Laplace transform of $$\cos \,\left( {\omega t} \right)$$ is $${s \over {{s^2} + {\omega ^2}.}}$$. The Laplace transform of $${e^{ - 2t}}\,\cos \left( {4t} \right)$$ is

The number of accidents occurring in a plant in a month follows Poisson distribution with mean as $$5.2.$$ The probability of occurrence of less than $$2$$ accidents in the plant during a randomly selected month is _________.

Which one of the following equations is a correct identity for arbitrary $$3 \times 3$$ real matrices $$P,Q$$ and $$R$$?

The value of the integral $$\int\limits_0^2 {{{{{\left( {x - 1} \right)}^2}\sin \left( {x - 1} \right)} \over {{{\left( {x - 1} \right)}^2} + \cos \left( {x - 1} \right)}}dx} $$ is

The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is

A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. $$500$$ and a standard deviation of Rs. $$50.$$ The percentage of savings account holders, who maintain an average daily balance more than Rs. $$500$$ is

The solution of the initial value problem $$\,\,{{dy} \over {dx}} = - 2xy;y\left( 0 \right) = 2\,\,\,$$ is

Consider an ordinary differential equation $${{dx} \over {dt}} = 4t + 4.\,\,$$ If $$x = {x_0}$$ at $$t=0,$$ the increment in $$x$$ calculated using Runge-Kutta fourth order multi-step method with a step size of $$\Delta t = 0.2$$ is

If $$z$$ is a complex variable, the value of $$\int\limits_5^{3i} {{{dz} \over z}} $$ is

For the truss shown in the figure, the forces F₁ and F₂ are 9 kN and 3 kN, respectively. The force (in kN) in the member QS is

A uniform slender rod (8 m length and 3 kg mass) rotates in a vertical plane about a horizontal axis 1m from its end as shown in the figure. The magnitude of the angular acceleration (in rad/s² ) of the rod at the position shown is _______

A wardrobe (mass 100 kg, height 4 m, width 2 m, depth 1 m), symmetric about the Y-Y axis, stands on a rough level floor as shown in the figure. A force P is applied at mid-height on the wardrobe so as to tip it about point Q without slipping. What are the minimum values of the force (in Newton) and the static coefficient of friction µ between the floor and the wardrobe, respectively?

A ladder AB of length 5 m and weight (W) 600 N is resting against a wall. Assuming frictionless contact at the floor (B) and the wall (A), the magnitude of the force P (in newton) required to maintain equilibrium of the ladder is __________400

In a statically determinate plane truss, the number of joints $$(j)$$ and the number of members $$(m)$$ are related by

Consider the following statements regarding streamline(s):
(i) It is a continuous line such that the tangent at any point on it shows the velocity vector at that point
(ii) There is no flow across streamlines
(iii) $${{dx} \over u} = {{dy} \over v} = {{dz} \over w}$$ is the differential equation of a streamline, where $$u,v$$ and $$w$$ are velocities in directions $$x,y$$ and $$z,$$ respectively
In an unsteady flow, the path of a particle is a streamline

Which one of the following combinations of the statements is true?

Consider a velocity field $$\overrightarrow V = K\left( {y\widehat i + x\widehat k} \right),$$ where $$K$$ is a constant. The vorticity, $${\Omega _z},$$ is

A flow field which has only convective acceleration is

A flow field which has only convective acceleration is

As the temperature increases, the thermal conductivity of a gas

Match Group $$A$$ with Group $$B$$

Group-A
$$P:$$$$\,\,\,\,\,\,\,$$ Biot number
$$Q:$$$$\,\,\,\,\,\,\,$$ Grashof number
$$R:$$$$\,\,\,\,\,\,\,$$ Prandtl number
$$S:$$$$\,\,\,\,\,\,\,$$ Reynolds number

Group-B
$$1:$$$$\,\,\,\,\,\,\,$$ Ratio of buoyancy to viscous force
$$2:$$$$\,\,\,\,\,\,\,$$ Ratio of inertia force to viscous force
$$3:$$$$\,\,\,\,\,\,\,$$ Ratio of momentum to thermal diffusivities
$$4:$$$$\,\,\,\,\,\,\,$$ Ratio of internal thermal resistance to boundary layer thermal resistance

Two infinite parallel plates are placed at a certain distance apart. An infinite radiation shield is inserted between the plates without touching any of them to reduce heat exchange between the plates. Assume that the emissivities of plates and radiation shield are equal. The ratio of the net heat exchange between the plates with and without the shield is

Demand during lead time with associated probabilities is shown below:

Expected demand during lead time is ___________

The precedence relations and duration (in days) of activities of a project network are given in the table. The total float (in days) of activities $$e$$ and $$f,$$ respectively, are

At a work station, $$5$$ jobs arrive every minute. The mean time spent on each job in the work station is $$1/8$$ minute. The mean steady state number of jobs in the system is __________

A shaft is subjected to pure torsional moment. The maximum shear stress developed in the shaft is $$100$$ $$MPa.$$ The yield and ultimate strengths of the shaft material in tension are $$300$$ $$MPa$$ and $$450$$ $$MPa,$$ respectively. The factor of safety using maximum distortion energy (Von - Mises) theory is __________.

A bolt of major diameter $$12$$ $$mm$$ is required to clamp two steel plates. Cross sectional area of the threaded portion of the bolt is $$84.3$$ $$m{m^2}$$ . The length of the threaded portion in grip is $$30$$ $$mm,$$ while the length of the unthreaded portion in grip is $$8$$ $$mm.$$ Young's modulus of material is $$200$$ $$GPa.$$ The effective stiffness (in $$MN/m$$) of the bolt in the clamped zone is _______.

