In a manufacturing plant, the probability of making a defective bolts is $$0.1$$. The mean and standard deviation of defective bolts in a total of $$900$$ bolts are respectively

Find the solution of the differential equation $$\,{{{d^2}u} \over {d{t^2}}} + {\lambda ^2}y = \cos \left( {wt + k} \right)$$ with initial conditions $$\,y\left( 0 \right) = 0,\,\,{{dy\left( 0 \right)} \over {dt}} = 0.$$ Here $$\lambda ,$$ $$w$$ and $$k$$ are constants. Use either the method of undetermined coefficients (or) the operator $$\left( {D = {\raise0.5ex\hbox{$\scriptstyle d$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {dt}$}}} \right)$$

The ratio of tension on the tight side to that on the slack side in a flat belt drive is

A steel wheel of 600 mm diameter rolls on a horizontal steel rail. It carries a load of 500 N. The coefficient of rolling resistance is 0.3 mm. The force in N, necessary to roll the wheel along the rail is

In given figure, if the pressure of gas in bulb $$A$$ is $$50$$ $$cm$$ $$Hg$$ vacuum and $${P_{atm}} = 76\,cm\,Hg,$$ then height of column $$H$$ is equal to

Navier Stoke's equation represents the conservation of

A thin metal plate is exposed to solar radiation. The air and the surroundings are at $${30^ \circ }C\,.$$ The heat transfer coefficient by free convection from the upper surface of the plate is $$17.4$$ $$\,W/{m^2}K$$. The plate has an absorptivity of $$0.9$$ at solar wavelength and an emissivity of $$0.1$$ at the long wavelength. Neglecting any heat loss from the lower surface, determine the incident solar radiation intensity in $$kW/{m^2}$$ , if the measured equilibrium temperature of the plate is $${50^ \circ }C\,.$$ Stefan Boltzmannn constant is $$5.67 \times {10^{ - 8}}\,\,W/{m^2}{K^4}$$

Air enters a counter-flow heat exchanger at $${7^ \circ }C$$ and leaves at $${40^ \circ }C$$. Water enters at $${30^ \circ }C$$ and leaves at $${50^ \circ }C$$. The $$LMTD$$ in deg $$C$$ is

A company places orders for supply of two items $$A$$ and $$B.$$ The order cost for each of the items is Rs.$$300$$ /order. The inventory carrying cost is $$18\% $$ of the unit price per year per unit. The unit prices of the items are Rs.$$40$$ and Rs.$$50$$ respectively. The annual demands are $$10,000$$ and $$20,000$$ respectively.

(a) Find the economic order quantities and the minimum total cost
(b) A supplier is willing to give a $$1\% $$ discount on price, if both the items are ordered from him and if the order quantities for each item are $$1000$$ units or more. Is it profitable to avail the discount?

Given below are the data for a project network.
(a) Draw the network for the above project capturing the precedence relationships.
(b) Find the critical path and its duration.

In a time series forecasting model, the demand for five time periods was $$10, 13,$$ $$15,$$ $$18$$ and $$22.$$ A linear regression fit resulted in an equation $$F = 6.9 + 2.9$$ $$t$$ where $$F$$ is the forecast for period $$t$$. The sum of absolute deviations for the five data is

In a single server infinite population queuing model, arrivals follow a Poisson distribution with mean $$\lambda = 4$$ per hour. The service times are exponential with mean service time equal to $$12$$ minutes. The expected length of the queue will be

Solve the following linear programming problem by simplex method

$$\eqalign{ & Maximize\,\,\,\,\,\,4{x_1} + 6{x_2} + {x_3} \cr & Subject\,\,to\,\,\,\,\,\,2{x_1} - {x_2} + 3{x_3}\, \le 5 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1},{x_2},{x_3} \ge 0 \cr} $$

$$(a)$$$$\,\,\,\,\,\,\,$$ What is the solution to the above problem?

$$(b)$$$$\,\,\,\,\,\,\,$$ Add the constant $${x_2} \le 2$$ to the simplex table of part $$(a)$$ and find the solution.

Given below is a basic feasible solution to a transportation problem with three supply points $$(A,B,C)$$ and three demand points $$(P, Q, R)$$ that minimizes cost of transportation in the standard tabular format.

$$(a)$$$$\,\,\,\,\,\,\,\,$$ Compute the cost corresponding to the present solution
$$(b)$$$$\,\,\,\,\,\,\,\,$$ It is optimal?
$$(c)$$$$\,\,\,\,\,\,\,\,$$ Does an alternate optimum exist ?

The forecasts for a product for the next three months are given as $$750,850$$ and $$1000$$ units. The number of regular time days and overtime days available are $$22,18,$$ $$22$$ and $$4,$$ $$4,5$$ respectively. With the existing number of employees, the company can produce $$38$$ units per day. To meet the high demand in the third month, the company decides to hire people to increase the daily production to $$45$$ units.

