All 65 questions of GATE 2021 ME (Set 1)

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.

(iv) After his surgery, Raja could hardly walk.

Which of the above sentences are grammatically CORRECT?

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 that building is facing _________

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 $50\text{ cm}^2$.

What is the area of the circle in $\text{cm}^2$?

If

“$\oplus$” means “$-$”,

“$\otimes$” means “$\div$”,

“$\Delta$” means “$+$”,

“$\nabla$” means “$\times$”,

then, the value of the expression $\Delta\,2\oplus3\Delta\big((4\otimes2)\nabla4\big)=$

“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 following can be logically inferred from the information presented in the above statement?

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. Oxpeckers also feed on the parasitic ticks found on rhinos.

In the symbiotic relationship described above, the primary benefits for oxpeckers and rhinos respectively are,

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, rotated or flipped to assemble with the above piece.

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 in farm Q is $5:14:13$. All the hens, ducks and goats are sent from farm Q to farm P.

The new ratio of hens:ducks:goats in farm P 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 degree to those without a management degree in each of these companies is provided in the table above. The total number of executives across all companies is $10,000$.

The total number of management degree holders among the executives in companies C2 and C5 together is_____.

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 distinct seating arrangements possible is:

If $y(x)$ satisfies the differential equation

$(\sin x)\frac{dy}{dx}+y\cos x=1,$

subject to the condition $y(\pi/2)=\pi/2$, then $y(\pi/6)$ is

The value of $\displaystyle\lim_{x\to0}\left(\frac{1-\cos x}{x^2}\right)$ is

The Dirac-delta function $(\delta(t-t_0))$ for $t,t_0\in\mathbb{R}$, has the following property

$\int_a^b\varphi(t)\delta(t-t_0)\,dt=\begin{cases}\varphi(t_0)&a<t_0<b\\0&\text{otherwise}\end{cases}$

The Laplace transform of the Dirac-delta function $\delta(t-a)$ for $a>0$; $\mathcal{L}(\delta(t-a))=F(s)$ is

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

$\frac{y(t_{n+1})-y(t_n)}{h}=-\pi y(t_n)$

where $h$ is the time step, $t_n=nh$, and $n=0,1,2,\ldots$. This numerical scheme is stable for all values of $h$ in the interval _________.

Consider a binomial random variable $X$. If $X_1,X_2,\ldots,X_n$ are independent and identically distributed samples from the distribution of $X$ with sum $Y=\sum_{i=1}^n X_i$, then the distribution of $Y$ as $n\to\infty$ can be approximated as

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

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 is

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

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

Consider the surface roughness profile as shown in the figure.

The center line average roughness ($R_a$, in $\mu\text{m}$) 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 $1500\text{ kPa}$, has a specific volume of $2.75\text{ m}^3/\text{kmol}$ and compressibility factor ($Z$) of $0.95$. The temperature of steam is______ $^\circ\text{C}$ (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 temperature of the ball using a Heisler chart?

An infinitely long pin fin, attached to an isothermal hot surface, transfers heat at a steady rate of $\dot Q_1$ to the ambient air. If the thermal conductivity of the fin material is doubled, while keeping everything else constant, the rate of steady-state heat transfer from the fin becomes $\dot Q_2$. The ratio $\dot Q_2/\dot Q_1$ is

The relative humidity of ambient air at $300\text{ K}$ is $50\%$ with a partial pressure of water vapour equal to $p_v$. The saturation pressure of water at $300\text{ K}$ is $p_{sat}$. The correct relation for the air-water mixture is

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 virtual crank of _______

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 $x=\dfrac{L}{2}$ 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 customer demand during the lead time period is $95\%$. The lead time period is $5$ days and daily customer demand can be assumed to follow the Gaussian (normal) distribution with mean $50$ units and a standard deviation of $10$ units. Using $\phi^{-1}(0.95)=1.64$, where $\phi$ represents the cumulative distribution function of the standard normal random variable, the amount of safety stock that must be maintained by Robot Ltd. to achieve this demand fulfillment probability for the lead time period is _________________ units (round off to two decimal places).

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 $4.2\text{ MPa}$. The density of sea water is $1050\text{ kg/m}^3$, the atmospheric pressure is $101\text{ kPa}$, and the acceleration due to gravity is $9.8\text{ m/s}^2$. The depth $H$ is ___________ $\text{m}$ (round off to the nearest integer).

