$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to

A real square matrix $$A$$ is called skew-symmetric if

The number of linear independent eigenvectors of matrix $$A = \left[ {\matrix{ 2 & 1 & 0 \cr 0 & 2 & 0 \cr 0 & 0 & 3 \cr } } \right]$$ is ________.

$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$ is

The value of the line integral $$\,\,\oint\limits_C {\overrightarrow F .\overrightarrow r ds,\,\,\,} $$ where $$C$$ is a circle of radius $${4 \over {\sqrt \pi }}\,\,$$ units is ________.

Here, $$\,\,\overrightarrow F x,y = y\widehat i + 2x\widehat j\,\,$$ and $$\,\overrightarrow r $$ is the UNIT tangent vector on the curve $$C$$ at an arc length s from a reference point on the curve. $$\widehat i$$ and $$\widehat j$$ are the basis vectors in the $$X-Y$$ Cartesian reference. In evaluating the line integral, the curve has to be traversed in the counter-clockwise direction.

Three cards were drawn from a pack of $$52$$ cards. The probability that they are a king, a queen, and a jack is

The area (in percentage) under standard normal distribution curve of variable $$Z$$ within limits from $$-3$$ to $$+3$$ is ________.

The root of the function $$f\left( x \right) = {x^3} + x - 1$$ obtained after first iteration on application of Newton-Raphson scheme using an initial guess of $${x_0} = 1$$ is

Solutions of Laplace's equation having continuous second-order partial derivatives are called

A circular disc of radius 100 mm and mass 1 kg, initially at rest at position A, rolls without slipping down a curved path as shown in figure. The speed v of the disc when it reaches position B is _________ m/s.
Acceleration due to gravity g = 10 m/s²

A rigid rod $$(AB)$$ of length $$L = \sqrt 2 $$ m is undergoing translational as well as rotational motion in the $$ x-y$$ plane (see the figure). The point $$A$$ has the velocity $${V_1} = \widehat i + 2\widehat jm/s.$$ The end $$B$$ is constrained to move only along the $x$$ direction.

The magnitude of the velocity $${V_2}$$ (in $$m/s$$) at the end $$B$ is __________

An inextensible massless string goes over a frictionless pulley. Two weights of 100 N and 200 N are attached to the two ends of the string. The weights are released from rest, and start moving due to gravity. The tension in the string (in N) is __________ .

A force $$F$$ is acting on a bent bar which is clamped at one end as shown in the figure. The CORRECT free body diagram is

For a two-dimensional flow, the velocity field is $$\overrightarrow u = {x \over {{x^2} + {y^2}}}\widehat i + {y \over {{x^2} + {y^2}}}\widehat j,$$ where $$\widehat i$$ and $$\widehat j\,\,$$ are the basis vectors in the $$x$$-$$y$$ Cartesian coordinate system .
Identify the CORRECT statements from below.
(1) The flow is incompressible
(2) The flow is unsteady
(3) $$y$$-component of acceleration, $${a_y} = {{ - y} \over {{{\left( {{x^2} + {y^2}} \right)}^2}}}$$
(4) $$x$$-component of acceleration , $${a_x} = {{ - \left( {x + y} \right)} \over {{{\left( {{x^2} + {y^2}} \right)}^2}}}$$

The water jet exiting from a stationary tank through a circular opening of diameter $$300$$ $$mm$$ impinges on a rigid wall as shown in the figure. Neglect all minor losses and assume the water level in the tank to remain constant. The net horizontal force experienced by the wall is ___________ $$kN.$$

Density of water is $$1000\,\,kg/{m^3}.$$

Acceleration due to gravity $$g = 10\,\,m/{s^2}.$$

Consider a fully developed steady laminar flow of an incompressible fluid with viscosity $$\mu $$ through a circular pipe of radius $$R.$$ Given that the velocity at a radial location of $$R/2$$ from the center-line of the pipe is $${U_1},$$ the shear stress at the wall is $$K\mu {U_1}/R,$$ where $$K$$ is _________________.

A channel of width $$450$$ $$mm$$ branches into two sub-channels having width $$300$$ $$mm$$ and $$200$$ $$mm$$ as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is $$0.9$$ $$3$$ $$m/s,$$ and the velocity in the sub-channel of width $$200$$ $$mm$$ is $$3$$ $$m/s,$$ the velocity in the sub-channel of width $$300$$ $$mm$$ is _____________ $$m/s$$.

Assume both inlet and outlet to be at the same elevation.

