Laplace transform of $$\cos \left( {\omega t} \right)$$ is

The Laplace transform of $${e^{i5t}}$$ where $$i = \sqrt { - 1} ,$$

At least one eigenvalue of a singular matrix is

At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has

Curl of vector $$\,V\left( {x,y,x} \right) = 2{x^2}i + 3{z^2}j + {y^3}k\,\,$$ at $$x=y=z=1$$ is

The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {z^2} = 9\,\,$$ is __________.

The chance of a student passing an exam is $$20$$% The chance of a student passing the exam and getting above $$90$$% marks in it is $$5$$%. Given that a student passes the examination, the probability that the student gets above $$90$$% marks is

Consider the following differential equation $${{dy} \over {dt}} = - 5y;$$ initial condition: $$y=2$$ at $$t=0.$$
The value of $$y$$ at $$t=3$$ is

The values of function $$(x)$$ at $$5$$ discrete points are given below:

Using Trapezodial rule with step size of $$0.1,$$ the value of $$\,\int\limits_0^{0.4} {f\left( x \right)\,dx\,\,} $$ is __________.

The initial velocity of an object is $$40$$ $$m/s.$$ The acceleration a of the object is given by the following expression: a$$=-0.1v$$
Where v is the instantaneous velocity of the object. The velocity of the object after $$3$$ seconds will be _______

For the truss shown in the figure, the magnitude of the force (in kN) in the member SR is

A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s) of 2 kg mass ball after the impact will be ____________

Within a boundary layer for a steady incompressible flow, the Bernoulli equation

The head loss for a laminar incompressible flow through a horizontal circular pipe is $${h_1}$$. Pipe length and fluid remaining the same, if the average flow velocity doubles and the pipe diameter reduces to half its previous value, the head loss is $${h_2}.$$ The ratio $${h_2}/{h_1}$$ is

If the fluid velocity for a potential flow is given by $$V\left( {x,y} \right) = u\left( {x,y} \right)i + v\left( {x,y} \right)j$$ with usual notations, then the slope of potential line at $$(x, y)$$ is

For a fully developed laminar flow of water (dynamic viscosity $$0.001$$ $$Pa$$-s) through a pipe of radius $$5$$ $$cm,$$ the axial pressure gradient is $$-10$$ $$Pa/m$$. The magnitude of axial velocity (in $$m/s$$) at a radial location of $$0.2$$ $$cm$$ is ____________

If a foam insulation is added to a $$4cm$$ outer diameter pipe as shown as figure, the critical radius of insulation (in $$cm$$) is ________.

A cylindrical uranium fuel rod of radius 5 mm in a nuclear reactor is generating heat at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The rod is cooled by a liquid (convective heat transfer coefficient $$1000W/{m^2}$$-$$K$$) at $${25^ \circ }C$$. At steady state, the surface temperature (in $$K$$) of the rod is

In the laminar flow of air $$(Pr=0.7)$$ over a heated plate, if $$\delta $$ and $${\delta _T}$$ denote, respectively the hydrodynamic and thermal boundary layer thickness, then

The total emissive power of a surface is $$500$$ $$W/{m^2}$$ at a temperature $${T_1}$$ and $$1200$$ $$W/{m^2}$$ at a temperature $${T_2}$$, where the temperatures are in Kelvin. Assuming the emissivity of the surface to be constant, the ratio of the temperatures $${{{T_1}} \over {{T_2}}}$$ is

A balanced counter-flow heat exchanger has a surface area of $$20\,\,{m^2}$$ and overall heat transfer coefficient of $$20$$ $$W/{m^2}$$-$$K$$. Air $$\left( {{C_p} = 1000J/kg - K} \right)$$ entering at $$0.4$$ $$kg/s$$ and $$280$$ $$K$$ is to preheated by the air leaving the system at $$0.4$$ $$kg/s$$ and $$300$$ $$K.$$ The outlet temperature (in $$K$$) of the preheated air is

Annual demand of a product is $$50,000$$ units and the ordering cost is Rs. $$7,000$$ per order. Considering the basic economic order quantity model, the economic order quantity is $$10,000$$ units. When the annual inventory cost is minimized, the annual inventory holding cost (in Rs.) is ___________

A project consists of $$7$$ activities. The network along with the time durations (in days) for various activities is shown in the figure.

