GATE ME 2005

## GATE ME

$$A$$ is a $$3 \times 4$$ matrix and $$AX=B$$ is an inconsistent system of equations. The highest possible rank of $$A$$

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Which one of the following is an eigen vector of the matrix $$\left[ {\matrix{
5 & 0 & 0 & 0 \cr
0 &

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$$\int\limits_{ - a}^a {\left[ {{{\sin }^6}\,x + {{\sin }^7}\,x} \right]dx} $$ is equal to

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By a change of variables $$x(u, v) = uv,$$ $$\,\,y\left( {u,v} \right) = {v \over u}$$ in a double integral, the integr

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Changing the order of integration in the double integral
$${\rm I} = \int\limits_0^8 {\int\limits_{{\raise0.5ex\hbox{$\

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Stokes theorem connects

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A lot had $$10$$% defective items. Ten items are chosen randomly from this lot. The probability that exactly $$2$$ of th

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A single die is thrown two times. What is the probability that the sum is neither $$8$$ nor $$9?$$

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Using given data points tabulated below, a straight line passing through the origin is fitted using least squares method

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If $${x^2}\left( {{{d\,y} \over {d\,x}}} \right) + 2xy = {{2\ln x} \over x}$$ and $$y(1)=0$$ then what is $$y(e)$$?

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The complete solution of the ordinary differential equation $${{{d^2}y} \over {d\,{x^2}}} + p{{dy} \over {dx}} + qy = 0$

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Which of the following is a solution of the differential equation $$\,{{{d^2}y} \over {d{x^2}}} + p{{dy} \over {dx}} + \

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Starting from $$\,{x_0} = 1,\,\,$$ one step of Newton - Raphson method in solving the equation $${x^3} + 3x - 7 = 0$$ gi

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A simple pendulum of length 5m, with a bob of mass 1 kg, is in simple harmonic motion. As it passes through its mean pos

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The time variation of the position of a particle in rectilinear motion is given by $$x = 2{t^3} + {t^2} + 2t$$. If 'V' i

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A shell is fired from a cannon. At the instant the shell is just about to leave the barrel, its velocity relative to the

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An elevator (lift) consists of the elevator cage and a counter weight, of mass m each. The cage and the counterweight ar

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Two books of mass 1 kg each are kept on a table, one over the other. The coefficient of friction on every pair of contac

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A 1 kg mass of clay, moving with a velocity of 10 m/s, strikes a stationary wheel and sticks to it. The solid wheel has

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The velocity components in the $$x$$ and $$y$$ directions of a two dimensional potential flow are $$u$$ and $$v$$, respe

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A leaf is caught in a whirlpool. At a given instant, the leaf is at a distance of $$120$$ $$m$$ from the centre of the w

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A venturimeter of $$20$$ $$mm$$ throat diameter is used to measure the velocity of water in a horizontal pipe of $$40$$

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A $$U$$-tube manometer with a small quantity of mercury is used to measure the static pressure difference between two lo

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In case of one dimensional heat conduction in a medium with constant properties, $$T$$ is the temperature at position $$

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Heat flows through a composite slab, as shown below. The depth of the slab is $$1$$ $$m.$$ the $$k$$ values are in $$W/m

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A small copper ball of $$5$$ $$mm$$ diameter at $$500$$ $$K$$ is dropped into an oil bath whose temperature is $$300$$ $

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An un-insulated air conditioning duct of rectangular cross section $$1\,\,m \times 0.5\,m,$$ carrying air at $${20^ \cir

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An un-insulated air conditioning duct of rectangular cross section $$1\,\,m \times 0.5\,m,$$ carrying air at $${20^ \cir

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The following figure was generated from experimental data relating spectral black body emissive power to wave length at

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A solid cylinder (surface $$2$$) is located at the center of a hollow sphere (surface $$1$$). The diameter of the sphere

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Hot oil is cooled from $${80^ \circ }C$$ to $${50^ \circ }C$$ in an oil cooler which uses air as the coolant. The air te

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The distribution of lead-time demand fro an item is as follows:
The reorder level is $$1.25$$ times the expected valu

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A project has six activities $$(A$$ to $$F)$$ with respective activity durations $$7,5,6,6,8,4$$ days. The network has t

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The sales of a product during the last four years were $$860, 880, 870$$ and $$890$$ units. The forecast for the fourth

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Consider a single server queuing model with Poisson arrivals $$\left( {\lambda = 4/hour} \right)$$ and exponential serv

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Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1

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Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1

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A company has two factories $${S_1},$$ $${S_2}$$ and two warehouses $${D_1},$$ $${D_2}$$ . the supplies from $${S_1}$$ a

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Which one of the following is criterion in the design of hydrodynamic journal bearings?

