GATE 2021 Mechanical Engineering

Consider the following sentences: (i) After his surgery, Raja hardly could walk. (ii) After his surgery, Raja could barely walk. (iii) After his surgery, Raja barely could walk. (i...

If y(x) satisfies the differential equation (sin x) dy/dx + y cos x = 1, subject to the condition y(π/2) = π/2, then y(π/6) is

The value of lim (x→0) (1-cos x)/x^2 is

Ms. X came out of a building through its front door to find her shadow due to the morning sun falling to her right side with the building to her back. From this, it can be inferred...

In the above figure, O is the center of the circle and, M and N lie on the circle. The area of the right triangle MON is 50 cm². What is the area of the circle in cm² ?

The Dirac-delta function ($\delta(t - t_0)$) for $t, t_0 \in \mathbb{R}$, has the following property $\int_a^b \varphi(t)\delta(t - t_0)dt = \begin{cases} \varphi(t_0) & a 0$; $\ma...

If "⊕" means "-", "⊗" means "÷", "Δ" means "+", "∇" means "x", then, the value of the expression Δ 2 ⊕ 3 Δ ((4 ⊗ 2) ∇ 4) =

The ordinary differential equation $\frac{dy}{dt} = -\pi y$ subject to an initial condition $y(0) = 1$ is solved numerically using the following scheme: $\frac{y(t_{n+1}) - y(t_n)}...

Consider a binomial random variable X. If X1, X2, ..., Xn are independent and identically distributed samples from the distribution of X with sum Y = $\Sigma_{i=1}^n X_i$, then the...

"The increased consumption of leafy vegetables in the recent months is a clear indication that the people in the state have begun to lead a healthy lifestyle" Which of the followin...

The loading and unloading response of a metal is shown in the figure. The elastic and plastic strains corresponding to 200 MPa stress, respectively, are

Oxpeckers and rhinos manifest a symbiotic relationship in the wild. The oxpeckers warn the rhinos about approaching poachers, thus possibly saving the lives of the rhinos. Oxpecker...

In a machining operation, if a cutting tool traces the workpiece such that the directrix is perpendicular to the plane of the generatrix as shown in figure, the surface generated i...

A jigsaw puzzle has 2 pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, r...

The number of hens, ducks and goats in farm P are 65, 91 and 169, respectively. The total number of hens, ducks and goats in a nearby farm Q is 416. The ratio of hens:ducks:goats i...

The correct sequence of machining operations to be performed to finish a large diameter through hole is

The distribution of employees at the rank of executives, across different companies C1, C2, ..., C6 is presented in the chart given above. The ratio of executives with a management...

In modern CNC machine tools, the backlash has been eliminated by

Five persons P, Q, R, S and T are sitting in a row not necessarily in the same order. Q and R are separated by one person, and S should not be seated adjacent to Q. The number of d...

Consider the surface roughness profile as shown in the figure. The center line average roughness (Ra, in µm) of the measured length (L) is

In which of the following pairs of cycles, both cycles have at least one isothermal process?

Superheated steam at 1500 kPa, has a specific volume of 2.75 m³/kmol and compressibility factor (Z) of 0.95. The temperature of steam is ______ °C (round off to the nearest integer...

A hot steel spherical ball is suddenly dipped into a low temperature oil bath. Which of the following dimensionless parameters are required to determine instantaneous center temper...

An infinitely long pin fin, attached to an isothermal hot surface, transfers heat at a steady rate of Q₁ to the ambient air. If the thermal conductivity of the fin material is doub...

The relative humidity of ambient air at 300 K is 50% with a partial pressure of water vapour equal to $p_v$. The saturation pressure of water at 300 K is $p_{sat}$. The correct rel...

Consider a reciprocating engine with crank radius R and connecting rod of length L. The secondary unbalance force for this case is equivalent to primary unbalance force due to a vi...

A cantilever beam of length, L, and flexural rigidity, EI, is subjected to an end moment, M, as shown in the figure. The deflection of the beam at x = L/2 is

A prismatic bar PQRST is subjected to axial loads as shown in the figure. The segments having maximum and minimum axial stresses, respectively, are

Shear stress distribution on the cross-section of the coil wire in a helical compression spring is shown in the figure. This shear stress distribution represents

Robot Ltd. wishes to maintain enough safety stock during the lead time period between starting a new production run and its completion such that the probability of satisfying the c...

A pressure measurement device fitted on the surface of a submarine, located at a depth H below the surface of an ocean, reads an absolute pressure of 4.2 MPa. The density of sea wa...

Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate move...

A rigid insulated tank is initially evacuated. It is connected through a valve to a supply line that carries air at a constant pressure and temperature of 250 kPa and 400 K respect...

