The sum of the eigen values of the matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \right]$$ is

For what value of $$x$$ will the matrix given below become singular? $$\left[ {\matrix{ 8 & x & 0 \cr 4 & 0 & 2 \cr {12} & 6 & 0 \cr } } \right]$$

The volume of an object expressed in spherical co-ordinates is given by
$$V = \int\limits_0^{2\pi } {\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}} {\int\limits_0^1 {{r^2}} \,Sin\phi \,drd\phi \,d\theta .} } $$ The value of the integral

If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \over {dx}} = \,\_\_\_\_\_.$$

The following data about the flow of liquid was observed in a continuous chemical process plant

Mean flow rate of the liquid is

From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be kings, if the card is NOT replaced?

The figure below shown a pair of pin jointed gripper tongs holding an object weighing $$2000N$$. The coefficient of friction (µ) at the gripping surface is $$0.1$$ $$XX$$ is the line of action of the input force and $$Y - Y$$ is the line of application of gripping force. If the pin joint is assumed to be frictionless, the magnitude of force $$F$$ required to hold the weight is

A rigid body shown in the fig.(a) has a mass of 10kg. It rotates with a uniform angular velocity $$'\omega '$$ A balancing mass of 20 kg is attached as shown in fig.(b). The percentage increase in mass moment of inertia as a result of this addition is

The figure shows a pin-jointed plane truss loaded at the point M by hanging a mass of 100 kg. The member LN of the truss is subjected to a load of v

An ejector mechanism consists of a helical compression spring having a spring constant of k = 981 × 10³ N/m. It is pre-compressed by 100mm from its free state. If it is used to eject mass of 100kg heald on it, the mass will move up through a distance of

The pressure gauges $${G_1}$$ and $${G_2}$$ installed on the system show pressures of $${P_{G1}} = 5.00$$ bar and $${P_{G2}} = 1.00$$ bar. The value of unknown pressure $$P$$ is

For air flow over a flat plate, velocity $$(U)$$ and boundary layer thickness $$\left( \delta \right)$$ can be expressed respectively, as $$${U \over {{U_\infty }}} = {3 \over 2}{y \over \delta } - {1 \over 2}{\left( {{y \over \delta }} \right)^3}\,\,\,\,;\,\,\,\,\delta = {{4.64x} \over {\sqrt {{{{\mathop{\rm Re}\nolimits} }_x}} }}$$$
If the free stream velocity is $$2$$ $$m/s$$, and air has Kinematic viscosity of $$1.5 \times {10^{ - 5}}{m^2}/s$$ and density of $$1.23$$ $$kg/{m^3}$$, then wall shear stress at $$x=1$$ $$m$$, is

A closed cylinder having a radius $$R$$ and height $$H$$ is filled with oil of density $$\rho .$$ If the cylinder is rotated about its axis at an angular velocity of $$\omega $$ , then thrust at the bottom of the cylinder is

For a fluid flow through a divergent pipe of length $$L$$ having inlet and outlet radii of $${R_1}$$ and $${R_2}$$ respectively and a constant flow rate of $$Q,$$ assuming the velocity to be axial and uniform at any cross- section , the acceleration at the exit is

An incompressible fluid (kinematic viscosity $$ = 7.4 \times {10^{ - 7}}\,\,{m^2}/s,$$ specific gravity, $$0.88$$) is held between two parallel plates. If the top plate is moved with a velocity of $$0.5$$ $$m/s$$ while the bottom one is held stationary, the fluid attains a linear velocity profile in the gap of $$0.5$$ $$mm$$ between these plates; the shear stress in Pascal on the surface of bottom plate is :

A fluid flow is represented by the velocity field $$\overrightarrow V = ax\,\overrightarrow i + ay\,\overrightarrow j ,$$ where a constant . The equation of stream line passing through a point $$(1, 2)$$ is

The pressure gauges $${G_1}$$ and $${G_2}$$ installed on the system shown pressures of $${P_{G1}} = 5.00$$ bar and $${P_{G2}} = 1.00$$ bar. The value of unknown pressure $$P$$ is

In a condenser, water enters at $${30^ \circ }C$$ and flows at the rate $$1500$$ $$Kg/hr.$$ The condensing steam is at a temperature of $${120^ \circ }C$$ and cooling water leaves the condenser at $${80^ \circ }C$$. Specific heat of water is $$4.187 kJ/kg$$ $$K.$$ If the overall heat transfer coefficient is $$2000$$ $$W/{m^2}K,$$ the heat transfer area is