Ball bearings are rated by a manufacturer for a life of $${10^6}$$ revolutions. The catalogue rating of a particular bearing is $$16$$ $$kN.$$ If the design load is $$2$$ $$kN,$$ the life of the bearing will be $$P \times {10^6}$$ revolutions, where $$P$$ is equal to ___________.

A cylindrical riser of $$6$$ $$cm$$ diameter and $$6$$ $$cm$$ height has to be designed for a sand casting mould for producing a steel rectangular plate casting of $$7cm \times 10cm \times 2cm$$ dimensions having the total solidification time of $$1.36$$ minute. The total solidification time (in minute) of the riser is ___________________

In a rolling process, the maximum possible draft, defined as the difference between the initial and the final thickness of the metal sheet, mainly depends on which pair of the following parameters?
$$P:$$ Strain
$$Q:$$ Strength of the work material
$$R:$$ Roll diameter
$$S:$$ Roll velocity
$$T:$$ Coefficient of friction between roll and work.

Within the Heat Affected Zone (HAZ) in a fusion welding process, the work material undergoes

A butt weld joint is developed on steel plates having yield and ultimate tensile strength of $$500$$ $$MPa$$ and $$700$$ $$MPa,$$ respectively. The thickness of the plates is $$8mm$$ and width is $$20mm.$$ Improper selection of welding parameters caused an undercut of $$3mm$$ depth along the weld. The maximum transverse tensile load (in $$kN$$) carrying capacity of the developed weld joint is _________________

Better surface finish is obtained with a large rake angle because

A cast iron block of $$200$$ $$mm$$ length is being shaped in a shaping machine with a depth of cut of $$4$$ $$mm,$$ feed of $$0.25$$ $$mm/stroke$$ and the tool principal cutting edge angle of $${30^ \circ }.$$ Number of cutting strokes per minutes is $$60.$$ Using specific energy for cutting as $$1.49J/m{m^3},$$ the average power consumption (in Watt) is ___________

Two separate slab milling operations, $$1$$ and $$2,$$ are performed with identical milling cutters. The depth of cut in operation $$2$$ is twice that in operation $$1.$$ The other cutting parameters are identical. The ratio of maximum uncut chip thicknesses in operations $$1$$ and $$2$$ is ___________

A $$GO-No$$ $$GO$$ plug gauge is to be designed for measuring a hole of nominal diameter $$25$$ $$mm$$ with a hole tolerance of $$ \pm 0.015\,\,mm.$$ Considering $$10\% $$ of work tolerance to be the gauge tolerance and no wear condition, the dimension (in $$mm$$) of the $$GO$$ plug gauge as per the unilateral tolerance system is

If the Poisson's ratio of an elastic material is $$0.4$$, the ratio of modulus of rigidity to Young's modulus is _____ .

The number of independent elastic constants required to define the stress-strain relationship for an isotropic elastic solid is ________

A frame is subjected to a load $$P$$ as shown in the figure. The frame has a constant flexural rigidity $$EI$$. The effect of axial load is neglected. The deflection at point $$A$$ due to the applied load $$P$$ is

A thin gas cylinder with an internal radius of 100 mm is subject to an internal pressure of 10 MPa. The maximum permissible working stress is restricted to 100 MPa. The minimum cylinder wall thickness (in mm) for safe design must be ____________.

Torque and angular speed data over one cycle for a shaft carrying a flywheel are shown in the figures. The moment of inertia (in kg.m² ) of the flywheel is _______

A point mass is executing simple harmonic motion with an amplitude of $$10$$ mm and frequency of $$4$$ Hz. The maximum acceleration (m/s²) of the mass is _________.

A single degree of freedom system has a mass of 2kg, stiffness 8 N/m and viscous damping ratio 0.02. The dynamic magnification factor at an excitation frequency of 1.5 rad/s is ___________

Steam with specific enthalpy $$\left( h \right)\,\,3214\,\,kJ/kg$$ enters an adiabatic turbine operating at steady state with a flow rate $$10kg/s.$$ As it expands, at a point where $$h$$ is $$2920$$ $$kJ/kg,$$ $$1.5$$ $$kg/s$$ is extracted for heating purposes. The remaining $$8.5kg/s$$ further expands to the turbine exit, where $$h=2374$$ $$kJ/kg$$. Neglecting changes in kinetic and potential energies, the net power output (in $$kW$$) of the turbine is __________.

A reversed Carnot cycle refrigerator maintains a temperature of $$ - {5^ \circ }C.$$ The ambient air temperature is $${35^ \circ }C.$$ The heat gained by the refrigerator at a continuous rate is $$2.5$$$$kJ/s.$$ The power (in watt) required to pump this heat out continuously is _____________.

A closed system contains $$10$$ $$kg$$ of saturated liquid ammonia at $${10^ \circ }C.$$ Heat addition required to convert the entire liquid into saturated vapour at a constant pressure is $$16.2$$ $$MJ.$$ If the entropy of the saturated liquid is $$0.88$$ $$KJ/kg.K,$$ the entropy (in $$kJ/kg.K$$) of saturated vapour is __________

Two identical metal blocks $$L$$ and $$M$$ (specific heat $$=0.4$$ $$kJ/kg.K$$), each having a mass of $$5$$ $$kg,$$ are initially at $$313$$ $$K.$$ A reversible refrigerator extracts heat from block $$L$$ and rejects heat to block $$M$$ until the temperature of block $$L$$ reaches $$293$$ $$K.$$ The final temperature (in $$K$$) of block $$M$$ is ___________.

Kaplan water turbine is commonly used when the flow through its runner is