The figure shows an arrangement of a heavy propeller shaft in a ship. The combined polar mass moment of inertia of the propeller and the shaft is 100 kg.m². The propeller rotates a...

Consider a single degree of freedom system comprising a mass M, supported on a spring and a dashpot as shown in the figure. If the amplitude of the free vibration response reduces...

Consider a vector p in 2-dimensional space. Let its direction (counter-clockwise angle with the positive x-axis) be θ. Let p be an eigenvector of a 2 × 2 matrix A with correspondin...

Let C represent the unit circle centered at origin in the complex plane, and complex variable, z = x + iy. The value of the contour integral $\oint_C \frac{\cosh 3z}{2z} dz$ (where...

A set of jobs A, B, C, D, E, F, G, H arrive at time t = 0 for processing on turning and grinding machines. Each job needs to be processed in sequence - first on the turning machine...

The fundamental thermodynamic relation for a rubber band is given by $dU = TdS + \tau dL$, where T is the absolute temperature, S is the entropy, $\tau$ is the tension in the rubbe...

Consider a two degree of freedom system as shown in the figure, where PQ is a rigid uniform rod of length, b and mass, m. Assume that the spring deflects only horizontally and forc...

A right solid circular cone standing on its base on a horizontal surface is of height H and base radius R. The cone is made of a material with specific weight w and elastic modulus...

A tappet valve mechanism in an IC engine comprises a rocker arm ABC that is hinged at B as shown in the figure. The rocker is assumed rigid and it oscillates about the hinge B. The...

Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop...

Let $f(x) = x^2 - 2x + 2$ be a continuous function defined on $x \in [1, 3]$. The point $x$ at which the tangent of $f(x)$ becomes parallel to the straight line joining $f(1)$ and...

Activities A, B, C and D form the critical path for a project with a PERT network. The means and variances of the activity duration for each activity are given below. All activity...

A true centrifugal casting operation needs to be performed horizontally to make copper tube sections with outer diameter of 250 mm and inner diameter of 230 mm. The value of accele...

The resistance spot welding of two 1.55 mm thick metal sheets is performed using welding current of 10000 A for 0.25 s. The contact resistance at the interface of the metal sheets...

An orthogonal cutting operation is performed using a single point cutting tool with a rake angle of 12° on a lathe. During turning, the cutting force and the friction force are 100...

In a grinding operation of a metal, specific energy consumption is 15 J/mm³. If a grinding wheel with a diameter of 200 mm is rotating at 3000 rpm to obtain a material removal rate...

A 200 mm wide plate having a thickness of 20 mm is fed through a rolling mill with two rolls. The radius of each roll is 300 mm. The plate thickness is to be reduced to 18 mm in on...

The XY table of a NC machine tool is to move from P(1,1) to Q(51,1); all coordinates are in mm. The pitch of the NC drive leadscrew is 1 mm. If the backlash between the leadscrew a...

Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the workstation is expo...

An uninsulated cylindrical wire of radius 1.0 mm produces electric heating at the rate of 5.0 W/m. The temperature of the surface of the wire is 75 °C when placed in air at 25 °C....

A solid sphere of radius 10 mm is placed at the centroid of a hollow cubical enclosure of side length 30 mm. The outer surface of the sphere is denoted by 1 and the inner surface o...

Consider a steam power plant operating on an ideal reheat Rankine cycle. The work input to the pump is 20 kJ/kg. The work output from the high pressure turbine is 750 kJ/kg. The wo...

In the vicinity of the triple point, the equation of liquid-vapour boundary in the P – T phase diagram for ammonia is ln P = 24.38 – 3063/T, where P is pressure (in Pa) and T is te...

A cylindrical jet of water (density = 1000 kg/m³) impinges at the center of a flat, circular plate and spreads radially outwards, as shown in the figure. The plate is resting on a...

A single jet Pelton wheel operates at 300 rpm. The mean diameter of the wheel is 2 m. Operating head and dimensions of jet are such that water comes out of the jet with a velocity...

An air-conditioning system provides a continuous flow of air to a room using an intake duct and an exit duct, as shown in the figure. To maintain the quality of the indoor air, the...

Two smooth identical spheres each of radius 125 mm and weight 100 N rest in a horizontal channel having vertical walls. The distance between vertical walls of the channel is 400 mm...

An overhanging beam PQR is subjected to uniformly distributed load 20 kN/m as shown in the figure. The maximum bending stress developed in the beam is ________ MPa (round off to on...

The Whitworth quick return mechanism is shown in the figure with link lengths as follows: OP = 300 mm, OA = 150 mm, AR = 160 mm, RS = 450 mm. The quick return ratio for the mechani...