For a certain two-dimensional incompressible flow, velocity field is given by $$2xy\widehat i - {y^2}\widehat j.$$ The streamlines for this flow are given by the family of curves

Steady one-dimensional heat conduction takes place across the faces $$1$$ and $$3$$ of a composite slab consisting of slabs $$A$$ and $$B$$ in perfect contact as shown in the figure, where $${k_A},\,\,{k_B}$$ denote the respective thermal conductivities. Using the data as given in the figure, the interface temperature $${T_2}$$ $$\left( {in\,{\,^ \circ }C\left. \, \right)} \right.$$ is ______________.

A cylindrical steel rod, $$0.01$$ $$m$$ in diameter and $$0.2$$ $$m$$ in length is first heated to $${750^ \circ }C$$ and then immersed in a water bath at $${100^ \circ }C.$$ The heat transfer coefficient is $$250\,W/{m^2}$$-$$K.$$ The density, specific heat and thermal conductivity of steel are $$\rho = 7801\,\,kg/{m^3},$$ $$c = 473$$ $$J/kg$$-$$K,$$ and $$k = 43$$ $$W/m$$-$$K,$$ respectively. The time required for the rod to reach $${300^ \circ }C$$ is ________ seconds.

Grashof number signifies the ratio of

Two large parallel plates having a gap of $$10$$ $$mm$$ in between them are maintained at temperatures $${T_1} = 1000\,\,K$$ and $${T_2} = 400\,\,K$$
Given emissivity values, $${\varepsilon _1} = 0.5,\,\,{\varepsilon _2} = 0.25$$ and Stefan-Boltzmann constant $$\sigma = 5.67 \times {10^{ - 8}}\,$$ $$\,W/{m^2}$$-$$K,$$ the heat transfer between the plates (in $$kW/{m^2}$$) is _____________.

For a heat exchanger, $$\Delta {T_{\max }}$$ is the maximum temperature difference and $$\Delta {T_{\min }}$$ is the minimum temperature difference between the two fluids. $$LMTD$$ is the log mean temperature difference. $${C_{\min }}$$ and $${C_{\max }}$$ are the minimum and the maximum heat capacity rates. The maximum possible heat transfer $$\left( {{Q_{\max }}} \right)$$ between the two fluids is

In $$PERT$$ chart, the activity time distribution is

The demand for a two-wheeler was $$900$$ units and $$1030$$ units in April $$2015$$ and May $$2015,$$ respectively. The forecast for the month of April $$2015$$ was $$850$$ units. Considering a smoothing constant of $$0.6,$$ the forecast for the month of June $$2015$$ is

A firm uses a turning center, a milling center and a grinding machine to produce two parts. The table below provides the machining time required for each part and the maximum machining time available on each machine. The profit per unit on parts $${\rm I}$$ and $${\rm II}$$ are Rs. $$40$$ and Rs. $$100,$$ respectively. The maximum profit per week of the firm is Rs. _______________

A bolted joint has four bolts arranged as shown in figure. The cross sectional area of each bolt is $$25\,m{m^2}.$$. A torque $$T = 200$$ $$N$$-$$m$$ is acting on the joint. Neglecting friction due to clamping force, maximum shear stress in a bolt is ______ $$MPa.$$

Which of the bearings given below SHOULD NOT be subjected to a thrust load?

Engineering strain of a mild steel sample is recorded as $$0.100\% .$$ The true strain is

Equal amounts of a liquid metal at the same temperature are poured into three moulds made of steel, copper and aluminum. The shape of the cavity is a cylinder with $$15$$ $$mm$$ diameter. The size of the moulds are such that the outside temper-ature of the moulds do not increase appreciably beyond the atmospheric temperature during solidification. The sequence of solidification in the mould from the fastest to slowest is

(Thermal conductivities of steel, copper and aluminum are $$60.5, 401$$ and $$237$$ $$W/m$$-$$K,$$ respectively. Specific heats of steel, copper and aluminum are $$434, 385$$ and $$903$$ $$J/kg$$-$$K,$$ respectively. Densities of steel, copper and aluminum are $$7854, 8933$$ and $$2700$$ $$kg/{m^3},$$ respectively.)

Spot welding of two steel sheets each $$2$$ $$mm$$ thick carried out successfully by passing $$4$$ $$kA$$ of current for $$0.2$$ seconds through the electrodes. The resulting weld nugget formed between the sheets is $$5$$ $$mm$$ in diameter. Assuming cylindrical shape for the nugget, the thickness of the nugget is ______________$$mm.$$

For an orthogonal cutting operation, tool material is $$HSS,$$ rake angle is $${22^ \circ },$$ chip thickness is $$0.8$$ $$mm,$$ speed is $$48$$ $$m/min$$ and feed is $$0.4$$ $$mm/rev$$. The shear plane angle (in degrees) is

In a single point turning operation with cemented carbide tool and steel work piece, it is found that the Taylor's exponent is $$0.25.$$ If the cutting speed is reduced by $$50\% $$ then the tool life changes by ____________ times.

