If $$F(s)$$ is the Laplace transform of the function $$f(t)$$ then Laplace transform of $$\int\limits_0^t {f\left( x \right)dx} $$ is

The solution of $${{d\,y} \over {d\,x}} = {y^2}$$ with initial value $$y(0)=1$$ is bounded in the interval is

Let $$X$$ and $$Y$$ be two independent random variables. Which one of the relations $$b/w$$ expectation $$(E),$$ variance $$\left( {{V_{ar}}} \right)$$ and covariance $$\left( {{C_{ov}}} \right)$$ given below is FALSE?

The area of a triangle formed by the tips of vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is

$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$

If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.

The minimum value of function $$\,\,y = {x^2}\,\,$$ in the interval $$\,\,\left[ {1,5} \right]\,\,$$ is

The number of linearly independent eigen vectors of $$\left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$$ is

If a square matrix $$A$$ is real and symmetric then the eigen values

The partial differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} + {{\partial \phi } \over {\partial x}} + {{\partial \phi } \over {\partial y}} = 0\,\,\,\,$$ has

If $$\phi (x,y)$$ and $$\psi (x,y)$$ are function with continuous 2^nd derivatives then $$\phi (x,y)\, + \,i\psi (x,y)$$ can be expressed as an analytic function of x +iy ($$i = \sqrt { - 1} $$) when

A block of mass $$M$$ is related from point $$P$$ on a rough inclined plane with inclination angle $$\theta $$, shown in the figure below. The coefficient of friction is $$\mu $$. If $$\mu < \tan \theta $$, then the time taken by the block to reach anorher point $$Q$$ on the inclined plane, where $$PQ=S,$$ is

During inelastic collision of two particles, which one of the following is conserved?

Consider a steady incompressible flow through a channel as shown below.

The velocity profile is uniform with a value of $${u_0}$$ at the inlet section $$A$$. The velocity profile at section B down stream is

$$$u\left\{ {\matrix{ {{V_m}{y \over \delta },} & {0 \le y \le \delta } \cr {{V_m},} & {\delta \le y \le H - \delta } \cr {{V_m}{{H - y} \over \delta },} & {H - \delta \le y \le H} \cr } } \right.$$$

The ratio $${{{P_A} - {P_B}} \over {{1 \over 2}\rho {u_0}^2}}$$ (where $${{P_A}}$$ and $${{P_B}}$$ are the pressure at section $$A$$ and $$B$$ respectively and $$\rho $$ is the density of the fluid ) is

Consider a steady incompressible flow through a channel as shown below.

The velocity profile is uniform with a value of $${u_0}$$ at the inlet section $$A$$. The velocity profile at section B down stream is

$$$u\left\{ {\matrix{ {{V_m}{y \over \delta },} & {0 \le y \le \delta } \cr {{V_m},} & {\delta \le y \le H - \delta } \cr {{V_m}{{H - y} \over \delta },} & {H - \delta \le y \le H} \cr } } \right.$$$

The ratio $${{{V_m}} \over {{u_0}}}$$ is

Consider an incompressible laminar boundary layer flow over a flat plate of length $$L$$, aligned with the direction of an incoming uniform free stream. If $$F$$ is the ratio of the drag force on the front half of the plate to the drag force on the rear half, then

Consider steady laminar incompressible axi-symmetric fully developed viscous flow through a straight circular pipe of constant cross - sectional area at a Reynolds number of $$5.$$ The ratio of inertia force to viscous force on a fluid particle is

Oil flows through a $$200mm$$ diameter horizontal cast iron pipe (friction factor, $$f=0.0225$$) of length $$500m.$$ The volumetric flow rate is $$0.2{m^3}/s.$$ The head loss (in $$m$$) due to friction is (assume $$g=9.81$$ $$m/{s^2}$$)

Which combination of the following statements about steady incompressible forced vortex flow is correct?

P: Shear stress is zero at all points in the flow.
Q: Velocity is directly proportional to the radius from the centre of the vortex.
R: Total mechanical energy per unit mass is constant in the entire flow field.
S: Total mechanical energy per unit mass is constant in the entire flow field.

In a steady flow through a nozzle, the flow velocity on the nozzle axis is given by $$u = {u_0}\left( {1 + 3x/L} \right),\,\,$$ where $$x$$ is the distance along the axis of the nozzle from its inlet plane and $$L$$ is the length of the nozzle. The time required for a fluid particle on the axis to travel from the inlet to the exit plane of the nozzle is .