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 moves with a constant velocity of $U=4\text{ m/s}$. Separation between the plates is $5\text{ mm}$. There is no pressure gradient in the direction of flow. The density of fluid is $800\text{ kg/m}^3$, and the kinematic viscosity is $1.25\times10^{-4}\text{ m}^2/\text{s}$. The average shear stress in the fluid is _________ $\text{Pa}$ (round off to the nearest integer).

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 $250\text{ kPa}$ and $400\text{ K}$ respectively. Now the valve is opened and air is allowed to flow into the tank until the pressure inside the tank reaches to $250\text{ kPa}$ at which point the valve is closed. Assume that the air behaves as a perfect gas with constant properties

$(c_p=1.005\text{ kJ/kg.K},\ c_v=0.718\text{ kJ/kg.K},\ R=0.287\text{ kJ/kg.K})$.

Final temperature of the air inside the tank is ___________ $\text{K}$ (round off to one decimal place).

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 $100\ \mathrm{kg\,m^2}$. The propeller rotates at $\omega=12\ \mathrm{rad/s}$. The waves acting on the ship hull induces a rolling motion as shown in the figure with an angular velocity of $5\ \mathrm{rad/s}$. The gyroscopic moment generated on the shaft due to the motion described is __________ $\mathrm{N\,m}$ (round off to the nearest integer).

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 from $8\ \mathrm{mm}$ to $1.5\ \mathrm{mm}$ in $3$ cycles, the damping ratio of the system is __________ (round off to three decimal places).

Consider a vector $\boldsymbol{p}$ in $2$-dimensional space. Let its direction (counter-clockwise angle with the positive $x$-axis) be $\theta$. Let $\boldsymbol{p}$ be an eigenvector of a $2\times2$ matrix $A$ with corresponding eigenvalue $\lambda$, $\lambda>0$. If we denote the magnitude of a vector $\boldsymbol{v}$ by $\|\boldsymbol{v}\|$, identify the VALID statement regarding $\boldsymbol{p}'$, where $\boldsymbol{p}'=A\boldsymbol{p}$.

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 $\displaystyle\oint_C\frac{\cosh 3z}{2z}\,dz$ (where integration is taken counter clockwise) is

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 and second on the grinding machine, and the grinding must occur immediately after turning. The processing times of the jobs are given below.

If the makespan is to be minimized, then the optimal sequence in which these jobs must be processed on the turning and grinding machines is

The fundamental thermodynamic relation for a rubber band is given by $dU=T\,dS+\tau\,dL$, where $T$ is the absolute temperature, $S$ is the entropy, $\tau$ is the tension in the rubber band, and $L$ is the length of the rubber band. Which one of the following relations is CORRECT:

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 force $F$ is applied horizontally at $Q$. For this system, the Lagrangian, $L$ is

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 $E$. The vertical deflection at the mid-height of the cone due to self-weight is given by

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 mass moment of inertia of the rocker about B is $10^{-4}\ \mathrm{kg\,m^2}$. The rocker arm dimensions are $a=3.5\ \mathrm{cm}$ and $b=2.5\ \mathrm{cm}$. A pushrod pushes the rocker at location A, when moved vertically by a cam that rotates at $N\ \mathrm{rpm}$. The pushrod is assumed massless and has a stiffness of $15\ \mathrm{N/mm}$. At the other end C, the rocker pushes a valve against a spring of stiffness $10\ \mathrm{N/mm}$. The valve is assumed massless and rigid.

Resonance in the rocker system occurs when the cam shaft runs at a speed of _________ rpm (round off to the nearest integer).

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 opens. The probability that a customer arrives within the next $3$ minutes is

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 $f(3)$ is

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 durations follow the Gaussian (normal) distribution, and are independent of each other.

The probability that the project will be completed within $40$ days is _________ (round off to two decimal places).

(Note: Probability is a number between $0$ and $1$).

A true centrifugal casting operation needs to be performed horizontally to make copper tube sections with outer diameter of $250\ \mathrm{mm}$ and inner diameter of $230\ \mathrm{mm}$. The value of acceleration due to gravity, $g=10\ \mathrm{m/s^2}$. If a $G$-factor (ratio of centrifugal force to weight) of $60$ is used for casting the tube, the rotational speed required is __________ rpm (round off to the nearest integer).