The minimum time (in days) for completion of the project is _____________

Sales data of a product is given in the following table:

Regarding forecast for the month of June, which one of the following statements is TRUE?

A manufacturer has the following data regarding a product:
Fixed cost per month $$=$$ Rs. $$50,000$$
Variable cost per unit $$=$$ Rs. $$200$$
Selling price per unit $$=$$ Rs. $$300$$
Production capacity $$=$$ $$1500$$ units per month

If the production is carried out at $$80\% $$ of the rated capacity, then the monthly profit (in Rs.) is ______________

The uniaxial yield stress of a material is $$300$$ $$MPa.$$ According to Von mises criterion, the shear yield stress $$(in$$ $$MPa)$$ of the material is ___________.

A cube and a sphere made of cast iron (each of volume $$1000\,\,c{m^3}$$) were cast under identical conditions. The time taken for solidifying the cube was $$4$$ $$s.$$ The solidification time (in $$s$$) for the sphere is _____________________

The flowing stress (in $$MPa$$) of a material is given by $$\sigma = 500{\varepsilon ^{0.1}},$$
When $$\varepsilon $$ is true strain. The Young's modulus of elasticity of the material is $$200$$ $$GPa.$$ A block of thickness $$100$$ $$mm$$ made of this material is compressed to $$95$$ $$mm$$ thickness and then the load is removed. The final dimension of the block (in $$mm$$) is ____________

In a two stage wire drawing operation, the fractional reduction (ratio of change in cross sectional area to initial cross sectional area ) in the first stage is $$0.4.$$ The fractional reduction in the second stage is $$0.3.$$ The overall fractional reduction is

During a $$TIG$$ welding process, the arc current and arc voltage were $$50$$ $$A$$ and $$60$$ $$V,$$ respectively, when the welding speed was $$150$$ $$mm/min.$$ In another process, the $$TIG$$ welding is carried out at a welding speed of $$120mm/min$$ at the same arc welding voltage and heat input to the material so that weld quality remains the same. The welding current (in $$A$$) for this process is

A single point cutting tool with $${0^0}$$ rake angle is used in an orthogonal machining process. At a cutting speed of $$180$$ $$m/min$$ the thrust force is $$490N.$$ If the coefficient of friction between the tool and the chip is $$0.7.$$ Then the power consumption (in $$kW$$) for the machining operation is ____________

Which one of the following statements is TRUE?

During the development of a product, an entirely new process plan is made based on design logic, examination of geometry and tolerance information. This type of process planning is known as

A drill is positioned at point $$P$$ and it has to proceed to point $$Q.$$ The coordinates of point $$Q$$ in the incremental system of defining position of a point in $$CNC$$ part program will be

A rod is subjected to an uniaxial load within linear elastic limit. When the change in the stress is $$200$$ MPa, the change in the strain is $$0.001.$$ If the Poisson's ratio of the rod is $$0.3$$ the modulus of rigidity (in GPa) is ________ .

In a plane stress condition, the components of stress at a point are $${\sigma _x} = 20$$ MPa, $${\sigma _y} = 80$$ MPa and $${\tau _{xy}} = 40$$ MPa, The maximum shear stress (in MPa) at the point is

A cantilever beam OP is connected to another beam PQ with a pin joint as shown in figure. A load of $$10$$ kN is applied at the mid-point of PQ. The magnitude of bending moment (in kN-m) at fixed end $$O$$ is

A hollow shaft (d₀ = $$2$$d_i, where d₀ and d_i are the outer and inner diameters respectively) needs to transmit $$20$$ kW power at $$3000$$ RMP. If the maximum permissible shear stress is $$30$$MPa, d₀ is

The deflection of a spring with 20 active turns under a load of 1000 N is 10 mm. The spring is made into two pieces each of 10 active coils and placed in parallel under the same load. The deflection of the system is

