The laplace transform of the periodic function $$f(t)$$ described by the curve below
$$i.e.\,\,f\left( t \right) = \left\{ {\matrix{ {\sin \,t,} & {if\left( {2n - 1} \right)\pi < t < 2n\pi \left( {n = 1,2,3,...} \right)} \cr 0 & {otherwise\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr } } \right.$$ is ___________.

Simpson's rule for integration gives exact result when $$f(x)$$ is a polynomial function of degree less than or equal to ________.

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

The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

The differential equation $${y^{11}} + y = 0\,$$ is subjected to the conditions $$y(0) = 0,$$ $$\,\,\,y\left( \lambda \right) = 0.\,\,$$ In order that the equation has non-trivial solutions, the general value of $$\lambda $$ is.

The differential $$\,\,\,{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} + \sin y = 0\,\,$$ is

Given the differential equation $${y^1} = x - y$$ with initial condition $$y(0)=0.$$ The value of $$y(0.1)$$ calculated numerically upto the third place of decimal by the $${2^{nd}}$$ order Runge-Kutta method with step size $$h=0.1$$ is

The cylinder shown below rolls without slipping. In which direction does the friction force act? Towards which of the following points is the acceleration of the point of contact A on the cylinder directed?

A velocity field is given as $$$\overrightarrow V = 3{x^2}y\widehat i - 6xyz\widehat k$$$
Where $$x,y,z$$ are in $$m$$ and $$V$$ $$m/s.$$ Determine if

(i) It represents an incompressible flow
(ii) The flow is irrotational
(iii) The flow is steady .

Shown below are three pipe sections through which water flows as shown. Option $$(a)$$ to $$(d)$$ below pertain to the direction of the net force on the pipe section due to the flow of water.

State which of the options indicate the correct direction of force for pipe $$1,$$ pipe $$2$$ and pipe $$3.$$

Shown below are three cylindrical gates which restrain water in $$2$$-$$D$$ channel. Which gate experiences the maximum vertical component, the minimum vertical component and the maximum horizontal component of the hydrostatic force?

Shown below are three tanks, tank $$1$$ without an orifice tube and tanks $$2$$ and $$3$$ with orifice tubes as shown. Neglecting losses and assuming the diameter of orifice to be much less than that of the tank, write expressions for the exit velocity in each of the three tanks.

The velocity profile across a boundary layer on a flat plate may be approximated as linear $$${V_x}\left( {x,y} \right) = {{{V_0}y} \over {\delta \left( x \right)}}$$$

Where $${{V_0}}$$ is the velocity far away and $$\delta \left( x \right)$$ is the boundary layer thickness at a distance $$x$$ from the leading edge, as shown below.

(a)$$\,\,\,\,\,\,$$ Use an appropriate control volume to determine the rate of mass influx into the
$$\,\,\,\,\,\,$$$$\,\,$$$$\,\,\,\,\,\,$$boundary layer up to $$x.$$

(b)$$\,\,\,\,\,\,$$ Obtain the momentum thickness into the boundary layer up to $$x.$$

(c)$$\,\,\,\,\,\,$$ In which direction (up or down) does the shear stress act on the face $$AB$$ of
$$\,\,\,\,\,\,$$$$\,\,\,\,\,\,\,$$the fluid element shown near the plate?

An electric hot plate is maintained at a temperature of $${350^ \circ }C,$$ and is used to keep a solution boiling at $${95^ \circ }C.$$ The solution is contained in cast iron vessel of wall thickness $$25mm,$$ which is enameled inside to a thickness of $$0.8mm$$. The heat transfer coefficient for the boiling solution is $$5.5$$ $$kW/{m^2}K$$ and the thermal conducti-vities of cast iron and enamel are $$50$$ and $$1.05W/mK,$$ respectively. Calculate the $$OHTC$$ and the rate of heat transfer per unit area.

The radiative heat transfer rate per unit area $$\left( {W/{m^2}} \right)$$ between two plane parallel gray surfaces (emissivity $$=0.9$$) maintained at $$400K$$ and $$300K$$ is ($${{\sigma _b} = }$$ Stephen Boltzmann constant $$5.67 \times {10^{ - 8}}\,\,W/{m^2}{K^4}$$)

For a small project with five jobs, the following data is given
(i) Draw the project network in activity on arc mode
(ii) Under $$PERT$$ assumptions, determine the distribution of project duration.

The outside diameter of a hollow shaft is twice its inside diameter. The ratio of its torque carrying capacity to that of a solid shaft of the same material and the same outside diameter is

The yield strength of a steel shaft is twice its endurance limit. Which of the following torque fluctuations represent the most critical situation according to Soderberg criterion?

Centrifugally casted products have

An aluminium cube of $$10$$ $$cm$$ side has to be cast along with a cylindrical riser of height equal to its diameter. The riser is not insulated on any surface. If the volume shrinkage of aluminium during solidification is $$6\% $$, calculate shrinkage volume of cube on solidification and minimum size of the riser so that it can provide the shrinkage volume.