A spherical thermocouple junction of diameter $$0.706 mm$$ is to be used for the measurement of temperature of a gas stream. The convective heat transfer co-efficient on the bead surface is $$400W/{m^2}K.$$ Thermo-physical properties of thermocouple material are
$$k = 20W/mK,$$ $$C = 400J/kgK$$ and $$\rho = 8500\,\,kg/{m^3}.$$ If the thermocouple initially at $${30^ \circ }C$$ is placed in a hot stream of $${300^ \circ }C$$ the time taken by the bead to reach $${298^ \circ }C$$, is

One dimensional unsteady state heat transfer equation for a sphere with heat generation at the rate $$'{q_g}',$$ can be written as

A company produces two types of toys: $$P$$ and $$Q.$$ Production time of $$Q$$ is twice that of $$P$$ and the company has a maximum of $$2000$$ time units per day. The supply of raw material is just sufficient to produce $$1500$$ toys (of any type) per day. Toy type $$Q$$ requires an electric switch which is available @ $$600$$ pieces per day only. The company makes a profit of Rs.$$3$$ and Rs.$$5$$ on type $$P$$ and $$Q$$ respectively. For maximization of profits, the daily production quantities of $$P$$ and $$Q$$ toys should respectively be

An electronic equipment manufacturer has decided to add a component sub-assembly operation that can produce $$80$$ units during a regular $$8$$-hour shift. This operation consists of three activities as below .

For line balancing the number of work stations required for the activities $$M, E$$ and $$T$$ would respectively be

For a product, the forecast and the actual sales for December $$2002$$ were $$25$$ and $$20$$ respectively. If the exponential smoothing constant $$(\alpha )$$ is taken as $$0.2,$$ then forecast sales for January $$2003$$ would be

A company has an annual demand of $$1000$$ units, ordering cost of Rs.$$100$$/order and carrying cost of Rs.$$100$$/unit –year. If the stock-out costs are estimated to be nearly Rs.$$400$$ each time the company runs out-of-stock, the safety stock justified by the carrying cost will be

A maintenance service facility has Poisson arrival rates, negative exponential service time and operates on a ‘first come first served’ queue discipline. Breakdowns occur on an average of $$3$$ per day with a range of zero to eight. The maintenance crew can service an average of $$6$$ machines per day with a range of zero to seven. The mean waiting time for an item to be serviced would be

In $$PERT$$ analysis a critical activity has

There are two products P and Q with the following characteristics

The economic order quantity (EOQ) of products P and Q will be in the ratio.

A compacting machine shown in the figure below is used to create a desired thrust force by using a rack and pinion arrangement. The input gear is mounted on the motor shaft. The gears have involutes teeth of $$2$$ $$mm$$ module.

If the pressure angle of the rack is $${20^ \circ },$$ the force acting along the line of action between the rack and the gear teeth is

Two mating spur gears have $$40$$ and $$120$$ teeth respectively. The pinion rotates at $$1200$$ $$rpm$$ and transmits a torque of $$20N.m.$$ The torque transmitted by gear is

A compacting machine shown in the figure below is used to create a desired thrust force by using a rack and pinion arrangement. The input gear is mounted on the motor shaft. The gears have involutes teeth of $$2$$ $$mm$$ module.

If the drive efficiency is $$80\% ,$$ the torque required on the input shaft to create $$1000N$$ output thrust is

Gray cast iron blocks $$200 \times 100 \times 10\,\,mm$$ are to be cast in sand moulds. Shrinkage allowance for pattern making is $$1$$%. The ratio of the volume of pattern to that of the casting will be

Misrun is a casting defect which occurs due to

In a rolling process, sheet of $$25$$ $$mm$$ thickness is rolled to $$20$$ $$mm$$ thickness. Roll is of diameter $$600$$ $$mm$$ and it rotates at $$100$$ $$rpm.$$ The roll strip contact length will be

$$10$$ $$mm$$ diameter holes are to be punched in a steel sheet of $$3$$ $$mm$$ thickness. Shear strength of the material is $$400\,\,N/m{m^2}$$ and penetration is $$40\% $$. Shear provided on the punch is $$2$$ $$mm.$$ The blanking force during the operation will be

In a $$2-D$$ $$CAD$$ package, clockwise circular arc of radius, $$5,$$ specified from $${P_1}\left( {15,10} \right)$$ to $${P_2}\left( {10,15} \right)$$ will have its center at