A short shoe drum (radius 260 mm) brake is shown in the figure. A force of 1 kN is applied to the lever. The coefficient of friction is 0.4. The magnitude of the torque applied by...

A machine part in the form of cantilever beam is subjected to fluctuating load as shown in the figure. The load varies from 800 N to 1600 N. The modified endurance, yield and ultim...

A cantilever beam of rectangular cross-section is welded to a support by means of two fillet welds as shown in figure. A vertical load of 2 kN acts at free end of the beam. Conside...

Consider an n×n matrix A and a non-zero n × 1 vector p. Their product Ap = α²p, where α ∈ R and α ∉ {−1, 0, 1}. Based on the given information, the eigen value of A² is:

Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P...

Consider the following sentences: (i) The number of candidates who appear for the GATE examination is staggering. (ii) A number of candidates from my class are appearing for the GA...

If the Laplace transform of a function f(t) is given by (s+3)/((s+1)(s+2)), then f(0) is

A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM. The immediate next time that they will beep together is _________

The mean and variance, respectively, of a binomial distribution for n independent trials with the probability of success as p, are

If ⊕ ÷ ⊙ = 2; ⊙ ÷ Δ = 3; ⊙ + Δ = 5; Δ × ⊗ = 10, Then, the value of (⊗ − ⊕)², is:

The Cast Iron which possesses all the carbon in the combined form as cementite is known as

The size distribution of the powder particles used in Powder Metallurgy process can be determined by

The front door of Mr. X's house faces East. Mr. X leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to...

Given below are two statements 1 and 2, and two conclusions I and II. Statement 1: All entrepreneurs are wealthy. Statement 2: All wealthy are risk seekers. Conclusion I: All risk...

In a CNC machine tool, the function of an interpolator is to generate

A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and...

The machining process that involves ablation is

The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is _____

A PERT network has 9 activities on its critical path. The standard deviation of each activity on the critical path is 3. The standard deviation of the critical path is

Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is agai...

The allowance provided in between a hole and a shaft is calculated from the difference between

In forced convective heat transfer, Stanton number (St), Nusselt number (Nu), Reynolds number (Re) and Prandtl number (Pr) are related as

The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to we...

For a two-dimensional, incompressible flow having velocity components u and v in the x and y directions, respectively, the expression $\frac{\partial(u^2)}{\partial x} + \frac{\par...

Which of the following is responsible for eddy viscosity (or turbulent viscosity) in a turbulent boundary layer on a flat plate?

A two dimensional flow has velocities in x and y directions given by u = 2xyt and v = -y^2t, where t denotes time. The equation for streamline passing through x = 1, y = 1 is

A plane truss PQRS (PQ = RS, and ∠PQR = 90°) is shown in the figure. The forces in the members PR and RS, respectively, are

Consider the mechanism shown in the figure. There is rolling contact without slip between the disc and ground. Select the correct statement about instantaneous centers in the mecha...

The controlling force curves P, Q and R for a spring controlled governor are shown in the figure, where r₁ and r₂ are any two radii of rotation. The characteristics shown by the cu...

The von Mises stress at a point in a body subjected to forces is proportional to the square root of the

Value of $\int_{4}^{5.2} \ln x \, dx$ using Simpson's one-third rule with interval size 0.3 is

Value of $(1 + i)^8$, where $i = \sqrt{-1}$, is equal to

Consider adiabatic flow of air through a duct. At a given point in the duct, velocity of air is 300 m/s, temperature is 330 K and pressure is 180 kPa. Assume that the air behaves a...

Consider an ideal vapour compression refrigeration cycle working on R-134a refrigerant. The COP of the cycle is 10 and the refrigeration capacity is 150 kJ/kg. The heat rejected by...

A rigid tank of volume 50 m³ contains a pure substance as a saturated liquid vapour mixture at 400 kPa. Of the total mass of the mixture, 20% mass is liquid and 80% mass is vapour....

An object is moving with a Mach number of 0.6 in an ideal gas environment, which is at a temperature of 350 K. The gas constant is 320 J/kg.K and ratio of specific heats is 1.3. Th...

A column with one end fixed and one end free has a critical buckling load of 100 N. For the same column, if the free end is replaced with a pinned end then the critical buckling lo...

A steel cubic block of side 200 mm is subjected to hydrostatic pressure of 250 N/mm². The elastic modulus is 2 × 10⁵ N/mm² and Poisson ratio is 0.3 for steel. The side of the block...

The value of $\int_0^{\pi/2} \int_0^{\cos\theta} r \sin\theta \, dr \, d\theta$ is

Let the superscript T represent the transpose operation. Consider the function $f(x) = \frac{1}{2} x^T Qx - r^T x$, where x and r are $n \times 1$ vectors and Q is a symmetric $n \...