Oil in a hydraulic cylinder is compressed from an initial volume $$2{m^3}$$ to $$1.96{m^3}$$. If the pressure of oil in the cylinder changes from $$40$$ $$MPa$$ to $$80$$ $$MPa$$ during compression, the bulk modulus of elasticity of oil is .........

Newton's law of viscosity states that the shear stress in a fluid is proportional to :

The temp distribution within the Laminar thermal boundary layer over a heated isothermal flat plate is given by
$$\left( {T - {T_w}} \right)/\left( {{T_\infty } - {T_w}} \right) = \left( {3/2} \right)\,\,\left( {y/{\delta _t}} \right) - \left( {1/2} \right){\left( {y/{\delta _t}} \right)^3},$$
where $${{T_w}}$$ and $${{T_ \propto }}$$ are the temp of plate and free stream respectively, and $$'y'$$ is the normal distance measuread from the plate. The ratio of Average to the local Nussult number based on the thermal boundary layer thickness $${{\delta _t}}$$ is given by

In a counter flow heat exchanger, hot fluid enters at $${65^ \circ }C$$ and cold fluid leaves at $${30^ \circ }C.$$ mass flow rate of the hot fluid is $$1$$ $$Kg/s$$ and that of cold fluid is $$2$$ $$kg/s$$. Specific heat of the hot fluid is $$10$$ $$kgK$$ and that of cold fluid is $$5$$ $$kj/kgK.$$ The $$LMTD$$ for the heat ecchanger is

The average heat transfer coefficient on a thin hot vertical plate suspended in still air can be determined from observations of the change in plate temperature with time as it cools. Assume the plate temp to be uniform at any instant of time and radiation heat exchange with the surroundings is negligible. The ambient temperature is $${25^ \circ }C,$$ the plate has a total surface area of $$0.1{m^2}$$ and a mass of $$4$$ kg.

The specific heat of the plate material is $$2.5KJ/KgK.$$ The convective heat transfer coefficient in $$W/{m^2}K,$$ at instant when the plate temp is $${225^ \circ }C$$ and the change in plate temp with time $$dT/dt=-0.02K/s,$$ is

Building has to be maintained at $${21^ \circ }C$$ (dry bulb) and $${14.5^ \circ }C$$ (wet bulb). The dew point temp under these conditions is $${10.17^0}C$$. The outside temp is $$ - {23^0}C$$ (dry bulb) and internal and external surface heat transfer coefficients are $$8\,\,W/{m^2}K$$ and $$23$$ $$W/{m^2}K$$ respectively. If the building wall has a thermal conductivity of $$1.2$$ $$W/mK,$$ the minimum thickness (m) of wall required to prevent condensation is

Consider steady one-dimensional heat flow in a plate of $$20mm$$ thickness with a uniform heat generation of $$80MW/{m^3}.$$ The left and right faces are kept at constant temperatures of $${160^0}C$$ and $${120^0}C$$ respectively. The plate has a constant thermal conductivity of $$200W/mK.$$

The minimum temp within the plate in degree $$C$$ is

Consider steady one-dimensional heat flow in a plate of $$20mm$$ thickness with a uniform heat generation of $$80MW/{m^3}.$$ The left and right faces are kept at constant temperatures of $${160^0}C$$ and $${120^0}C$$ respectively. The plate has a constant thermal conductivity of $$200W/mK.$$

The location of maximum temp within the plate from left face is

Capacities of production of an item over $$3$$ consecutive months in regular time are $$100,$$ $$100$$ and $$80$$ and in overtime are $$20,$$ $$20$$ and $$40.$$ The demands over those $$3$$ months are $$90,$$ $$130$$ and $$110.$$ The cost of production in regular time and overtime are respectively Rs. $$20$$ per item and Rs. $$24$$ per item. Inventory carrying cost is Rs. $$2$$ per item per month. The levels of starting and final inventory are nil. Backorder is not permitted. For minimum cost of plan, the level of planned production in overtime in the third month is

In a machine shop, pins of $$15mm$$ diameter are produced at a rate of $$1000$$ per month and the same is consumed at a rate of $$500$$ per month. The production and consumption continue simultaneously till the maximum inventory is reached. Then inventory is allowed to reduced to zero due to consumption. The lot size of production is $$1000.$$ If backlog is not allowed, the maximum inventory levels is

