GATE 2015 Mechanical Engineering

Under certain cutting conditions, doubling the cutting speed reduces the tool life to $${\left( {{1 \over {16}}} \right)^{th}}$$ of the original. Taylor’s tool life index $$(n)$$ f...

The Laplace Transform of $$f\left( t \right) = {e^{2t}}\sin \left( {5t} \right)\,u\left( t \right)$$ is

An orthogonal turning operation is carried out under the following conditions; rake angle $$ = {5^ \circ },$$ spindle rotational speed $$=400$$ $$rpm,$$ axial feed $$=0.4$$ $$m/min...

The probability of obtaining at least two $$'SIX'$$ in throwing a fair dice $$4$$ times is

The velocity field on an incompressible flow is given by $$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j\,$$ $$ + \left( {{c_1}x + {c_...

If any two columns of a determinant $$P = \left| {\matrix{ 4 & 7 & 8 \cr 3 & 1 & 5 \cr 9 & 6 & 2 \cr } } \right|$$ are interchanged, which one of the following statements regarding...

The solidification time of a casting is proportional to $${\left( {{V \over A}} \right)^2},$$ Where $$V$$ is the volume of the casting and $$A$$ is the total casting surface area l...

Simpson's $${1 \over 3}$$ rule is used to integrate the function $$f\left( x \right) = {3 \over 5}{x^2} + {9 \over 5}\,\,$$ between $$x=0$$ and $$x=1$$ using the least number of eq...

A triangular facet in a $$CAD$$ model has vertices: $${P_1}\left( {0,0,0} \right);\,\,{P_2}\left( {1,1,0} \right)$$ and $$\,{P_3}\left( {1,1,1} \right).$$ The area of the facet is

Air ( $${\rho = 1.2\,\,kg/{m^3}}$$ and kinematic viscosity, $${v = 2 \times {{10}^{ - 5}}{m^2}/s}$$ ) with a velocity of $$2m/s$$ flows over the top surface of a flat plate of leng...

In a slab rolling operation, the maximum thickness reduction $$\left( {\Delta {h_{\max }}} \right)$$ is given by $$\Delta {h_{\max }} = {\mu ^2}R,$$ where $$R$$ is the radius of th...

For flow of viscous fluid over a flat plate, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is

Find the solution of $${{{d^2}y} \over {d{x^2}}} = y$$ which passes through origin and the point $$\left( {ln2,{3 \over 4}} \right)$$

A $$DC$$ welding power source has a linear voltage-current $$(V$$-$$I)$$ characteristic with open circuit voltage of $$80V$$ and a short circuit current of $$300A.$$ For maximum ar...

The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is

The Blasius equation related to boundary layer theory is a

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram's selection is $$1/6$$ and that of Ramesh is $$1/8$$. What is the probabili...

Holes of diameter $${25.0^{\matrix{ { + 0.040} \cr { + 0.020} \cr } }}\,\,mm$$ are assembled interchangeably with the pins of diameter $${25.0^{\matrix{ { + 0.005} \cr { + 0.008} \...

Water $$\left( {\rho = 1000kg/{m^3}} \right)$$ flows through a venturimeter with inlet diameter $$80$$ $$mm$$ and throat diameter $$40$$ $$mm.$$ The inlet and throat gauge pressure...

In a linear arc welding process, the heat input per unit length is inversely proportional to

A $$10$$ $$mm$$ diameter electrical conductor is covered by an insulation of $$2$$ $$mm$$ thickness. The conductivity of the insulation is $$0.08$$ $$W/m$$-$$K$$ and the convection...

Steam enters a well insulated turbine and expands isentropically throughout. At an intermediate pressure, $$20$$ percent of the mass is extracted for process heating and the remain...

Given two complex numbers $${z_1} = 5 + \left( {5\sqrt 3 } \right)i$$ and $${z_2} = {2 \over {\sqrt 3 }} + 2i,$$ the argument of $${{{z_1}} \over {{z_2}}}$$ in degrees $$i$$

A machine element is subjected to the following bi-axial state of stress: $$\,\,{\sigma _x} = 80MPa;\,\,{\sigma _y} = 20MPa;\,\,{\tau _{xy}} = 40MPa.$$ If the shear strength of the...