Consider the following differential equation $(1 + y) \frac{dy}{dx} = y$. The solution of the equation that satisfies the condition $y(1) = 1$ is

A factory produces m (i = 1,2,..., m) products, each of which requires processing on n (j = 1, 2, ..., n) workstations. Let aij be the amount of processing time that one unit of th...

Ambient pressure, temperature, and relative humidity at a location are 101 kPa, 300 K, and 60%, respectively. The saturation pressure of water at 300 K is 3.6 kPa. The specific hum...

A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length, L, and flexural rigidity, EI. Neglecting the effect of axial stress and t...

A power transmission mechanism consists of a belt drive and a gear train as shown in the figure. Diameters of pulleys of belt drive and number of teeth (T) on the gears 2 to 7 are...

A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to reduce the force trans...

Consider the system shown in the figure. A rope goes over a pulley. A mass, m, is hanging from the rope. A spring of stiffness, k, is attached at one end of the rope. Assume rope i...

Find the positive real root of x³ - x - 3 = 0 using Newton-Raphson method. If the starting guess (x₀) is 2, the numerical value of the root after two iterations (x₂) is ________ (r...

Daily production capacity of a bearing manufacturing company is 30000 bearings. The daily demand of the bearing is 15000. The holding cost per year of keeping a bearing in the inve...

A cast product of a particular material has dimensions 75 mm × 125 mm × 20 mm. The total solidification time for the cast product is found to be 2.0 minutes as calculated using Chv...

A spot welding operation performed on two pieces of steel yielded a nugget with a diameter of 5 mm and a thickness of 1 mm. The welding time was 0.1 s. The melting energy for the s...

A surface grinding operation has been performed on a Cast Iron plate having dimensions 300 mm (length) x 10 mm (width) x 50 mm (height). The grinding was performed using an alumina...

In a pure orthogonal turning by a zero rake angle single point carbide cutting tool, the shear force has been computed to be 400 N. The cutting velocity, Vc = 100 m/min, depth of c...

The thickness, width and length of a metal slab are 50 mm, 250 mm and 3600 mm, respectively. A rolling operation on this slab reduces the thickness by 10% and increases the width b...

A 76.2 mm gauge block is used under one end of a 254 mm sine bar with roll diameter of 25.4 mm. The height of gauge blocks required at the other end of the sine bar to measure an a...

The demand and forecast of an item for five months are given in the table. Month | Demand | Forecast ---|---|--- April | 225 | 200 May | 220 | 240 June | 285 | 300 July | 290 | 270...

A shell and tube heat exchanger is used as a steam condenser. Coolant water enters the tube at 300 K at a rate of 100 kg/s. The overall heat transfer coefficient is 1500 W/m².K, an...

Ambient air flows over a heated slab having flat, top surface at y = 0. The local temperature (in Kelvin) profile within the thermal boundary layer is given by T(y) = 300 + 200 exp...

Water flows out from a large tank of cross-sectional area A₁ = 1 m² through a small rounded orifice of cross-sectional area A₀ = 1 cm², located at y = 0. Initially the water level,...

Consider the open feed water heater (FWH) shown in the figure given below: Specific enthalpy of steam at location 2 is 2624 kJ/kg, specific enthalpy of water at location 5 is 226.7...

A high velocity water jet of cross section area = 0.01 m² and velocity = 35 m/s enters a pipe filled with stagnant water. The diameter of the pipe is 0.32 m. This high velocity wat...

A vertical shaft Francis turbine rotates at 300 rpm. The available head at the inlet to the turbine is 200 m. The tip speed of the rotor is 40 m/s. Water leaves the runner of the t...

An adiabatic vortex tube, shown in the figure given below is supplied with 5 kg/s of air (inlet 1) at 500 kPa and 300 K. Two separate streams of air are leaving the device from out...

A block of negligible mass rests on a surface that is inclined at 30° to the horizontal plane as shown in the figure. When a vertical force of 900 N and a horizontal force of 750 N...

The wheels and axle system lying on a rough surface is shown in the figure. Each wheel has diameter 0.8 m and mass 1 kg. Assume that the mass of the wheel is concentrated at rim an...

A cantilever beam with a uniform flexural rigidity (EI = 200 x 10⁶ N.m²) is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding t...

The torque provided by an engine is given by T(θ) = 12000 + 2500 sin(2θ) N.m, where θ is the angle turned by the crank from inner dead center. The mean speed of the engine is 200 r...

The figure shows the relationship between fatigue strength (S) and fatigue life (N) of a material. The fatigue strength of the material for a life of 1000 cycles is 450 MPa, while...