The Laplace Transform of $$f\left( t \right) = {e^{2t}}\sin \left( {5t} \right)\,u\left( t \right)$$ is

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram's selection is $$1/6$$ and that of Ramesh is $$1/8$$. What is the probability that only one of them will be selected?

If any two columns of a determinant $$P = \left| {\matrix{ 4 & 7 & 8 \cr 3 & 1 & 5 \cr 9 & 6 & 2 \cr } } \right|$$ are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?

Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, = \,\,\sin t,\,\,\,z\left( t \right)\,\, = \,\,{2 \over \pi }t,\,\,\,0 \le t \le {\pi \over 2}.$$ The length of the curve is ________.

The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is

Consider an ant crawling along the curve $$\,{\left( {x - 2} \right)^2} + {y^2} = 4,$$ where $$x$$ and $$y$$ are in meters. The ant starts at the point $$(4, 0)$$ and moves counter $$-$$clockwise with a speed of $$1.57$$ meters per second. The time taken by the ant to reach the point $$(2, 2)$$ is _________ (in seconds).

The velocity field on an incompressible flow is given by
$$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j\,$$ $$ + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k,\,\,$$
where $${{a_1} = 2}$$ and $${{c_3} = - 4.}$$ The value of $${{b_2}}$$ is ________.

The probability of obtaining at least two $$'SIX'$$ in throwing a fair dice $$4$$ times is

Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?

Find the solution of $${{{d^2}y} \over {d{x^2}}} = y$$ which passes through origin and the point $$\left( {ln2,{3 \over 4}} \right)$$

Simpson's $${1 \over 3}$$ rule is used to integrate the function $$f\left( x \right) = {3 \over 5}{x^2} + {9 \over 5}\,\,$$ between $$x=0$$ and $$x=1$$ using the least number of equal sub-intervals. The value of the integral is __________.

Given two complex numbers $${z_1} = 5 + \left( {5\sqrt 3 } \right)i$$ and $${z_2} = {2 \over {\sqrt 3 }} + 2i,$$ the argument of $${{{z_1}} \over {{z_2}}}$$ in degrees $$i$$

For the truss shown in figure, the magnitude of the force in member PR and the support reaction at R are respectively

A ball of mass 0.1 kg, initially at rest, is dropped from height of 1m. Ball hits the ground and bounces off the ground. Upon impact with the ground, the velocity reduces by 20%. The height (in m) to which the ball will rise is _______

Two identical trusses support a load of 100 N as shown in the figure. The length of each truss is 1.0m; cross-sectional area is 200 mm² ; Young’s modulus E = 200GPa. The force in the truss AB (in N) is _____ .

Water $$\left( {\rho = 1000kg/{m^3}} \right)$$ flows through a venturimeter with inlet diameter $$80$$ $$mm$$ and throat diameter $$40$$ $$mm.$$ The inlet and throat gauge pressures are measured to be $$400$$ $$kPa$$ and $$130$$ $$kPa$$ respectively. Assuming the venturimeter to be horizontal and neglecting friction, the inlet velocity (in $$m/s$$) is ___________.

Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of

Air ( $${\rho = 1.2\,\,kg/{m^3}}$$ and kinematic viscosity, $${v = 2 \times {{10}^{ - 5}}{m^2}/s}$$ ) with a velocity of $$2m/s$$ flows over the top surface of a flat plate of length $$2.5m.$$ If the average value of friction coefficient is $${C_f} = {{1.328} \over {\sqrt {{{{\mathop{\rm Re}\nolimits} }_x}} }},\,\,$$ the total drag force (in $$N$$) per unit width of the plate is ____________

Match the following pairs:

The velocity field on an incompressible flow is given by
$$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j$$ $$$ + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k,$$$

Where $${a_1} = 2$$ and $${c_3} = - 4.$$ The value of $${b_2}$$ is _____________.

A $$10$$ $$mm$$ diameter electrical conductor is covered by an insulation of $$2$$ $$mm$$ thickness. The conductivity of the insulation is $$0.08$$ $$W/m$$-$$K$$ and the convection coefficient at the insulation surface is $$10$$ $$W/{m^2}$$ -$$K.$$ Addition of further insulation of the same material will

The Blasius equation related to boundary layer theory is a

For flow of viscous fluid over a flat plate, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is

For flow through a pipe of radius $$R,$$ the velocity and temperature distribution are as follows:
$$U\left( {r,x} \right) = {C_1}$$ and $$T\left( {r,x} \right) = {C_2}\left[ {1 - {{\left( {{r \over R}} \right)}^3}} \right],$$
where $${C_1}$$ and $${C_2}$$ are constants. The bulk mean temperature is given by

$${T_m} = {2 \over {{U_m}{R^2}}}\int\limits_0^R {u\left( {r,x} \right)T\left( {r,x} \right)rdr,} $$
with $${{U_m}}$$ being the mean velocity of flow. The value of $${T_m}$$ is

Following data refers to the activities of a project, where, node $$1$$ refers to the start and node $$5$$ refers to the end of the project.