A well insulated rigid container of volume $$1{m^3}$$ contains $$1.0$$ $$kg$$ of an ideal gas $$\left[ {{C_p} =1000\,\,\,J/\left( {kg.K} \right)} \right.$$ and $$\left. {{C_v} = 80...

Temperature of nitrogen in a vessel of volume $$2{m^3}$$ is $$288$$ $$K.$$ $$A$$ $$U$$-tube manometer connected to the vessel shows a reading of $$70$$ $$cm$$ of mercury (level hig...

For flow through a pipe of radius $$R,$$ the velocity and temperature distribution are as follows: $$U\left( {r,x} \right) = {C_1}$$ and $$T\left( {r,x} \right) = {C_2}\left[ {1 -...

For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy,

A Carnot engine $$(CE$$-$$1)$$ works between two temperature reservoirs $$A$$ and $$B,$$ where $${T_A} = 900K$$ and $${T_B} = 500K.$$ A second Carnot engine $$(CE$$-$$2)$$ works be...

The function of interpolator in a $$CNC$$ machine controller is to

Which one of the following is the most conservative fatigue failure criterion?

Consider an ant crawling along the curve $$\,{\left( {x - 2} \right)^2} + {y^2} = 4,$$ where $$x$$ and $$y$$ are in meters. The ant starts at the point $$(4, 0)$$ and moves counter...

Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, = \,\,\sin t,\,\,\,z\left( t \...

The velocity field on an incompressible flow is given by $$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j$$ $$$ + \left( {{c_1}x + {c_2...

For a canteen, the actual demand for disposable cups was $$500$$ units in January and $$600$$ units in February. The forecast for the month of January was $$400$$ units. The foreca...

A ball of mass 0.1 kg, initially at rest, is dropped from height of 1m. Ball hits the ground and bounces off the ground. Upon impact with the ground, the velocity reduces by 20%. T...

Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of...

A precision instrument package (m = 1kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to t...

Consider a steel (Young’s modulus $$E = 200$$ $$GP$$a) column hinged on both sides. Its height is $$1.0$$ $$m$$ and cross-section is $$10 mm$$ $$ \times $$ $$20 mm$$. The lowest Eu...

A cylindrical uranium fuel rod of radius 5 mm in a nuclear reactor is generating heat at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The rod is cooled by a liquid (convective heat...

A single point cutting tool with $${0^0}$$ rake angle is used in an orthogonal machining process. At a cutting speed of $$180$$ $$m/min$$ the thrust force is $$490N.$$ If the coeff...

The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {z^2} = 9\,\,$$ is __________.

In the laminar flow of air $$(Pr=0.7)$$ over a heated plate, if $$\delta $$ and $${\delta _T}$$ denote, respectively the hydrodynamic and thermal boundary layer thickness, then

Consider the following differential equation $${{dy} \over {dt}} = - 5y;$$ initial condition: $$y=2$$ at $$t=0.$$ The value of $$y$$ at $$t=3$$ is

The Laplace transform of $${e^{i5t}}$$ where $$i = \sqrt { - 1} ,$$

The total emissive power of a surface is $$500$$ $$W/{m^2}$$ at a temperature $${T_1}$$ and $$1200$$ $$W/{m^2}$$ at a temperature $${T_2}$$, where the temperatures are in Kelvin. A...

Within a boundary layer for a steady incompressible flow, the Bernoulli equation

During the development of a product, an entirely new process plan is made based on design logic, examination of geometry and tolerance information. This type of process planning is...

In a two stage wire drawing operation, the fractional reduction (ratio of change in cross sectional area to initial cross sectional area ) in the first stage is $$0.4.$$ The fracti...

The flowing stress (in $$MPa$$) of a material is given by $$\sigma = 500{\varepsilon ^{0.1}},$$ When $$\varepsilon $$ is true strain. The Young's modulus of elasticity of the mater...