The maximum level of inventory of item is $$100$$ and it is achieved with infinite replenishment rate. The inventory becomes zero over one and half month due to consumption at a uniform rate. This cycle continuous throughout the year. Ordering cost is Rs.$$100$$ per order and inventory carrying cost is Rs. $$10$$ per item per month. Annual cost (in Rs. ) of the plan, neglecting material cost, is

A gear set has a pinion with $$20$$ teeth and a gear with $$40$$ teeth. The pinion runs at $$30rev/s$$ and transmits a power of $$20KW$$. The teeth are on the $${20^ \circ }$$ full-depth system and have a module of $$5mm.$$ The length of the line of action is $$19mm.$$

The resultant force on the contacting gear tooth is $$N$$ is

A block brake shows below has a free width of $$300$$ $$mm$$ and a mean coefficient of friction of $$0.25$$. For an activating force of $$400$$ $$N$$, the braking torque in $$N$$-$$m$$ is

A natural feed journal bearing of diameter $$50mm$$ and length $$50mm$$ operating at $$20$$ revolutions/sec carries a load of $$2kN$$. The lubricant used has a viscosity of $$20mPa$$-$$s$$. The radial clearance is $$50\mu m.$$ The Sommerfeld number for the bearings is

A ball bearing operating at a load $$F$$ has $$8000$$hours of life. The life of the bearing, in hours, when the load is doubled to $$2F$$ is

A gear set has a pinion with $$20$$ teeth and a gear with $$40$$ teeth. The pinion runs at $$30rev/s$$ and transmits a power of $$20KW$$. The teeth are on the $${20^ \circ }$$ full-depth system and have a module of $$5mm.$$ The length of the line of action is $$19mm.$$

The contact ratio of the contacting tooth is

A gear set has a pinion with $$20$$ teeth and a gear with $$40$$ teeth. The pinion runs at $$30rev/s$$ and transmits a power of $$20KW$$. The teeth are on the $${20^ \circ }$$ full-depth system and have a module of $$5mm.$$ The length of the line of action is $$19mm.$$

The center distance for the above gear set in $$mm$$ is

A bolted joint in shown below. The maximum shear stress, in $$MPa,$$ in bolts $$A$$ and $$B$$ respectively are

A thin spherical pressure vessel of $$200mm$$ diameter and $$1mm$$ thickness is subjected to an internal pressure varying from $$4$$ to $$8MPa.$$ Assume that the yield, ultimate and endurance strength of material are $$600,800,400MPa$$respectively. The factor of safety as per Goodman's relation is

Which of the following engineering materials is the most suitable candidate for hot chamber die casting

If a particular $$Fe$$-$$C$$ alloy contains less than $$0.83$$% carbon, it is called

A low carbon steel bar of $$147$$ $$mm$$ diameter with length of $$630$$ $$mm$$ is being turned with uncoated carbide insert. The observed tool lives are $$24$$ and $$12$$ for cutting velocities of $$90$$ $$m/min$$ and $$120$$ $$m/min$$ respectively. The feed and depth of cut are $$0.2$$ $$mm/rev$$ and $$2$$ $$mm$$ respectively. Use the unmachined diameter to calculate the cutting velocity.

Neglect over travel or approach of the tool. When tool life is $$20min,$$ the machining time in $$min$$ for a single pass is

Which type of motor is used in axial or spindle drives of $$CNC$$ m/c tools

A hole is specified as $${40^{\matrix{ {0.050} \cr {0.000} \cr } }}\,\,mm.$$ The mating shaft has a clearance fit with minimum clearance of $$0.01mm.$$ The tolerance on the shaft is $$0.04mm.$$ The maximum clearance in $$mm$$ between the hole and the shaft is

The force requirement in a blanking operation of low carbon steel sheet is $$5.0$$ $$kN.$$ The thickness of the sheet is $$'t'$$ and diameter of the blanked part is $$'d'.$$ For the same work material, if the diameter of the blanked part is increased to $$1.5d$$ and thickness is reduced to $$0.4t$$, the new blanking force in $$kN$$ is

In orthogonal turning of low carbon steel pipe with principal cutting edge angle of $$90,$$ the main cutting force is $$1000$$$$N$$ and the feed force is $$800N.$$ The shear angle is $$25$$ and orthogonal rake angle is zero. Employing Merchants theory, the ratio of friction force to the normal force acting on the cutting tool is

In orthogonal turning of medium carbon steel, the specific machining energy is $$2$$ $$kJ/m{m^3},$$ the cutting velocity, feed and depth of cut are $$120m/min,$$ $$0.2mm/rev$$ and $$2mm$$ respectively. The main cutting force $$N$$ is