During the execution of a CNC part program block
$$NO20\,\,\,\,GO2\,\,\,\,X45.0\,\,\,\,Y25.0\,\,\,\,R5.0.$$ The type of tool motion will be

A hole of diameter $${25.00^{ + 0.01}}\,\,mm$$ is to be inspected by using GO / NO GO gages. The size of the GO plug gage should be

Through holes of $$10$$ $$mm$$ diameter are to be drilled in steel plate of $$20$$ $$mm$$ thickness. Drill spindle speed is $$300$$ $$rpm,$$ feed $$0.2$$ $$mm/rev$$ and drill point angle is $${120^0}.$$ Assuming drill over travel of $$2mm$$ the time for producing a hole will be

A standard machine tool and an automatic machine tool are being compared for the production of a component. Following data refers to the two machines.

The breakeven production batch size above which the automatic machine tool will be economical to use, will be

In an orthogonal cutting test on mild steel, the following data were obtained
Cutting speed: $$40$$ $$m/min,$$
Depth of cut: $$0.3$$ $$mm,$$
Tool rake angle : $$ + {5^ \circ },$$
Chip thickness: $$1.5$$ $$mm,$$
Cutting force: $$900$$ $$N,$$
Thrust force: $$450$$ $$N$$
Using Merchant's analysis, the Friction angle during the machining will be

In a machining operation, doubling the cutting speed reduces the tool life to $$1/{8^{th}}$$ of the original value. The exponent $$n$$ in Taylor's tool life equation $$V{T^n} = C,$$ is

Two 1 mm thick steel sheets are to be spot welded at a current of $$5000$$ $$A.$$ Assuming effective resistance to be $$200$$ micro-ohms and current flow time of $$0.2$$ second, heat generated during the process will be

A uniform, slender cylindrical rod is made of a homogeneous and isotropic material. The rod rests on a frictionless surface. The rod is heated uniformly. If the radial and longitudinal thermal stresses are represented by $${\sigma _r}$$ and $${\sigma _z}$$ respectively, then

A solid circular shaft of $$60$$ mm diameter transmits a torque of $$1600$$ N - m. The value of maximum shear developed is

A steel beam of breadth 120 mm and height 750 mm is loaded as shown in the figure. Assume modulus of elasticity as 200 GPa.

The beam is subjected to a maximum bending moment of

A torque of $$10$$ $$N-m$$ is transmitted through a stepped shaft as shown in figure. The torsional stiffnesses of individual sections of lengths $$MN, NO$$ and $$OP$$ are $$20$$ $$N-m/rad,$$ $$30$$ $$N-m/rad$$ and $$60$$ $$N-m/rad$$ respectively. The angular deflection between the ends $$M$$ and $$P$$ of the shaft is

A steel beam of breadth 120 mm and height 750 mm is loaded as shown in the figure. Assume modulus of elasticity as 200 GPa.

The value of maximum deflection of the beam is

In terms of Poisson's ratio $$\left( \mu \right)$$ the ratio of Young's Modulus $$(E)$$ to Shear Modulus $$(G)$$ of elastic materials is

The figure below shows a steel rod of $$25$$ mm² cross sectional area. It is loaded at four points, K, L, M and N. Assume E_steel $$=$$ $$200$$ GPa. The total change in length of the rod due to loading is

The Mohr's circle of plane stress for a point in a body is shown. The design is to be done on the basis of the maximum shear stress theory for yielding. Then, yielding will just begin if the designer chooses a ductile material whose yield strength is

The figure shows the state of stress at a certain point in a stressed body. The magnitudes of normal stresses in the $$x$$ and $$y$$ directions are $$100$$ MPa and $$20$$ MPa respectively. The radius of Mohr's stress circle representing this state of stress is

The figure below shows a planar mechanism with single degree of freedom. The instant center $${{\rm I}_{24}}$$ for the given configuration is located at a position.

Match the following
Type of Mechanism
$$P.$$ Scott - Russel mechanism
$$Q.$$ Geneva mechanism
$$R.$$ Off-set slider- crank mechanism
$$S.$$ Scotch Yoke mechanism

Motion achieved
$$1.$$ Intermittent motion
$$2.$$ Quick return motion
$$3.$$ Simple harmonic motion
$$4.$$ Straight line motion

In the figure shown, the relative velocity of link $$1$$ with respect to link $$2$$ is $$12$$ m/sec. Link $$2$$ rotates at a constant speed of $$120$$ rpm. The magnitude of Coriolis component of acceleration of link $$1$$ is

Match the following with respect to spatial mechanism.