The resistance spot welding of two $1.55\ \mathrm{mm}$ thick metal sheets is performed using welding current of $10000\ \mathrm{A}$ for $0.25\ \mathrm{s}$. The contact resistance at the interface of the metal sheets is $0.0001\ \Omega$. The volume of weld nugget formed after welding is $70\ \mathrm{mm^3}$. Considering the heat required to melt unit volume of metal is $12\ \mathrm{J/mm^3}$, the thermal efficiency of the welding process is ________ % (round off to one decimal place).

An orthogonal cutting operation is performed using a single point cutting tool with a rake angle of $12^\circ$ on a lathe. During turning, the cutting force and the friction force are $1000\ \mathrm{N}$ and $600\ \mathrm{N}$, respectively. If the chip thickness and the uncut chip thickness during turning are $1.5\ \mathrm{mm}$ and $0.75\ \mathrm{mm}$, respectively, then the shear force is __________ $\mathrm{N}$ (round off to two decimal places).

In a grinding operation of a metal, specific energy consumption is $15\ \mathrm{J/mm^3}$. If a grinding wheel with a diameter of $200\ \mathrm{mm}$ is rotating at $3000\ \mathrm{rpm}$ to obtain a material removal rate of $6000\ \mathrm{mm^3/min}$, then the tangential force on the wheel is __________ $\mathrm{N}$ (round off to two decimal places).

A $200\ \mathrm{mm}$ wide plate having a thickness of $20\ \mathrm{mm}$ is fed through a rolling mill with two rolls. The radius of each roll is $300\ \mathrm{mm}$. The plate thickness is to be reduced to $18\ \mathrm{mm}$ in one pass using a roll speed of $50\ \mathrm{rpm}$. The strength coefficient $(K)$ of the work material flow curve is $300\ \mathrm{MPa}$ and the strain hardening exponent, $n$ is $0.2$. The coefficient of friction between the rolls and the plate is $0.1$. If the friction is sufficient to permit the rolling operation then the roll force will be __________ $\mathrm{kN}$ (round off to the nearest integer).

The $XY$ table of a NC machine tool is to move from $P(1,1)$ to $Q(51,1)$; all coordinates are in $\mathrm{mm}$. The pitch of the NC drive leadscrew is $1\ \mathrm{mm}$. If the backlash between the leadscrew and the nut is $1.8^\circ$, then the total backlash of the table on moving from $P$ to $Q$ is __________ $\mathrm{mm}$ (round off to two decimal places).

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 exponentially distributed with a mean of $4$ minutes. The expected number of jobs at the workstation at any given point of time is __________ (round off to the nearest integer).

An uninsulated cylindrical wire of radius $1.0\ \mathrm{mm}$ produces electric heating at the rate of $5.0\ \mathrm{W/m}$. The temperature of the surface of the wire is $75\ ^\circ\mathrm{C}$ when placed in air at $25\ ^\circ\mathrm{C}$. When the wire is coated with PVC of thickness $1.0\ \mathrm{mm}$, the temperature of the surface of the wire reduces to $55\ ^\circ\mathrm{C}$. Assume that the heat generation rate from the wire and the convective heat transfer coefficient are same for both uninsulated wire and the coated wire. The thermal conductivity of PVC is __________ $\mathrm{W/(m\,K)}$ (round off to two decimal places).

A solid sphere of radius $10\ \mathrm{mm}$ is placed at the centroid of a hollow cubical enclosure of side length $30\ \mathrm{mm}$. The outer surface of the sphere is denoted by $1$ and the inner surface of the cube is denoted by $2$. The view factor $F_{22}$ for radiation heat transfer is __________ (rounded off to two decimal places).

Consider a steam power plant operating on an ideal reheat Rankine cycle. The work input to the pump is $20\ \mathrm{kJ/kg}$. The work output from the high pressure turbine is $750\ \mathrm{kJ/kg}$. The work output from the low pressure turbine is $1500\ \mathrm{kJ/kg}$. The thermal efficiency of the cycle is $50\%$. The enthalpy of saturated liquid and saturated vapour at condenser pressure are $200\ \mathrm{kJ/kg}$ and $2600\ \mathrm{kJ/kg}$, respectively. The quality of steam at the exit of the low pressure turbine is __________ % (round off to the nearest integer).