In orthogonal turning of a low carbon steel bar of diameter $$150mm$$ with uncoated carbide tool, the cutting velocity is $$90m/min,$$ The feed is $$0.24mm/rev$$ and the depth of cut is $$2mm.$$ The chip thickness obtained is $$0.48$$$$mm.$$ If the orthogonal rake angle is zero, and the principle cutting edge angle is $$90,$$ the shear angle in degrees is

A low carbon steel bar of $$147$$ $$mm$$ diameter with length of $$630$$ $$mm$$ is being turned with uncoated carbide insert. The observed tool lives are $$24$$ and $$12$$ for cutting velocities of $$90$$ $$m/min$$ and $$120$$ $$m/min$$ respectively. The feed and depth of cut are $$0.2$$ $$mm/rev$$ and $$2$$ $$mm$$ respectively. Use the unmachined diameter to calculate the cutting velocity.

When tool life is $$20min.$$ The cutting velocity in $$m/min$$ is

Two metallic sheets, each of $$2.0$$ $$mm$$ thickness are welded in lap joint configuration by resistance spot welding at a welding current of $$10$$ $$KA$$ and welding time of $$10$$ milliseconds. A spherical fusion zone extending up to the full thickness of each sheet is formed. The properties of the metallic sheets are given below:
Ambient temp $$=293$$ $$K$$
Melting point temp $$=1793$$ $$K$$
Density $$ = 7000kg/{m^3}$$
Latent heat of fusion $$=300$$ $$kJ/kg$$
Specific heat $$=800$$ $$J/kgK$$

Assume:
(i) Contact resistance along sheet - sheet interface is $$500$$ micro-ohm and along
$$\,\,\,\,\,\,$$electrode-sheet interface is zero.:
(ii) No conductive heat loss through the bulk sheet material; and
(iii) The complete weld fusion zone is at the melting temperature, the melting
$$\,\,\,\,\,\,\,\,\,$$efficiency (in %) of the process is .........

A $$DC$$ welding machine with a linear power source characteristic provides open circuit voltage of $$80V$$ and short circuit current of $$800A,$$ during welding with the machine , the measured are current is $$500A$$ corresponding to an arc length of $$5.0$$ $$mm$$ and the measured are current $$460A$$ corresponding to an arc length of $$7.0$$ $$mm.$$ The linear voltage $$(E)$$ & arc length $$(L)$$ characteristic of the welding arc can be given as (where $$E$$ is in Volts and $$L$$ is in $$mm$$)

Which of the following is a solid state welding process.

In open die forging, a disc of diameter $$200mm$$ height $$60mm$$ is compressed without any barreling effect. The final diameter of the disc is $$400mm.$$ The true strain is

The thickness of metallic sheet is reduced from an initial value of $$16mm$$ to a final value of $$10mm$$ in one single pass rolling with a pair of cylindrical rollers each of diameter of $$400mm.$$ The bite angle in degrees will be

Match the correct combination for following metal working processes.

Volume of a cube of a side $$''l''$$ and volume of sphere of radius $$' r '$$ are equal. Both the cube and the sphere are solid and same material. They are being cast. The ratio of the solidification time of the cube to the same of the sphere is

A $$200$$ $$mm$$ long down sprue has an area of cross section of $$650$$ $$m{m^2}$$ where the pouring basin meets the down sprue (i.e. at the beginning of the down sprue). A constant head of molten metal is maintaining by the pouring basin. The molten metal flow rate is $$6.5 \times {10^5}\,\,\,m{m^3}/s.$$ considering the end of down sprue to be open to atmosphere and an acceleration due to gravity of $${10^4}mm/{s^2},$$ the area of the down sprue in $$m{m^2}$$ at its end (avoiding aspiration effects) should be

A stepped steel shaft shown below is subjected to $$10$$ N-m torque. If the modulus of rigidity is $$80$$ GPa, the strain energy in the shaft in N-mm is

A uniformly loaded propped cantilever beam and its free body diagram are shown below. The reactions are

A machine frame shown in the figure below is subjected to a horizontal force of $$600$$ $$N$$ parallel to $$Z$$-direction.

The maximum principal stresses in $$MP$$a and the orientation of the corresponding principal plane in degrees are respectively

A machine frame shown in the figure below is subjected to a horizontal force of $$600$$ $$N$$ parallel to $$Z$$-direction.

