The particular solution for the differential equation $${{{d^2}y} \over {d{t^2}}} + 3{{dy} \over {dx}} + 2y = 5\cos x$$ is

In the Gauss - elimination for a solving system of linear algebraic equations, triangularization leads to

The eigen values of $$\left[ {\matrix{ 1 & 1 & 1 \cr 1 & 1 & 1 \cr 1 & 1 & 1 \cr } } \right]$$ are

If a function is continuous at a point its first derivative

The expression curl $$\left( {grad\,f} \right)$$ where $$f$$ is a scalar function is

The one dimensional heat conduction partial difference equation $$\,\,{{\partial T} \over {\partial t}} = {{{\partial ^2}T} \over {\partial {x^2}}}\,\,$$ is

$${i^i}$$, where $$i\, = \,\sqrt { - 1} $$ is given by

A wheel of mass m and radius r is in accelerated rolling motion without slip under a steady axial torque $$T$$. If the coefficient of kinetic friction is $$\mu $$, the friction force from the ground on the wheel is

AB and CD tow uniform and identical bars of mass 10kg each, as shown. The hinges at A and B are frictionless. The assembly is released from rest and motion occurs in the vertical plane. At the instant that the hinge B passes the point B, the angle between the two bars will be

A rod of length 1m is sliding in a corner as shown in. At an instant when the rod makes an angle of 60 degrees with the horizontal plane, the velocity of point A on the rod is 1m/s. The angular velocity of the rod at this instant is

Match 4 correct pairs between List $$I$$ and List $$II$$. No credit will be given for partially correct matching

List $$I$$
(a) Collision of particles
(b) Stability
(C) Satellite motion
(D) Spinning top

List $$II$$
(1) Euler’s equation of motion
(2) Minimum kinetic energy
(3) Minimum potential energy
(4) Impulse-momentum principle
(5) Conservation of moment of momentum

A ball A of mass m falls under gravity from a height h and strikes another ball B of mass m which supported at rest on a spring of stiffness k. Assume perfectly elastic impact. Immediately after the impact

For laminar flow through a long pipe, the pressure drop per unit length increases.

A fluid is one which can be defined as a substance that:

The dimension of surface tension is :

A mercury manometer is used to measure the static pressure at a point in a water pipe as shown in Fig. The level difference of mercury in the two limbs is $$10$$ $$mm.$$ The gauge pressure at that point is

For a given heat flow and for the same thickness, the temp drop across the material will be maximum for

For a current carrying wire of $$20$$ $$mm$$ diameter and having $$k=0.5W/mK$$ is exposed to air $$\left( {h = 20W/{m^2}K} \right),$$ maximum heat dissiption occurs when thickness of insulation is

Match List - $${\rm I}$$ with List - $${\rm II}$$ and select the correct answer using the code given below the Lists:

List - $${\rm I}$$
$$A.$$ Grashof number
$$B.$$ Schmid number
$$C.$$ Weber number
$$D.$$ Fourier number

List - $${\rm II}$$
$$1.$$ Mass diffusion
$$2.$$ Transient heat conduction
$$3.$$ Free convection
$$4.$$ Forced convection
$$5.$$ Surface tension
$$6.$$ Radiation

In an assembly line for assembling toys, five workers are assigned tasks which take times of $$10, 8, 6, 9$$ and $$10$$ minutes respectively. The balance delay for the line is

If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that

A job shop incurs a cost of Rs.$$60$$/- per day for each day a job is in the shop. At the beginning of a month there are five jobs in the shop with the following data:

Which schedule will minimize the total cost?
What is the minimum total cost?
Which jobs (if any) fail to meet their due dates?

A solid shaft can resist a bending moment of $$3.0$$ $$kNm$$ and a twisting moment of $$4.0$$ $$kNm$$ together, then the maximum torque that can be applied is

If the ratio of the diameter of river hole to the pitch of rivets is $$0.25$$, then the tearing Efficiency of the joint is

The bolts in a rigid flanged coupling connecting two shafts transmitting power are subjected to

In spur gears, the circle on which the involute is generated is called the

In thick film hydrodynamic journal bearings, the coefficient of friction

The pouring basin in a sand mold is $$200$$ $$mm$$ diameter $$100$$ $$mm$$ height. The mould cavity is a cube of $$125$$ $$mm$$ side and must be filled in $$25$$ seconds. The maximum velocity must ensure laminar flow into the mould cavity and in the gates of circular cross-section. Design the sprue to avoid aspiration effects.

Assume
(a) constant temperature,
(b) no loss in the velocity head and
(c) diameter of gate $$=8$$ times diameter at the sprue base.

Properties of the molten metal are $$\gamma = 0.9\,\,m{m^2}$$ per second, $$\rho = 700\,\,kg/{m^3},$$ & $${C_p} = 33.6\,\,J/mol$$-$$K.$$ The diameter of sprue to avoid aspiration effect.