Type of Joint
$$P.$$ Revolute
$$Q.$$ Cylindrical
$$R.$$ Spherical

Motion constrained
$$1.$$ Three
$$2.$$ Five
$$3.$$ Four
$$4.$$ Two
$$5.$$ Zero

For a mechanism shown below, the mechanical advantage for the given configuration is

A uniform stiff rod of length $$300$$ $$mm$$ and having a weight of $$300$$ $$N$$ is pivoted at one end and connected to a spring at the other end. For keeping the rod vertical in a stable position the minimum value of spring constant $$K$$ needed is

A vibrating machine is isolated from the floor using springs. If the ratio of excitation frequency of vibration of machine to the natural frequency of the isolation system is equal to 0.5, the transmissibility ratio of isolation is

A mass M, of 20 kg is attached to the free end of a steel cantilever beam of length 1000 mm having a cross-section of 25 $$\, \times \,$$ 25 mm. Assume the mass of the cantilever to be negligible and E_steel = 200 Gpa. If the lateral vibration of this system is critically damped using a viscous damper, the damping constant of the damper is

A steel billet of $$2000$$ $$kg$$ mass is to be cooled from $$1250$$ $$K$$ to $$450$$ $$K.$$ The heat released during this process is to be used as a source of energy. The ambient temperature is $$303$$ $$K$$ and specific heat of steel is $$0.5$$ $$kJ/kg K.$$ the available energy of this billet is

Consider a steam power plant using a reheat cycle as shown. Steam leaves the boiler and enters the turbine at $$4$$ $$MPa,$$ $${350^ \circ }C$$ $$\left( {{h_3} = 3095\,\,kJ/kg} \right).$$ After expansion in the turbine to $$400$$ $$kPa$$ $$\left( {{h_4} = 2609\,\,kJ/kg} \right)$$, the steam is reheated to $${350^ \circ }C$$ $$\left( {{h_5} = 3170\,kJ/kg} \right),$$ and then expanded in a low pressure turbine to $$10$$ $$kPa$$ $$\left( {{h_6} = 2165\,kJ/kg} \right).$$

The enthalpy at the pump discharge $$({h_2})$$ is

A heat engine having an efficiency of $$70\% $$ is used to drive a refrigerator having a coefficient of performance of $$5.$$ The energy absorbed from low temperature reservoir by the refrigerator for each $$kJ$$ of energy absorbed from high temperature source by the engine is

Consider a steam power plant using a reheat cycle as shown. Steam leaves the boiler and enters the turbine at $$4$$ $$MPa,$$ $${350^ \circ }C$$ $$\left( {{h_3} = 3095\,\,kJ/kg} \right).$$ After expansion in the turbine to $$400$$ $$kPa$$ $$\left( {{h_4} = 2609\,\,kJ/kg} \right)$$, the steam is reheated to $${350^ \circ }C$$ $$\left( {{h_5} = 3170\,kJ/kg} \right),$$ and then expanded in a low pressure turbine to $$10$$ $$kPa$$ $$\left( {{h_6} = 2165\,kJ/kg} \right).$$

The thermal efficiency of the plant neglecting pump work is

A solar collector receiving solar radiation at the rate of $$0.6$$$$kW/{m^2}$$ transforms it to the internal energy of a fluid at an overall efficiency of $$50\% $$. The fluid heated to $$350$$ $$K$$ is used to run a heat engine which rejects heat at $$313$$ $$K.$$ If the heat engine is to deliver $$2.5$$ $$kW$$ power, then minimum area of the solar collector required would be

A gas contained in a cylinder is compressed, the work required for compression being $$5000$$ $$kJ.$$ During the process, heat interaction of $$200$$ $$kJ$$ causes the surroundings to the heated. The change in internal energy of the gas during the process is

A centrifugal pump is required to pump water to an open water tank situated $$4$$ $$km$$ away from the location of the pump through a pipe of diameter $$0.2$$ $$m$$ having Darcy’s friction factor of $$0.01.$$ The average speed of water in the pipe is $$2$$ $$m/s.$$ if it is maintain a constant head of $$5$$ $$m$$ in the tank, neglecting other minor losses, the absolute discharge pressure at the pump exist is ______________

At a hydro electric power plant site, available head and flow rate are $$24.5$$ $$m$$ and $$10.1{m^3}/s$$ respectively. If the turbine to be installed is required to run at $$4.0$$ revolution per second ($$rps$$) with an overall efficiency of $$90\% $$, the suitable type of turbine for this site is _______________.