Curl of vector $$\,V\left( {x,y,x} \right) = 2{x^2}i + 3{z^2}j + {y^3}k\,\,$$ at $$x=y=z=1$$ is

During a $$TIG$$ welding process, the arc current and arc voltage were $$50$$ $$A$$ and $$60$$ $$V,$$ respectively, when the welding speed was $$150$$ $$mm/min.$$ In another proces...

Which one of the following statements is TRUE?

Annual demand of a product is $$50,000$$ units and the ordering cost is Rs. $$7,000$$ per order. Considering the basic economic order quantity model, the economic order quantity is...

At least one eigenvalue of a singular matrix is

If the fluid velocity for a potential flow is given by $$V\left( {x,y} \right) = u\left( {x,y} \right)i + v\left( {x,y} \right)j$$ with usual notations, then the slope of potential...

Laplace transform of $$\cos \left( {\omega t} \right)$$ is

The head loss for a laminar incompressible flow through a horizontal circular pipe is $${h_1}$$. Pipe length and fluid remaining the same, if the average flow velocity doubles and...

Work is done on an adiabatic system due to which its velocity changes from $$10$$ $$m/s$$ to $$20$$ $$m/s,$$ elevation increases by $$20$$ $$m$$ and temperature increases by $$1$$...

The Vander-waals equation of state is $$\left( {p + {a \over {{v^2}}}} \right)\left( {v - b} \right) = RT,$$ where $$p$$ is pressure, $$v$$ is specific volume, $$T$$ is temperature...

One kg of air ($${R = 287\,\,J/kg}$$-$$K$$) under goes irreversible process between equilibrium state $$1\,\,\left( {{{20}^ \circ }C,\,\,0.9\,\,{m^3}} \right)$$ and equilibrium sta...

Which of the following statements regarding a Rankine cycle with reheating are TRUE ? (i) increase in average temperature of heat addition (ii) reduction in thermal efficiency (iii...

In a Rankine cycle, the enthalpies at turbine entry and outlet are $$3159$$ $$kJ/kg$$ and $$2187$$ $$kJ/kg$$ respectively. If the specific pump work is $$2$$ $$kJ/kg,$$ the specifi...

The $$COP$$ of a Carnot heat pump operating between $${6^ \circ }C$$ and $${37^ \circ }C$$ is ____________

The chance of a student passing an exam is $$20$$% The chance of a student passing the exam and getting above $$90$$% marks in it is $$5$$%. Given that a student passes the examina...

In a certain slider-crank mechanism, lengths of crank and connecting rod are equal. If the crank rotates with a uniform angular speed of 14 rad/s and the crank length is 300 mm, th...

For a fully developed laminar flow of water (dynamic viscosity $$0.001$$ $$Pa$$-s) through a pipe of radius $$5$$ $$cm,$$ the axial pressure gradient is $$-10$$ $$Pa/m$$. The magni...

The uniaxial yield stress of a material is $$300$$ $$MPa.$$ According to Von mises criterion, the shear yield stress $$(in$$ $$MPa)$$ of the material is ___________.

A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s...

A balanced counter-flow heat exchanger has a surface area of $$20\,\,{m^2}$$ and overall heat transfer coefficient of $$20$$ $$W/{m^2}$$-$$K$$. Air $$\left( {{C_p} = 1000J/kg - K}...

At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has

A cube and a sphere made of cast iron (each of volume $$1000\,\,c{m^3}$$) were cast under identical conditions. The time taken for solidifying the cube was $$4$$ $$s.$$ The solidif...

A manufacturer has the following data regarding a product: Fixed cost per month $$=$$ Rs. $$50,000$$ Variable cost per unit $$=$$ Rs. $$200$$ Selling price per unit $$=$$ Rs. $$300...

The initial velocity of an object is $$40$$ $$m/s.$$ The acceleration a of the object is given by the following expression: a$$=-0.1v$$ Where v is the instantaneous velocity of the...

A rope brake dynamometer attached to the crank shaft of an $$I.C.$$ engine measures a brake power of $$10$$ kW when the speed of rotation of the shaft is $$400$$ rad/s. The shaft t...

