GATE ME 2008

## GATE ME

The matrix $$\left[ {\matrix{
1 & 2 & 4 \cr
3 & 0 & 6 \cr
1 & 1 & P \cr
} } \ri

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The eigen vectors of the matrix $$\left[ {\matrix{
1 & 2 \cr
0 & 2 \cr
} } \right]$$ are written in

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For what values of 'a' if any will the following system of equations in $$x, y$$ and $$z$$ have a solution?
$$$2x+3y=4,$

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Consider the shaded triangular region $$P$$ shown in the figure. What is $$\int\!\!\!\int\limits_p {xy\,dx\,dy\,?} $$

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The value of $$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$ is

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In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is

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Which of the following integrals is unbounded?

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The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is

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Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?

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The divergence of the vector field $$\left( {x - y} \right)\widehat i + \left( {y - x} \right)\widehat j + \left( {x + y

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The directional derivative of the scalar function $$f(x, y, z)$$$$ = {x^2} + 2{y^2} + z\,\,$$ at the point $$P = \left(

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A coin is tossed $$4$$ times. What is the probability of getting heads exactly $$3$$ times?

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It is given that $$y'' + 2y' + y = 0,\,\,\,\,y\left( 0 \right) = 0,y\left( 1 \right) = 0.$$ What is $$y(0.5)$$?

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Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bulle

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The integral $$\oint {f(z)dz} $$ evaluated around the unit circle on the complex plane for $$f(z) = {{\cos z} \over z}$$

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A straight rod length $$L(t),$$ hinged at one end freely extensible at the other end, roates through an angle $$\theta \

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Consider a truss $$PQR$$ loaded at $$P$$ with a force $$F$$ as shown in the figure. The tension in the member $$QR$$ is

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A circular disk of radius $$'R'$$ rolls without slipping at a velocity $$v$$. The magnitude of the velocity at point $$'

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A journal bearing has a shaft diameter of $$40$$ $$mm$$ and a length of $$40mm.$$ The shaft is rotating at $$20$$ $$rad/

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A cubic block of side $$'L'$$ and mass $$'M'$$ is dragged over an oil film across table by a string connects to a hangin

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For a continuity equation given $$\nabla .\overrightarrow V = 0$$ to be valid, $$\overrightarrow V $$ where is the velo

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The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate co

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The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate co

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A two dimensional fluid element rotates like a rigid body. At a point with in the element, the pressure is $$1$$ unit. R

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Steady two dimensional heat conduction takes place in the body shown in the fig below. The normal temperature gradients

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For the three dimensional object shown in the fig below. Five faces are insulated. The sixth face $$(PQRS),$$ which is n

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For flow of fluid over a heated plate, the following fluid properties are known: Viscosity $$=0.001Pa.s;$$ Specific heat

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A hollow enclosure is formed between two infinitely concentric cylinders of radii $$1m$$ and $$2m$$ respectively. Radiat

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The $$LMTD$$ of a counter flow heat exchanger is $${20^ \circ }C$$. The cold fluid enters at $${20^ \circ }C$$ and the h

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For the network below, the objective is to find the length of the shortest path from node $$P$$ to node $$G.$$ Let $${d_

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A moving average system is used for forecasting weekly demand. $${F_1}\left( t \right)$$ and $${F_2}\left( t \right)$$ a

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In an $$M/M/1$$ queuing system, the number of arrivals in an interval of length $$T$$ is a Poisson random variable (i.e.

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Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
$$\eqalign{
& \,\,\,\,\,\,\,\,\,\,\,3x + 2

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Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
$$\eqalign{
& \,\,\,\,\,\,\,\,\,\,\,3x + 2

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For the standard transportation linear programme with $$m$$ sources and $$n$$ destinations and total supply equaling tot

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A set of $$5$$ jobs is to be processed on a single machine. The processing time (in days) is given in the table below. T

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A steel bar of $$10$$ $$mm\,\, \times \,\,50\,\,mm$$ is cantilevered with two $$M12$$ bolts ($$P$$ and $$Q$$) to support

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A steel bar of $$10$$ $$mm\,\, \times \,\,50\,\,mm$$ is cantilevered with two $$M12$$ bolts ($$P$$ and $$Q$$) to support

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A spur gate has a module of $$3mm,$$ number of teeth $$16,$$ a face width of $$36mm$$ and a pressure angle of $${20^ \ci

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Match the type of gears with their most appropriate description
Type of gear
(P)$$\,\,\,\,\,$$ Helical
(Q)$$\,\,\,\,\,$