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 $\mathrm{Pa}$) and $T$ is temperature (in $\mathrm{K}$). Similarly, the solid-vapour boundary is given by $\ln P=27.92-3754/T$. The temperature at the triple point is __________ $\mathrm{K}$ (round off to one decimal place).

A cylindrical jet of water (density $=1000\ \mathrm{kg/m^3}$) impinges at the center of a flat, circular plate and spreads radially outwards, as shown in the figure. The plate is resting on a linear spring with a spring constant $k=1\ \mathrm{kN/m}$. The incoming jet diameter is $D=1\ \mathrm{cm}$.

If the spring shows a steady deflection of $1\ \mathrm{cm}$ upon impingement of jet, then the velocity of the incoming jet is __________ $\mathrm{m/s}$ (round off to one decimal place).

A single jet Pelton wheel operates at $300\ \mathrm{rpm}$. The mean diameter of the wheel is $2\ \mathrm{m}$. Operating head and dimensions of jet are such that water comes out of the jet with a velocity of $40\ \mathrm{m/s}$ and flow rate of $5\ \mathrm{m^3/s}$. The jet is deflected by the bucket at an angle of $165^\circ$. Neglecting all losses, the power developed by the Pelton wheel is __________ $\mathrm{MW}$ (round off to two decimal places).

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 intake duct supplies a mixture of fresh air with a cold air stream. The two streams are mixed in an insulated mixing chamber located upstream of the intake duct. Cold air enters the mixing chamber at $5\ ^\circ\mathrm{C}$, $105\ \mathrm{kPa}$ with a volume flow rate of $1.25\ \mathrm{m^3/s}$ during steady state operation. Fresh air enters the mixing chamber at $34\ ^\circ\mathrm{C}$ and $105\ \mathrm{kPa}$. The mass flow rate of the fresh air is $1.6$ times of the cold air stream. Air leaves the room through the exit duct at $24\ ^\circ\mathrm{C}$.

Assuming the air behaves as an ideal gas with $c_p=1.005\ \mathrm{kJ/(kg\,K)}$ and $R=0.287\ \mathrm{kJ/(kg\,K)}$, the rate of heat gain by the air from the room is __________ $\mathrm{kW}$ (round off to two decimal places).

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

All dimensions are in $\mathrm{mm}$

The reaction at the point of contact between two spheres is _________ $\mathrm{N}$ (round off to one decimal place).

An overhanging beam $PQR$ is subjected to uniformly distributed load $20\ \mathrm{kN/m}$ as shown in the figure.

All dimensions are in $\mathrm{mm}$

The maximum bending stress developed in the beam is __________ $\mathrm{MPa}$ (round off to one decimal place).

The Whitworth quick return mechanism is shown in the figure with link lengths as follows: $\mathrm{OP}=300\ \mathrm{mm}$, $\mathrm{OA}=150\ \mathrm{mm}$, $\mathrm{AR}=160\ \mathrm{mm}$, $\mathrm{RS}=450\ \mathrm{mm}$.

The quick return ratio for the mechanism is _________ (round off to one decimal place).

A short shoe drum (radius $260\ \mathrm{mm}$) brake is shown in the figure. A force of $1\ \mathrm{kN}$ is applied to the lever. The coefficient of friction is $0.4$.

All dimensions are in $\mathrm{mm}$

The magnitude of the torque applied by the brake is __________ $\mathrm{N\,m}$ (round off to one decimal place).

A machine part in the form of cantilever beam is subjected to fluctuating load as shown in the figure. The load varies from $800\ \mathrm{N}$ to $1600\ \mathrm{N}$. The modified endurance, yield and ultimate strengths of the material are $200\ \mathrm{MPa}$, $500\ \mathrm{MPa}$ and $600\ \mathrm{MPa}$, respectively.

All dimensions are in $\mathrm{mm}$

The factor of safety of the beam using modified Goodman criterion is __________ (round off to one decimal place).

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 $2\ \mathrm{kN}$ acts at free end of the beam.

Considering that the allowable shear stress in weld is $60\ \mathrm{N/mm^2}$, the minimum size (leg) of the weld required is _________ $\mathrm{mm}$ (round off to one decimal place).