In a plane stress condition, the components of stress at a point are $${\sigma _x} = 20$$ MPa, $${\sigma _y} = 80$$ MPa and $${\tau _{xy}} = 40$$ MPa, The maximum shear stress (in...

A rod is subjected to an uniaxial load within linear elastic limit. When the change in the stress is $$200$$ MPa, the change in the strain is $$0.001.$$ If the Poisson's ratio of t...

A hollow shaft of $$1$$ m length is designed to transmit a power of $$30$$ kW at $$700$$ rpm. The maximum permissible angle of twist in the shaft is $${1^0}.$$ The inner diameter o...

In a spring-mass system the mass is m and the spring constant is $$k$$. The critical damping coefficient of the system is $$0.1$$ kg/s. In another spring mass system, the mass is $...

A single-degree-freedom spring mass system is subjected to a sinusoidal force of $$10$$ N amplitude and frequency $$\omega $$ along the axis of the spring. The stiffness of the spr...

A gas is stored in a cylindrical tank of inner radius $$7$$ m and wall thickness $$50$$ mm. The gage pressure of the gas is $$2$$ MPa. The maximum shear stress (in MPa) in the wall...

The deflection of a spring with 20 active turns under a load of 1000 N is 10 mm. The spring is made into two pieces each of 10 active coils and placed in parallel under the same lo...

A hollow shaft (d 0 = $$2$$d i , where d 0 and d i are the outer and inner diameters respectively) needs to transmit $$20$$ kW power at $$3000$$ RMP. If the maximum permissible she...

Using a unit step size, the value of integral $$\int\limits_1^2 {x\,\ln \,xdx\,\,\,} $$ by trapezoidal rule is ___________.

The ratio of momentum diffusivity $$\left( \upsilon \right)$$ to thermal diffusivity $$\left( \alpha \right),$$ is called

If $$P(X)$$ $$ = $$ $$1/4,$$ $$P(Y) = 1/3,$$ and $$\,\,P\left( {X \cap Y} \right) = 1/12,\,\,$$ the value of $$P(Y/X)$$ is

Orthogonal turning of a mild steel tube with a tool of rake angle $${10^0}$$ is carried out at a feed of $$0.14$$ $$mm/rev.$$ If the thickness of the chip produced is $$0.28mm,$$ t...

Saturated vapor is condensed to saturated liquid in a condenser. The heat capacity ratio is $${c_r} = {{{c_{\min }}} \over {{c_{\max }}}}.$$ The effectiveness $$(s)$$ of the conden...

Ratio of solidification time of a cylindrical casting (height $$=$$ radius) to that of a cubic casting of side two times the height of cylindrical casting is ____________

Which two of the following joining processes are autogeneous? $$(i)$$ Diffusion welding $$(ii)$$ Electro slag welding $$(iii)$$ Tungsten inert gas welding $$(iv)$$ Friction welding

In a $$CNC$$ milling operation, the tool has to machine the circular arc from point $$(20, 20)$$ to $$(10, 10)$$ at sequence number $$5$$ of the $$CNC$$ part program. If the center...

In a rolling operation using rolls of diameter $$500mm,$$ if a thick plate cannot be reduced to less than $$20mm$$ in one pass, the coefficient of friction between the roll and the...

A Prandtl tube (Pitot-static tube with $$C=1$$) is used to measure the velocity of water. The differential manometer reading is $$10$$ mm of liquid column with a relative density o...

A cylindrical tank with closed ends is filled with compressed air at a pressure of $$500 kP$$a. The inner radius of the tank is $$2 m,$$ and it has wall thickness of $$10 mm.$$ The...

For the linear programming problem: $$\eqalign{ & Maximize\,\,\,\,\,Z = 3{x_1} + 2{x_2} \cr & Subject\,\,to\,\,\,\, - 2{x_1} + 3{x_2} \le 9 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...