Light impurities in the molten metal are prevented from reaching the mold cavity by providing a

A wire of $$0.1mm$$ diameter is drawn from a rod of $$15mm$$ diameter dies giving reductions of $$20\% $$, $$40\% $$ and $$80\% $$ are available. For minimum error in the final size, the number of stages and reduction at each stage respectively would be

Preheating before welding is done to

The cross section weld bead is shown in fig the profile of the bead and the fusion zone are taken circular for convenience. Bead width and radii of curvature of circular profiles are given in fig. The bead height and depth of penetration are

Tool life of a $$10$$ hours is obtained when cutting with a single point tool at $$63m/min.$$ If Taylor's constant $$C=257.35,$$ tool life on doubling the velocity will be

The rake angle in drill

The rake angle in drill

A $$50$$ $$mm$$ diameter disc is to be punched out from a carbon steel sheet $$1.0$$ $$mm$$ thick. The diameter of the punch should be (Take clearance as $$3\% $$ of sheet thickness)

The fit on a hole-shaft system is specified as $$H7$$-$$s6.$$ The type of fit is

If the two principal strains at a point are 1000 $$ \times $$ 10^-6 and -600 $$ \times $$ 10^-6, then the maximum shear strain is

Two bars AB and BC are connected by a frictionless hinge at B. The assembly is supported and loaded as shown in Fig. Draw the shear force and bending moment diagrams for the combined beam AC, clearly labeling the important values. Also indicate your sign convention.

A thin cylinder of $$100$$mm internal diameter and $$5$$mm thickness is subjected to an internal pressure of $$10$$ MPa and a torque of $$2000$$ Nm. Calculate the magnitudes of the principal stresses.

A rod of length $$1$$ $$m$$ is sliding in a corner as shown in Fig.$$2.7.$$ At an instant when the rod makes an angle of $$60$$ degrees with the horizontal plane, the velocity of point $$A$$ on the rod is $$1$$ $$m/s$$. The angular velocity of the rod at this instant is

Below Fig show a planetary gear train. Gears 2, 4 and 5 have 24, 40 and 144 teeth respectively. Gear 5 is fixed. Gear 2 is rotating clockwise at 700 rpm. What will be the rpm of the arm and gear 4?

A mass of $$1$$ kg is suspended by means of $$3$$ springs as shown in Figure. The spring constants $$\,{K_1},\,{K_2}\,\,$$ and $$\,{K_3},$$ are respectively $$1$$ $$kN/m,$$ $$3$$ $$kN/m$$ and $$2$$ $$kN/m.$$ The natural frequency of the system is approximately )

A cylinder of mass 1 kg and radius 1 m is connected by two identical springs at a height of $$0.5m$$ above the center as shown in the Fig. The cylinder rolls without slipping. If the spring constant is $$30$$ $$kN/m$$ for each spring, find the natural frequency of the system for small oscillations.

When wet steam flows through a throttle valve and remains wet at exit

The specific heats of an ideal gas depend on its

Match List - $${\rm I}$$ with List - $${\rm II}$$

List - $${\rm I}$$
$$A.$$ Cetane number
$$B.$$ Approach and range
$$C.$$ $${\left( {{{\delta T} \over {\delta P}}} \right)_h} \ne 0$$
$$D.$$ $$dh = {c_p}\,\,dT,$$ even when pressure varies

List - $${\rm II}$$
$$1.$$ Ideal gas
$$2.$$ Vander Waals gas
$$3.$$ $$S.\,{\rm I}.$$ engine
$$4.$$ $$C.\,{\rm I}.$$ engine
$$5.$$ Cooling towers
$$6.$$ Heat exchangers

For reversible adiabatic compression in a steady flow process, the work transfer per unit mass is

A solar energy based heat engine which receives $$80$$ $$kJ$$ of heat at $${100^ \circ }C$$ and rejects $$70$$ $$kJ$$ of heat to the ambient at $${30^ \circ }C$$ is to be designed. The thermal efficiency of the heat engine is

For two cycles coupled in series, the topping cycle has an efficiency of $$30\% $$ and the bottoming cycle has an efficiency of $$20\% $$. The overall combined cycle efficiency is

An iron cube at a temperature of $$400$$ deg $$C$$ is dropped into an insulated bath containing $$10$$ $$kg$$ water at $${25^ \circ }C.$$ The water finally reaches a temperature of $${50^ \circ }C$$ at steady state. Given that the specific heat of water is equal to $$4186$$ $$J/kgK,$$ find the entropy changes for the iron cube and the water. Is the process irreversible ? If so, why?

A steam plant has the boiler efficiency of $$92\% $$, turbine efficiency (mechanical) of $$94\% $$, generator efficiency of $$92\% $$ and cycle efficiency of $$44\% $$. If $$6\% $$ of the generated power is used to run the auxiliaries, the overall plant efficiency is

In a hydroelectric station, water is available at the rate of $$175{m^3}/s$$ under a head of $$18m.$$ The turbines run at a speed of $$150$$ $$rpm$$ with overall efficiency of $$82\% $$. Find the number of turbines required if they have the maximum specific speed of $$460.$$