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A journal bearing has a shaft diameter of $$40mm$$ and a length of $$40mm.$$ The shaft is rotating at $$20$$ $$rad/s$$ a

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One tooth of gear having $$4$$ module and $$32$$ teeth is shown in the figure. Assume that the gear tooth and the corres

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A clutch has outer and inner diameters $$100mm$$ and $$40mm$$ respectively. Assuming a uniform pressure of $$2MPa$$ and

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While cooling, a cubical casting of side $$40mm$$ undergoes $$3\% ,\,\,4\% $$ and $$5\% $$ volume shrinkage during the l

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In a single pass rolling operation, a $$20mm$$ thick plate with plate width of $$100mm$$, is reduced to $$18mm$$ thick.

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In arc welding of a butt joint, the welding speed is to be selected such that highest cooling rate is achieved. Melting

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In a single point turning tool, the side rake angle and orthogonal rake angle are equal. $$\varphi $$ is the principle c

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Orthogonal turning is performed on a cylindrical work piece with shear strength of $$250Mpa.$$ The following conditions

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Orthogonal turning is performed on a cylindrical work piece with shear strength of $$250Mpa.$$ The following conditions

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Internal gear cutting operation can be performed by

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The figure shows an incomplete schematic of a conventional lathe to be used for cutting threads with different pitches.

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In the deep drawing of cups, blanks show a tendency to wrinkle up around the periphery (flange). The most likely cause

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A displacement sensor (a dial indicator) measures the lateral displacement of a mandrel on the taper hole inside a drill

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For generating coon's surface we require

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In the feed drive of a point to point open loop $$CNC$$ drive, a stepper motor rotating at $$200steps/rev$$ drives a tab

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In the feed drive of a point to point open loop $$CNC$$ drive, a stepper motor rotating at $$200steps/rev$$ drives a tab

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A two dimensional fluid element rotates like a rigid body. At a point within the element, the pressure is $$1$$ unit. Ra

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A solid circular shaft of diameter 100 mm is subjected to an axial stress of 50 MPa. It further subjected to a torque of

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For the component loaded with a force $$F$$ as shown in the figure, the axial stress at the corner point $$P$$ is

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A compression spring is made of music wire of $$20$$ mm diameter having a shear strength and shear modulus of $$800$$ MP

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A cantilever type gate hinged at $$Q$$ is shown in the figure. $$P$$ and $$R$$ are the centers of gravity of the cantile

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The rod $$PQ$$ of length $$L$$ and with flexural rigidity $$EI$$ is hinged at both ends. For what minimum force $$F$$ is

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A cylindrical container of radius $$R=1$$m, wall thickness $$1$$mm is filled with water upto a depth of $$2$$m and suspe

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A cylindrical container of radius $$R=1$$m, wall thickness $$1$$mm is filled with water upto a depth of $$2$$m and suspe

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The strain energy stored in the beam with flexural rigidity $$EI$$ and loaded as shown in the figure is

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A planar mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler's

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A uniform rigid rod of mass $$m =1kg$$ and length $$L=1$$ m is hinged at its centre & laterally supported at one e

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The natural frequency of the spring mass system shown in the figure is closest to

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In the figure shown, the system is a pure substance kept in a piston- cylinder arrangement. The system is initially a tw

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In the figure shown, the system is a pure substance kept in a piston- cylinder arrangement. The system is initially a tw

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In the figure shown, the system is a pure substance kept in a piston- cylinder arrangement. The system is initially a tw

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A balloon containing an ideal gas is initially kept in an evacuated and insulated room. The balloon ruptures and the gas

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A rigid, insulated tank is initially evacuated. The tank is connected with a supply line through which air ( assumed to

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In a steady state steady flow process taking place in a device with a single inlet and a single outlet, the work done pe

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$$2$$ moles of oxygen are mixed adiabatically with another $$2$$ moles of oxygen in mixing chamber, so that the final to

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A cycle device operates between three thermal reservoirs, as shown in the figure. Heat is transferred to/from the cyclic

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$$A$$ gas expands in a frictionless piston-cylinder arrangement. The expansion process is very slow, and is resisted by

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A thermal power plant operates on a regenerative cycle with a single open feed water heater, as shown in the figure. For

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Water having a density of $$1000\,\,kg/{m^3},$$ issues from a Nozzle with a velocity of $$10$$ $$m/s$$ and the jet strik

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