The critical path $$(CP)$$ in the network is

For a canteen, the actual demand for disposable cups was $$500$$ units in January and $$600$$ units in February. The forecast for the month of January was $$400$$ units. The forecast for the month of March considering smoothing coefficient as $$0.75$$ is_______________

A machine element is subjected to the following bi-axial state of stress: $$\,\,{\sigma _x} = 80MPa;\,\,{\sigma _y} = 20MPa;\,\,{\tau _{xy}} = 40MPa.$$ If the shear strength of the material is $$100 MPa,$$ the factor of safety as per Tresca’s maximum shear stress theory is

Which one of the following is the most conservative fatigue failure criterion?

A horizontal plate has been joined to a vertical post using four rivets arranged as shown in the figure. The magnitude of the load on the worst loaded rivet (in N) is ___________.

The solidification time of a casting is proportional to $${\left( {{V \over A}} \right)^2},$$ Where $$V$$ is the volume of the casting and $$A$$ is the total casting surface area losing heat. Two cubes of same material and size are cast using sand casting process. The top face of one of the cubes is completely insulated. The ratio of the solidification time for the cube with top face insulated to that of the other cube is

Match the following products with preferred manufacturing process:

In a slab rolling operation, the maximum thickness reduction $$\left( {\Delta {h_{\max }}} \right)$$ is given by $$\Delta {h_{\max }} = {\mu ^2}R,$$ where $$R$$ is the radius of the roll and $$\mu $$ is the coefficient of friction between the roll and the sheet. If $$\mu = 0.1,$$ the maximum angle subtended by the deformation zone at the center of the roll (bit angle in degrees) is ____________

In a linear arc welding process, the heat input per unit length is inversely proportional to

A $$DC$$ welding power source has a linear voltage-current $$(V$$-$$I)$$ characteristic with open circuit voltage of $$80V$$ and a short circuit current of $$300A.$$ For maximum arc power, the current (in Amperes) should be set as _______________

Under certain cutting conditions, doubling the cutting speed reduces the tool life to $${\left( {{1 \over {16}}} \right)^{th}}$$ of the original. Taylor’s tool life index $$(n)$$ for this tool workpiece combination will be ______________

An orthogonal turning operation is carried out under the following conditions; rake angle $$ = {5^ \circ },$$ spindle rotational speed $$=400$$ $$rpm,$$ axial feed $$=0.4$$ $$m/min$$ and radial depth of $$cut = 5mm$$. The chip thickness, $${t_c},$$ is found to be $$3mm.$$ The shear angle (in degrees) in this turning process is ________________

Holes of diameter $${25.0^{\matrix{ { + 0.040} \cr { + 0.020} \cr } }}\,\,mm$$ are assembled interchangeably with the pins of diameter $${25.0^{\matrix{ { + 0.005} \cr { + 0.008} \cr } }}\,\,mm.$$

The minimum clearance in the assembly will be

The function of interpolator in a $$CNC$$ machine controller is to

In the assembly shown below, the part dimensions are:
$$\eqalign{ & {L_1} = {22.0^{ \pm 0.01}}\,\,mm, \cr & {L_2} = {L_3} = {10.0^{ \pm 0.005}}\,\,mm, \cr} $$
Assuming the normal distributions of part dimensions, the dimension $${L_4}$$ in $$mm$$ for assembly condition would be,

A triangular facet in a $$CAD$$ model has vertices: $${P_1}\left( {0,0,0} \right);\,\,{P_2}\left( {1,1,0} \right)$$ and $$\,{P_3}\left( {1,1,1} \right).$$ The area of the facet is

Consider a stepped shaft subjected to a twisting moment applied at $$B$$ as shown in the figure. Assume shear modulus, $$G=77$$ GPa. The angle of twist at $$C$$ (in degrees) is ________

A cantilever beam with flexural rigidity of 200 N.m² is loaded as shown in the figure. The deflection (in mm) at the tip of the beam is _______ .

Consider a steel (Young’s modulus $$E = 200$$ $$GP$$a) column hinged on both sides. Its height is $$1.0$$ $$m$$ and cross-section is $$10 mm$$ $$ \times $$ $$20 mm$$. The lowest Euler critical buckling load (in $$N$$) is __________ .