In the notation $$(a/b/c) : (d/e/f)$$ for summarizing the characteristics of queuing situation, the letters $$‘b’$$ and $$‘d’$$ stand respectively for

The value of $$\int\limits_C {\left[ {\left( {3x - 8{y^2}} \right)dx + \left( {4y - 6xy} \right)dy} \right],\,\,} $$ (where $$C$$ is the region bounded by $$x=0,$$ $$y=0$$ and $$x+...

Let $$\phi $$ be an arbitrary smooth real valued scalar function and $$\overrightarrow V $$ be an arbitrary smooth vector valued function in a three dimensional space. Which one of...

Couette flow is characterized by

In a machining operation, if the generatrix and directrix both are straight lines, the surface obtained is

A shaft of length $$90$$ $$mm$$ has a tapered portion of length $$55$$ $$mm.$$ The diameter of the taper is $$80$$ $$mm$$ at one end and $$65$$ $$mm$$ at the other. If the taper is...

The value of $$\mathop {Lim}\limits_{x \to 0} \left( {{{ - \sin x} \over {2\sin x + x\cos x}}} \right)\,\,\,$$ is __________.

Which of the following statement are TRUE, when the cavitation parameter $$\sigma = 0?$$ (i) The local pressure is reduced to vapor pressure. (ii) Cavitation starts (iii) Boiling o...

Steam enters a turbine at $$30bar,$$ $${300^ \circ }C$$ $$\left( {u = 2750\,\,kJ/kg,\,\,h = 2993\,\,kJ/kg} \right)$$ and exits the turbine as saturated liquid at $$15$$ $$kPa$$ $$\...

A rigid container of volume $$0.5$$ $${m^3}$$ contains $$1.0$$ $$kg$$ of water at $${120^ \circ }C$$ $$\left( {{V_f} = 0.00106\,{m^3}/kg,\,\,{v_g} = 0.8908\,\,{m^3}/kg} \right).$$...

One side of a wall is maintained at $$400K$$ and the other at $$300K.$$ The rate of heat transfer through the wall is $$1000W$$ and the surrounding temperature is $${25^ \circ }C.$...

In full mould (cavity-less) casting process, the Pattern is made of

Three parallel pipes connected at the two ends have flow-rates $$\,\,\,{Q_1},\,\,\,{Q_2},\,\,\,\,$$ and $${Q_3}$$ respectively, and the corresponding frictional head losses are $${...

The lowest eigen value of the $$2 \times 2$$ matrix $$\left[ {\matrix{ 4 & 2 \cr 1 & 3 \cr } } \right]$$ is ______.

Newton-Raphson method is used to find the roots of the equation, $${\,{x^3} + 2{x^2} + 3x - 1 = 0}$$ If the initial guess is $${x_0} = 1,$$ then the value of $$x$$ after $${2^{nd}}...

For ball bearings, the fatigue life $$L$$ measured in number of revolutions and the radial load $$F$$ are related by $$F{L^{1/3}} = K,$$ where $$K$$ is a constant. It withstands a...

The annual requirement of rivets at a ship manufacturing company is $$2000$$ $$kg.$$ The rivets are supplied in units of 1 kg costing Rs. $$25$$ each. If it costs Rs. $$100$$ to pl...

A brick wall $$\,\left( {k = 0.9{W \over {m.K}}} \right)$$ of thickness $$0.18$$ $$m$$ separates the warm air in a room from the cold ambient air. On a particular winter day, the o...

For a given matrix $$P = \left[ {\matrix{ {4 + 3i} & { - i} \cr i & {4 - 3i} \cr } } \right],$$ where $$i = \sqrt { - 1} ,$$ the inverse of matrix $$P$$ is

The torque (in N-m) exerted on the crank shaft of a two stroke engine can be described as is the crank angle as $$\,\,T = 10000 + 1000\sin 2\theta - 1200\,\cos 2\theta ,\,\,$$ wher...

Which of the following statements are TRUE for damped vibrations? $$P.$$ For a system having critical damping, the value of damping ratio is unity and system does not undergo a vib...

For the same material and the mass, which of the following configurations of flywheel will have maximum mass moment of inertia about the axis of rotation $$OO’$$ passing through th...