In $$DC$$ welding, the straight polarity (electrode negative) results in

A milling cutter having $$8$$ teeth is rotating at $$150$$ $$rpm.$$ If the feed per tooth is $$0.1$$ $$mm,$$ the speed in $$mm$$ per minute is

A shaft of diameter $${20^{\matrix{ { + 0.05} \cr { - 0.15} \cr } }}\,\,mm$$ and a hole of diameter $${20^{\matrix{ { + 0.20} \cr { + 0.10} \cr } }}\,\,mm$$ when assembled would yield.

With reference to $$NC$$ machines, which of the following statements is wrong?

Determine the temperature rise necessary to induce buckling in a 1 m long circular rod of diameter 40 mm shown in the figure below. Assume the rod to be pinned at its ends and the coefficient of thermal expansion as 20 $$ \times $$ 10^-6/^oC. Assume uniform heating of the bar.

At a point in a stressed body the state of stress on two planes $$45$$⁰ apart is as shown below. Determine the two principal stresses

A concentrated load $$P$$ acts on a simply supported beam of span $$L$$ at a distance $${L \over 3}$$ from the left support. The bending moment at the point of application of the load is given by

A circular rod of diameter $$d$$ and length $$3d$$ is subjected to a compressive force $$F$$ acting at the top point as shown below. Calculate the stress at the bottom-most support point A.

The compound shaft shown is built-in at the two ends. It is subjected to a twisting moment $$T$$ at the middle. What is the ratio of the reaction torques $${{T_1}}$$ and $${{T_2}}$$ at the ends?

The number of degrees of freedom of a five link plane mechanism with five revolute pairs as shown in the figure is:

When a system executes an irreversible cycle:

The relationship $$\,{\left( {\partial T/{\partial _p}} \right)_h} = 0$$ holds good for :

At the point of a pure substance, the number of degrees of freedom is :

In the vicinity of the triple point, the vapour pressures of liquid and solid ammonia are respectively given by
$$\eqalign{ & \ln \,\,P = 15.16 - 3063/T \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,and \cr & \ln \,\,P = 18.70 - 3754/T \cr} $$

Where $$p$$ is in atmospheres and $$T$$ is in Kelvin. What is the temperature at the triple point.

A vessel of volume $$\,\,1.0\,\,{m^3}$$ contains a mixture of liquid water and steam in equilibrium at $$1.0$$ bar.Given that $$90\% $$ of the volume is occupied by the steam, find the dryness fraction of the mixture. Assume, at $$1.0$$ bar, $${\,{v_f} = 0.001\,\,{m^3}/kg}$$ and $${\,{v_g} = 1.7\,\,{m^3}/kg}.$$

An insulated rigid vessel contains a mixture of fuel and air. The mixture is ignited by a minute spark. The contents of the vessel experience.

A body of weight $$100$$ $$N$$ falls freely a vertical distance of $$50m$$.The atmospheric drag force is $$0.5N.$$ For the body, the work interaction is

The first law of thermodynamics takes the form $$W = - \Delta \,H$$ when applied to :

A rigid insulated cylinder has two compartments separated by a thin membrane. While one compartment contains one kmol nitrogen at a certain pressure and temperature, the other contains one kmol carbon dioxide at the same pressure and temperature. The membrane is ruptured and the two gases are allowed to mix.

Assume that the gases behave as ideal gases. Calculate the increase in entropy of the contents of the cylinder. Universal gas constant equal to $$8314.3$$ $$J/kmol$$ $$K.$$

A vertical cylinder with a freely floating piston contains $$0.1$$ $$kg$$ air at $$1.2$$ bar and a small electrical resistor. The resistor is wired to an external $$12$$ Volt battery. When a current of $$1.5$$ amps is passed through the resistor for $$90$$ $$secs,$$ the piston sweeps a volume of $$0.01$$ $${m^3}.$$ Assume $$(i)$$ piston and the cylinder are insulated and $$(ii)$$ air behaves as a ideal gas with $${C_v} = 700\,\,J/kg\,K.$$
Find the rise in temperature of air.

A reversible heat transfer demands:

Round the clock cooling of an apartment having a load of $$300$$ $$MJ/day$$ requires and air-conditioning plant of capacity about

During the phase change of a pure substance:

Which among the following relations is/are valid ONLY for reversible process undergone by a pure substance?

Figure below shows a reversible heat engine $${E_R}$$ having heat interactions with three constant temperature systems. Calculate the thermal efficiency of the heat engine.

The equivalent evaporation $$(kg/hr)$$ of a boiler producing $$2000$$ $$kg/hr$$ of steam with enthalpy content of $$2426$$ $$kJ/kg$$ from feed water at temperature $${40^ \circ }C$$ (liquid enthalpy $$=168$$ $$kJ/kg$$) is (Enthalpy of vaporization of water at $${100^0}C = 2258$$ $$kJ/kg$$).

In a single stage single row impulse turbine, the steam is entering at a velocity of $$1200$$ $$m/s$$ with a nozzle angle of $${20^ \circ }$$ and leaving the blade in the axial direction. The ratio of blade velocity to the tangential (whirl) velocity of steam is $$0.6.$$ Sketch the velocity diagram and calculate the work output.

Cavitation in a hydraulic reaction turbine is most likely to occur at the turbine

Specific speed of a Kaplan turbine ranges between