The normal and shear stresses in $$MP$$a at point $$P$$ are respectively

A disc of radius' r 'has a hole of radius 'r/2' cut-out as shown. The centroid of the remaining disc (shaded portion) at a radial distance from the centre ''O'' is

In a simply-supported beam loaded as shown below, the maximum bending moment in N-m is

A steel rod of length $$L$$ and diameter $$D$$, fixed at both ends, is uniformly heated to a temperature rise of $$\Delta T.$$ The Young's modulus is $$E$$ and the coefficient of linear expansion is $$'\alpha '\,.$$ The thermal stress in the rod is

The input link Q₂P of a four bar linkage is rotated at $$2$$ rad/s counter clockwise direction as shown below. The angular velocity of the coupler PQ in rad/s, at an instant when $$\angle {O_4}{O_2}P = {180^ \circ },$$ is

A quick return mechanism is shown in below. The crank OS is drive at $$2$$ rev/s in counter clockwise direction

The angular speed of PQ in rev/s when the block R attains maximum speed during forward stroke (stroke with slower speed) is

A quick return mechanism is shown in below. The crank OS is drive at $$2$$ rev/s in counter clockwise direction

If the quick return ratio is $$1:2$$, then the length of the crank in mm is

The natural frequency of the system shown below is

The equation of motion of a harmonic oscillator is given by $${{{d^2}x} \over {d{t^2}}} + 2\xi {\omega _n}{{dx} \over {dt}} + \omega _n^2\,x = 0\,\,\,$$ and the initial conditions at $$t=0$$ are $$\,x\left( 0 \right) = X,{{dx} \over {dt}}\left( 0 \right) = 0.\,\,$$
The amplitude of $$x(t)$$ after $$n$$ complete cycles is

For an under damped harmonic oscillator, resonance

The speed of an engine varies from $$210$$ rad/s to $$190$$ rad/s. During cycle the change in kinetic energy is found to be $$400$$ Nm. The inertia of the flywheel in kgm² is

Which combination of the following statements is correct?

The incorporation of re-heater in a steam power plant.

$$P:$$ always increases the thermal efficiency of the plant.
$$Q:$$ always increases the dryness function of steam at condenser inlet.
$$R:$$ always increases the mean temperature of heat addition
$$S:$$ always increases the specific work output.

Which of the following relationships is valid only for reversible process undergone by a closed system of simple compressible substance (neglect changes in kinetic and potential energy?

An irreversible heat engine extracts heat from a high temperature source at a rate of $$100kW$$ and rejects heat to a sink at a rate of $$50kW.$$ The entire work output of the heat engine is used to drive a reversible heat pump operating between a set of independent isothermal heat reservoirs at $${17^ \circ }C$$ and $${75^ \circ }C$$. The rate (in $$kW$$) at which the heat pump delivers heat to its high temperature sink is

A heat transformer is a device that transfers a part of the heat, supplied to it at an intermediate temperature, to a high temperature reservoir while rejecting the remaining part to a low temperature heat sink. In such a heat transformer, $$100kJ$$ of heat is supplied at $$350$$ $$K.$$ The maximum amount of heat in $$kJ$$ that can be transferred to $$400K,$$ when the rest is rejected to a heat sink at $$300K$$ is

Water has a critical specific volume of $$0.003155$$ $${m^3}/kg.$$ A closed and rigid steel tank of volume $$0.025{m^3}$$ contains a mixture of water and steam at $$0.1MPa.$$ The mass of the mixture is $$10$$ $$kg.$$ The tank is now slowly heated. The liquid level inside the tank.

Match the items in columns $$I$$ and $$II$$.

Column $$I$$
$$P:$$ Centrifugal compressor
$$Q:$$ Centrifugal pump
$$R:$$ Pelton wheel
$$S:$$ Kaplan turbine

Column $$II$$
$$1:$$ Axial flow
$$2:$$ Surging
$$3:$$ Priming
$$4:$$ Pure impulse

The inlet angle of runner blades of a Francis turbine is $${90^0}.$$ The blades are so shaped that the tangential component of velocity at blade outlet is zero. The flow velocity remains constant throughout the blade passage and is equal to half of the blade velocity at runner inlet. The blade efficiency of the runner is

A model of a hydraulic turbine is tested at a head of $$1/{4^{th}}$$ of that under which the full scale turbine works. The diameter of the model is half of that of the full scale turbine. If $$N$$ is the $$RPM$$ of the full scale turbine, then the $$RPM$$ of the model will be