GATE ME 2017 Set 1

## GATE ME

The product of eigenvalues of the matrix $$P$$ is $$P = \left[ {\matrix{
2 & 0 & 1 \cr
4 & { - 3} &a

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The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

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A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\

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For the vector $$\overrightarrow V = 2yz\widehat i + 3xz\widehat j + 4xy\widehat k,$$ the value of $$\,\nabla .\left( {

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A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________.

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The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\le

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Consider the following partial differential equation for $$u(x,y)$$ with the constant $$c>1:$$ $$\,{{\partial u} \ove

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$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$

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A Particle of unit mass is moving on a plane. Its trajectory, in polar coordinates, is given by $$r\left( t \right) = {t

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The following figure shows the velocity- time plot for a particle traveling along a straight line. The distance covered

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Two disks A and B with identical mass (m) and radius (R) are initially at rest. They roll down from the top of identical

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Consider the two-dimensional velocity field given by
$$\overrightarrow V = \left( {5 + {a_1}x + {b_1}y} \right)\wideha

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For a steady flow, the velocity field is $$\overrightarrow V = \left( { - {x^2} + 3y} \right)\widehat i + \left( {2xy}

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For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter, the average velocity in

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Consider steady flow of an incompressible fluid through two long and straight pipes of diameters $${d_1}$$ and $${d_2}$$

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The velocity profile inside the boundary layer for flow over a flat plate is given as $${u \over {{U_\infty }}} = \sin \

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Heat is generated uniformly in a long solid cylindrical rod ( diameter $$ = 10\,\,mm$$) at the rate of $$4 \times {10^7}

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Two black surfaces, $$AB$$ and $$BC,$$ of lengths $$5m$$ and $$6m,$$ respectively, are oriented as shown. Both surfaces

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Saturated steam at $${100^ \circ }C$$ condenses on the outside of a tube. Cold fluid enters the tube at $${20^ \circ }C$

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Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production

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Following data refers to the jobs $$(P, Q, R, S)$$ which have arrived at a machine for scheduling. The shortest possible

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Consider the schematic of a riveted lap joint subjected to tensile load $$F,$$ as shown below. Let $$d$$ be the diameter

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A machine element has an ultimate strength $$\left( {{\sigma _u}} \right)$$ of $$600\,\,N/m{m^2},$$ and endurance limit

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A sprue in a sand mould has a top diameter of $$20mm$$ and height of $$200mm.$$ The velocity of the molten metal at the

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In a metal forming operation when the material has just started yielding, the principal stresses are $${\sigma _1} = \,\

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In an arc welding process, welding speed is doubled. Assuming all other process parameters to be constant, the cross sec

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Two cutting tools with tool life equations given below are being compared:
$$\eqalign{
& Tool\,\,1:\,\,\,\,\,\,\,

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Metric thread of $$0.8$$ $$mm$$ pitch is to be cut on a lathe. Pitch of the lead screw is $$1.5$$ $$mm.$$ If the spindle

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A block of length $$200$$ $$mm$$ is machined by a slab milling cutter $$34$$ $$mm$$ in diameter. The depth of cut and ta

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A $$10$$ $$mm$$ deep cylindrical cup with diameter of $$15$$ $$mm$$ is drawn from a circular blank. Neglecting the varia

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Cylindrical pins of diameter $${15^{ \pm 0.020}}\,\,mm$$ are being produced on a machine. Statistical quality control te

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Assume that the surface roughness profile is triangular as shown schematically in the figure. If the peak to valley heig

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Circular arc on a part profile is being machined on a vertical $$CNC$$ milling machine. $$CNC$$ part program using metri

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In the engineering stress-strain curve for mild steel, the Ultimate Tensile Strength $$(UTS)$$ refers to

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The Poisson's ratio for a perfectly incompressible linear elastic material is

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A horizontal bar, fixed at one end (x = 0), has a length of 1 m, and cross-sectional area of 100 mm2. Its elastic modulu

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A point mass of $$100$$ kg is dropped onto a massless elastic bar (cross-sectional area = $$100$$ $$mm$$2 ,length = $$1$

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A rectangular region in a solid is in a state of plane strain. The $$(x,y)$$ coordinates of the corners of the undeforme

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Consider a beam with circular cross-section of diameter $$d.$$ The ratio of the second moment of area about the neutral

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A motor driving a solid circular steel shaft transmits $$40$$ $$kW$$ of power at $$500$$ $$rpm$$. If the diameter of the

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An initially stress-free massless elastic beam of length $$L$$ and circular cross-section with diameter $$d (d < <

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For an inline slider-crank mechanism, the lengths of the crank and connecting rod are 3 m and 4 m respectively. At the i

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In an epicyclic gear train, shown in the figure, the outer ring gear is fixed, while the sun gear rotates counterclockwi

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The damping ratio for a viscously damped spring mass system, governed by the relationship $$\,m{{{d^2}x} \over {d{t^2}}}

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A thin uniform rigid bar of length $$L$$ and mass $$M$$ is hinged at point $$O,$$ located at a distance of $${L \over 3}

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The molar specific heat at constant volume of an ideal gas is equal to $$2.5$$ times the universal gas constant ($$8.314

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Water (density $$ = 1000$$ $$kg/{m^3}$$) at ambient temperature flows through a horizontal pipe of uniform cross section

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One $$kg$$ of an ideal gas (gas constant, $$R = 400$$ $$J/kg.K;$$ specific heat at constant volume, $${c_v} = 1000\,J/kg

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A heat pump absorbs $$10$$ $$kW$$ of heat from outside environment at $$250$$ $$K$$ while absorbing $$15$$ $$kW$$ of wor

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Which one of the following is NOT a rotating machine?

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