre to the outside edge of a rotating disc of mass 200g and radius 20 cm. The disk was initially rotating at 22.0 rads1 . What will be its final angular velocity? What is the change in the kinetic energy of the system?

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Question:

. An automobile travelling at 80 km hr1 has tyres of radius 80 cm. On applying brakes, the car is brought to a stop in 30 complete turns of the tyres. What is the magnitude of the angular acceleration of the wheels? How far does the car move while the brakes are applie

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Question:

A truck of mass 2000 kg moving on a highway experiences an average frictional force of 800 N. If its speed increases from 25 ms1 to 35 ms1 over a distance of 500 m, what is the force generated by the truck

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Question:

A bullet of mass 20 g, travelling at a speed of 350 ms1 , strikes a steel plate at an angle of 30º with the plane of the plate. It ricochets off at the same angle, at a speed of 320 ms1 . What is the magnitude of the impulse that the steel plate gives to the bullet? If the collision with the plate takes place over a time $\Delta t=10^{-3}$ s, what is the average force exerted by the plate on the bullet?

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Question:

A box of mass 50 kg is placed on an inclined plane. When the angle of the plane is increased to 30º, the box begins to slide downwards. Calculate the coefficient of static friction between the plane and the box. Draw the free body diagram.

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