The following costs are given :
Cost of regular time production $$=$$ Rs. $$20$$ per unit
Cost of overtime production $$=$$ Rs. $$25$$ Per unit
Cost of hiring $$=$$ $$200{L^2}$$
where $$'L'$$ is the increase in daily capacity

Inventory $$=$$ Rs. $$10$$ per unit per month (based on average inventory)
Shortage (back-ordering cost) $$=$$ Rs. $$20$$ per unit per month

The beginning inventory is $$100$$ units. The company decides to produce $$800,$$ $$700$$ and $$900$$ units respectively in the three months. Compute the cost of the production plan.

A $$1.5$$ $$kW$$ motor is running at $$1440$$ rev/min. It is to be connected to a stirrer running at $$36$$ rev/min. the gearing arrangement suitable for this application is

The life of a ball bearing at a load of $$10$$ $$kN$$ is $$8000$$ hours. Its life in hours, if the load is increased to $$20$$ $$kN,$$ keeping all other conditions same, is

Cellular manufacturing is suitable for

Two castings, a cube and a slab of the same material solidify under identical conditions. The volumes of the castings are equal but the slab dimensions are in the ratio of $$1:2:4$$, find the ratio of the solidification times of the cube to that of the slab.

A cylindrical billet of $$100$$ $$mm$$ diameter is forged from $$50$$ $$mm$$ height to $$40$$ $$mm$$ at $${1000^0}$$$$C$$. The material has constant flow stress of $$80$$ $$MPa.$$ Find the work of deformation. If a $$10$$ $$kN$$ drop hamper is used to complete the reduction in one blow. What will be the height of fall.

Two different pairs of sheets of same material are welded by resistance-spot welding. In one pair, the average radius $$(r)$$ of each spherical bridge is $$0.2$$ $$mm$$ and the number of bridges per $$c{m^2}\,\,\left( n \right)$$ is $$25.$$ In another pair, the number of bridges per $$c{m^2}$$ is $$50$$ with the same $$'r'$$ of bridge. The contact resistance $${R_c}$$ per unit area is given by $${R_c} = 0.85$$ $$\left( {\rho /n\pi r} \right)$$ where $$'\,\rho \,'$$ is the resistivity of metal. If the voltage applied is $$5$$ volts and the resistivity of metal is $$2 \times {10^{ - 5}}$$ ohm-cm. The rate of heat generated per $$c{m^2}$$ in each case

A built up edge is formed while machining

A conventional lathe and a $$CNC$$ lathe are under consideration for machining a given part. The relevant data are shown below :

The machine preferred for producing $$100$$ pieces is

Deep hole drilling of small diameter, say $$0.2$$ $$mm$$ is done with $$EDM$$ by selecting the tool material as

Abrasive material used in grinding wheel selected for grinding ferrous alloys is

The abrasive material used in grinding wheel selected for grinding of ferrous alloys is

In computer aided drafting practice, an arc is defined by

Disposable patterns are made of

In finish machining of an island on a casting with $$CNC$$ milling machine, an end mill with $$10$$ $$mm$$ diameter is employed. The corner points of the island are represented by $$(0,0), (0,30), (50,30)$$ and $$(50,0)$$. By applying cutter radius right compensation, the trajectory of the cutter will be

A 1.5 kW motor is running at 1440 rev/min. Its is to be connected to a stirrer running at 36 rev/min. The gearing arrangement suitable for this application is

As shown in Figure, a mass of 100 kg is held between two springs. The natural frequency of vibration of the system, in cycles/s, is

A steam turbine receives steam steadily at $$10$$ $$bar$$ with an enthalpy of $$3000$$ $$kJ/kg$$ and discharges at $$1$$ bar with an enthalpy of $$2700$$ $$kJ/kg.$$ The work output is $$250$$ $$kJ/kg.$$ The changes in kinetic and potential energies are negligible. The heat transfer from the turbine casing to the surroundings is equal to

When an ideal gas with constant specific heats is throttled adiabatically, with negligible changes in kinetic and potential energies

A certain mass of a pure substance undergoes an irreversible process from state $$1$$ to state $$2,$$ the path of the process being a straight line on the $$T$$-$$s$$ diagram. Calculate heat transfer & work done. Some of the properties at the initial and final states are:
$${T_1} = 330\,K,\,\,{T_2} = 440\,K;\,\,{U_1} = 170\,kJ,\,\,$$
$$\,{U_2} = 190\,kJ;\,\,{H_1} = 220{\,_{}}kJ,\,\,{H_2} = 247\,kJ,$$
and $${S_1} = 0.23\,kJ/K$$ and $${S_2} = 0.3\,kJ/K$$ where $$T, U, H$$ and $$S$$ represent temperature, internal energy, enthalpy and entropy respectively.

Availability of a system at any given state is

Which of the following is a pressure compounded turbine?

For a compressible fluid, sonic velocity is

A simple impulse turbine expands steam frictionlessly from $$12$$ bar, $${250^0}C$$ with an enthalpy of $$2935$$ $$kJ/kg$$ to an enthalpy of $$2584$$ $$kJ/kg$$ at $$0.1$$ bar. Assuming that the nozzle makes an angle of $${20^0}$$ with the blade motion, and that the blades are symmetrical, find the blade velocity that produces maximum efficiency for a turbine speed of $$3600$$ rev/min. assume that the steam enters the nozzle with negligible velocity.