A rope brake dynamometer attached to the crank shaft of an $$I.C.$$ engine measures a brake power of $$10$$ kW when the speed of rotation of the shaft is $$400$$ rad/s. The shaft torque (in N-m) sensed by the dynamometer is ________

A hollow shaft of $$1$$ m length is designed to transmit a power of $$30$$ kW at $$700$$ rpm. The maximum permissible angle of twist in the shaft is $${1^0}.$$ The inner diameter of the shaft is $$0.7$$ times the outer diameter. The modulus of rigidity is $$80$$ GPa. The outside diameter (in mm) of the shaft is _______

A cantilever beam with square cross-section of 6 mm side is subjected to a load of 2kN normal to the top surface as shown in figure. The Young’s modulus of elasticity of the material of the beam is 210 GPa. The magnitude of slope (in radian) at Q (20 mm from the fixed end) is ________

A gas is stored in a cylindrical tank of inner radius $$7$$ m and wall thickness $$50$$ mm. The gage pressure of the gas is $$2$$ MPa. The maximum shear stress (in MPa) in the wall is

In a certain slider-crank mechanism, lengths of crank and connecting rod are equal. If the crank rotates with a uniform angular speed of 14 rad/s and the crank length is 300 mm, the maximum acceleration of the slider (in m/s²) is _________

The number of degrees of freedom of the planetary gear train shown in the figure is

In a spring-mass system the mass is m and the spring constant is $$k$$. The critical damping coefficient of the system is $$0.1$$ kg/s. In another spring mass system, the mass is $$2m$$ and the spring constant is $$8k$$. The critical damping coefficient (in kg/s) of this system is __________

A single-degree-freedom spring mass system is subjected to a sinusoidal force of $$10$$ N amplitude and frequency $$\omega $$ along the axis of the spring. The stiffness of the spring is $$150$$N/m, damping factor is $$0.2$$ and the undamped natural frequency is $$10$$$$\omega $$. At steady state, the amplitude of vibration (in m) is approximately

In a Rankine cycle, the enthalpies at turbine entry and outlet are $$3159$$ $$kJ/kg$$ and $$2187$$ $$kJ/kg$$ respectively. If the specific pump work is $$2$$ $$kJ/kg,$$ the specific steam consumption (in $$kg/k$$$$W$$-$$h$$) of the cycle based on net output is ________________

The Vander-waals equation of state is $$\left( {p + {a \over {{v^2}}}} \right)\left( {v - b} \right) = RT,$$ where $$p$$ is pressure, $$v$$ is specific volume, $$T$$ is temperature and $$R$$ is characteristic gas constant. The $$S{\rm I}$$ unit of a is

Work is done on an adiabatic system due to which its velocity changes from $$10$$ $$m/s$$ to $$20$$ $$m/s,$$ elevation increases by $$20$$ $$m$$ and temperature increases by $$1$$ $$K$$. The mass of the system is $$10$$ $$kg,$$ $${{C_v} = 100J\left( {kg.K} \right)}$$ and gravitational acceleration is $$10\,\,m/{s^2}.$$ If there is no change in any other component of the energy of the system, the magnitude of total work done (in $$kJ$$) on the system is _______________.

The $$COP$$ of a Carnot heat pump operating between $${6^ \circ }C$$ and $${37^ \circ }C$$ is ____________

One kg of air ($${R = 287\,\,J/kg}$$-$$K$$) under goes irreversible process between equilibrium state $$1\,\,\left( {{{20}^ \circ }C,\,\,0.9\,\,{m^3}} \right)$$ and equilibrium state $$2\,\,\left( {{{20}^ \circ }C,\,\,0.6\,\,{m^3}} \right).$$ The change in entropy $${s_2} - {s_1}$$ (in $$J/kg$$-$$K$$) is _________

Which of the following statements regarding a Rankine cycle with reheating are TRUE?
(i) increase in average temperature of heat addition
(ii) reduction in thermal efficiency
(iii) drier steam at the turbine exit