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A band brake consists of a lever attached to one end of the band. The other end of the band is fixed to the ground. The

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A band brake consists of a lever attached to one end of the band. The other end of the band is fixed to the ground. The

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A mould has a downsprue whose length is $$20$$ $$cm$$ and the cross sectional area at the base of the down sprue is $$1\

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Match List - $${\rm I}$$ (products) with List - $${\rm II}$$ (casting process) and select the correct answer using the c

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The strength of a brazed joint

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Spot welding of two $$1$$ $$mm$$ thick sheets of steel (density $$ = 8000kg/{m^3}$$ ) is carried out successfully by pas

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The figure below shows a graph, which qualitatively relates cutting speed and cost per piece produced.
The three curve

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Two tools $$P$$ and $$Q$$ have signature $${5^ \circ } - {5^ \circ } - {6^ \circ } - {6^ \circ } - {8^ \circ } - {30^ \c

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A $$600$$ $$mm$$ $$ \times \,\,30\,\,mm$$ flat surface of a plate is to be finish machined on a shaper. The plate has b

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A $$2$$ $$mm$$ thick metal sheet is to be bent at an angle of one radian with a bend radius of $$100$$ $$mm.$$ If the st

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In order to have interference fit, it is essential that the lower limit of the shaft should be

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Which among the $$NC$$ operations given below are continuous path operations? Arc Welding $$(A$$ $$W)$$, Milling $$(M),

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The tool of an $$NC$$ machine has to move along a circular arc from $$(5,5)$$ to $$(10,10)$$ while performing an operati

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A cantilever beam carries the anti-symmetric load shown, where wo is the peak intensity of the distributed load. Qualita

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A beam is made up of two identical bars AB and BC, by hinging them together at B. The end A is built-in (cantilevered) a

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A mass less beam has a loading pattern as shown in figure. The beam is rectangular cross-section with a width of 30 mm a

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A weighing machine consists of a 2 kg pan resting on a spring in this condition, with the pan resting on the spring, the

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A cantilever beam has the square cross section of 10 mm $$ \times $$ 10 mm. It carries a transverse load of 10 N. Consid

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A mass less beam has a loading pattern as shown in figure. The beam is rectangular cross-section with a width of 30 mm a

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The two shafts $$AB$$ and $$BC$$, of equal length and diameters $$d$$ and $$2d$$, are made of the same material. They ar

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Two identical cantilever beams are supported as shown, with their free ends in contact through a rigid roller. After the

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The number of degrees of freedom of a planar linkage with 8 links and 9 simple revolute joints is

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An instantaneous configuration of a four-bar mechanism, whose plane is horizontal, is shown in the figure below. At this

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An instantaneous configuration of a four-bar mechanism, whose plane is horizontal, is shown in the figure below. At this

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An instantaneous configuration of a four-bar mechanism, whose plane is horizontal, is shown in the figure below. At this

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There are four samples P, Q, R and S, with natural frequencies 64, 96, 128 and 256 Hz, respectively. There are mounted o

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In a spring-mass system, the mass is 0.1 kg and the stiffness of the spring is 1 kN/m. By introducing a damper, the freq

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The following table of properties was printed out for saturated liquid and saturated vapour of ammonia. The titles for o

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The following table of properties was printed out for saturated liquid and saturated vapour of ammonia. The titles for o

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Nitrogen at an initial state of $$10$$ bar, $$1{m^3}$$ and $$300$$ $$K$$ is expanded isothermally to a final volume of $

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The following four figures have been drawn to represent a fictitious thermodynamic cycle, on the $$P$$-$$v$$ and $$T$$-$

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A vapor absorption refrigeration system is a heat pump with three thermal reservoirs as shown in the figure. A refrigera

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In the velocity diagram shown below, $$u=$$ blade velocity, $$C=$$ absolute fluid velocity and $$W = $$ relative veloci

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