Internal gears are manufactured by

In a sheet metal of $$2$$ $$mm$$ thickness a hole of $$10$$ $$mm$$ diameter needs to be punched. The yield strength in tension of the sheet material is $$100$$ $$MPa$$ and its ultimate shear strength is $$80$$ $$MPa.$$ The force required to punch the hole (in $$kN$$) is ______________

Match the following part programming codes with their respective functions

Two optically flat plates of glass are kept at a small angle $$\theta $$ as shown in the figure. Monochromatic light is incident vertically.

If the wavelength of light used to get a fringe spacing of $$1$$ $$mm$$ is $$450$$ $$nm,$$ the wavelength of light (in $$nm$$) to get a fringe spacing of $$1.5$$ $$mm$$ is ________________________.

A point $$P\left( {1,\,\,\,3,\,\,\, - 5} \right)$$ is translated by $$2\widehat i + 3\widehat j - 4\widehat k$$ and then rotated counter clockwise by $${90^ \circ }$$ about the $$z$$-axis. The new position of the point is

A square plate of dimension L $$ \times $$ L is subjected to a uniform pressure load P = 250 MPa on its edges as shown in the figure. Assume plane stress conditions. The Young’s modulus E = 200 GPa.

The deformed shape is a square of dimension 𝐿 − 2$$\delta .$$ If 𝐿 = 2 m and $$\delta $$ = 0.001 m, the Poisson’s ratio of the plate material is __________ .

The state of stress at a point on an element is shown in figure (a). The same state of stress is shownin another coordinate system in figure (b).

The components $$\left( {{\tau _{xx,}}{\tau _{yy,}}{\tau _{xy}}} \right)$$ are given by

The cross-sections of two solid bars made of the same material are shown in the figure. The square cross-section has flexural (bending) rigidity $${{\rm I}_1}$$ while the circular cross-section has flexural rigidity $${{\rm I}_2}$$. Both sections have the same cross-sectional area. The ratio $${{\rm I}_1}$$/$${{\rm I}_2}$$ is

Two circular shafts made of same material, one solid $$(S)$$ and one hollow $$(H)$$, have the same length and polar moment of inertia. Both are subjected to same torque. Here. $${\theta _S}$$ is the twist and $${\tau _S}$$ is the maximum shear stress in the solid shaft, whereas $${\theta _H}$$ is the twist and $${\tau _H}$$ is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

A beam of length $$L$$ is carrying a uniformly distributed load $$w$$ per unit length. The flexural rigidity of the beam is $$EI$$. The reaction at the simple support at the right end is

A rigid link $$PQ$$ is undergoing plane motion as shown in the figure ($${V_P}$$ and $${V_Q}$$ are non-zero). $${V_{QP}}$$ is the relative velocity of point $$Q$$ with respect to point $$P$$.

Which one of the following is TRUE?

The number of degrees of freedom in a planner mechanism having $$n$$ links and $$j$$ simple hinge joints is

The static deflection of a spring under gravity, when a mass of 1 kg is suspended from it, is 1 mm. Assume the acceleration due to gravity g =10 m/s² . The natural frequency of this spring-mass system (in rad/s) is _____________

A single degree of freedom spring-mass system is subjected to a harmonic force of constant amplitude. For an excitation frequency of $$\sqrt {{{3k} \over m}} ,$$ the ratio of the amplitude of steady state response to the static deflection of the spring is __________

The INCORRECT statement about the characteristics of critical point of a pure substance is that

Steam at an initial enthalpy of $$100$$ $$kJ/kg$$ and inlet velocity of $$100$$ $$m/s,$$ enters an insulated horizontal nozzle. It leaves the nozzle at $$200$$ $$m/s.$$ The exit enthalpy (in $$kJ/kg$$) is ___________

The blade and fluid velocities for an axial turbine are as shown in the figure.

The magnitude of absolute velocity at entry is $$300$$ $$m/s$$ at an angle of $${65^0}$$ to the axial direction, while the magnitude of the absolute velocity at exit is $$150$$ $$m/s.$$ The exit velocity vector has a component in the downward direction. Given that the axial (horizontal) velocity is the same at entry and exit, the specific work (in $$kJ/kg$$) is ______________.