A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of point P is 10m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is ______

A pinion with radius $${r_1}$$, and inertia $${{\rm I}_1}$$ is driving a gear with radius $${r_2}$$ and inertia $${{\rm I}_2}$$ . Torque $${\tau _1}$$ is applied on pinion. The following are free body diagrams of pinion and gear showing important forces ($${F_1}$$and $${F_2}$$) of interaction. Which of the following relations hold true?

Consider a slider crank mechanism with nonzero masses and inertia. A constant torque $$\tau $$ is applied on the crank as shown in the figure. Which of the following plots best resembles variation of crank angle, $$\theta $$ versus time

A mobile phone has a small motor with an eccentric mass used for vibrator mode. The location of the eccentric mass on motor with respect to center of gravity $$(CG)$$ of the mobile and the rest of the dimensions of the mobile phone are shown. The mobile is kept on a flat horizontal surface.

Given in addition that the eccentric mass = $$2$$ grams, eccentricity = $$2.19$$ mm, mass of the mobile = $$90$$ grams, g = $$9.81$$ $$m/{s^2}.$$ Uniform speed of the motor in $$RPM$$ for which the mobile will get just lifted off the ground at the end $$Q$$ is approximately

A precision instrument package (m = 1kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument. Assume that the isolation is designed with its natural frequency significantly lesser than 60Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is _____________.

Considering massless rigid rod and small oscillations, the natural frequency (in rad/s) of vibration of the system shown in the figure is

For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy,

Temperature of nitrogen in a vessel of volume $$2{m^3}$$ is $$288$$ $$K.$$ $$A$$ $$U$$-tube manometer connected to the vessel shows a reading of $$70$$ $$cm$$ of mercury (level higher in the end open to atmosphere). The universal gas constant is $$8314$$ $$J/kmol$$-$$K,$$ atmospheric pressure is $$1.01325$$ bar, acceleration due to gravity is $$9.81$$ $$m/{s^2}$$ and density of mercury is $$13600$$ $$kg/{m^3}.$$ The mass of nitrogen (in $$kg$$) in the vessel is _________

A well insulated rigid container of volume $$1{m^3}$$ contains $$1.0$$ $$kg$$ of an ideal gas $$\left[ {{C_p} =1000\,\,\,J/\left( {kg.K} \right)} \right.$$ and $$\left. {{C_v} = 800J/\left( {kg.K} \right)} \right]$$ at a pressure of $${10^5}\,\,Pa.$$ A stirrer is rotated at constant $$rpm$$ in the container for $$1000$$ rotations and the applied torque is $$100$$ $$N$$-$$m.$$ The final temperature of the gas (in $$K$$) is _______________.

A Carnot engine $$(CE$$-$$1)$$ works between two temperature reservoirs $$A$$ and $$B,$$ where $${T_A} = 900K$$ and $${T_B} = 500K.$$ A second Carnot engine $$(CE$$-$$2)$$ works between temperature reservoirs $$B$$ and $$C,$$ where $${T_c} = 300\,\,K.$$ In each cycle of $$CE$$-$$1$$ and $$CE$$-$$2,$$ all the heat rejected by $$CE$$-$$1$$ to reservoir $$B$$ is used by $$CE$$-$$2.$$ For one cycle operation, if the net $$Q$$ absorbed by $$CE$$-$$1$$ from reservoir $$A$$ is $$150$$ $$MJ,$$ the net heat rejected to reservoir $$C$$ by $$CE$$-$$2$$ (in $$MJ$$) is ______________

Steam enters a well insulated turbine and expands isentropically throughout. At an intermediate pressure, $$20$$ percent of the mass is extracted for process heating and the remaining steam expands isentropically to $$9$$ $$kPa.$$ Inlet to turbine $$P=14$$ $$MPa,$$ $$T = {560^ \circ }C,$$ $$h=3486$$ $$kJ/kg,$$ $$s=6.6$$ $$kJ/(kg.K)$$

Intermediate stage: $$h=2776$$ $$kJ/kg$$
Exit of turbine : $$P=9kPa,$$ $${h_f} = 174\,\,kJ/kg,$$
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{h_g} = 2574\,\,kJ/kg,\,\,\,{s_f} = 0.6kJ/\left( {kg.K} \right); \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,{s_g} = 8.1\,\,kJ/(kg.K) \cr} $$

If the flow rate of steam entering the turbine is $$100$$ $$kg/s,$$ then the work output (in $$MW$$) is __________.