All 65 questions of GATE 2020 ME (Set 1)

He is known for his unscrupulous ways. He always sheds_____tears to deceive people.

Jofra Archer, the England fast bowler, is_____than accurate.

Select the word that fits the analogy:

Build : Building :: Grow : _____

I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does the phrase “having said that” mean in the given text?

Define $[x]$ as the greatest integer less than or equal to $x$, for each $x\in(-\infty,\infty)$. If $y=[x]$, then area under $y$ for $x\in[1,4]$ is _____.

Crowd funding deals with mobilisation of funds for a project from a large number of people, who would be willing to invest smaller amounts through web-based platforms in the project.

Based on the above paragraph, which of the following is correct about crowd funding?

P, Q, R and S are to be uniquely coded using $\alpha$ and $\beta$. If P is coded as $\alpha\alpha$ and Q as $\alpha\beta$, then R and S, respectively, can be coded as _____.

The sum of the first $n$ terms in the sequence $8,88,888,8888,\ldots$ is_____.

Select the graph that schematically represents BOTH $y=x^m$ and $y=x^{1/m}$ properly in the interval $0\leq x\leq1$, for integer values of $m$, where $m>1$.

The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is _____.

Multiplication of real valued square matrices of same dimension is

The value of

$\lim_{x\to1}\left(\frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}}\right)$

is

The Laplace transform of a function $f(t)$ is $\mathcal{L}(f)=\frac{1}{(s^2+\omega^2)}$. Then, $f(t)$ is

Which of the following function $f(z)$, of the complex variable $z$, is NOT analytic at all the points of the complex plane?

The members carrying zero force (i.e. zero-force members) in the truss shown in the figure, for any load $P>0$ with no appreciable deformation of the truss (i.e. with no appreciable change in angles between the members), are

A single-degree-of-freedom oscillator is subjected to harmonic excitation $F(t)=F_0\cos(\omega t)$ as shown in the figure.

The non-zero value of $\omega$, for which the amplitude of the force transmitted to the ground will be $F_0$, is

The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of $10\ \text{MPa}$. If one unit on the $\sigma-\tau$ plane is $1\ \text{MPa}$, the Mohr’s circle representation of the state-of-stress is given by

A four bar mechanism is shown below.

For the mechanism to be a crank-rocker mechanism, the length of the link PQ can be

A helical gear with $20^\circ$ pressure angle and $30^\circ$ helix angle mounted at the mid-span of a shaft that is supported between two bearings at the ends. The nature of the stresses induced in the shaft is

The crystal structure of $\gamma$ iron (austenite phase) is

Match the following.

The base of a brass bracket needs rough grinding. For this purpose, the most suitable grinding wheel grade specification is

In the Critical Path Method (CPM), the cost-time slope of an activity is given by

Froude number is the ratio of

Match the following non-dimensional numbers with the corresponding definitions:

The velocity field of an incompressible flow in a Cartesian system is represented by

$\vec{V}=2(x^2-y^2)\hat{i}+v\hat{j}+3\hat{k}$

Which one of the following expressions for $v$ is valid?

For an ideal gas, the value of the Joule-Thomson coefficient is

For an ideal gas, a constant pressure line and a constant volume line intersect at a point, in the Temperature ($T$) versus specific entropy ($s$) diagram. $C_P$ is the specific heat at constant pressure and $C_V$ is the specific heat at constant volume. The ratio of the slopes of the constant pressure and constant volume lines at the point of intersection is

For three vectors $\vec{A}=2\hat{j}-3\hat{k}$, $\vec{B}=-2\hat{i}+\hat{k}$ and $\vec{C}=3\hat{i}-\hat{j}$, where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system, the value of $(\vec{A}\cdot(\vec{B}\times\vec{C})+6)$ is _______________.

A flywheel is attached to an engine to keep its rotational speed between $100\ \text{rad/s}$ and $110\ \text{rad/s}$. If the energy fluctuation in the flywheel between these two speeds is $1.05\ \text{kJ}$ then the moment of inertia of the flywheel is _______________ $\text{kg}\cdot\text{m}^2$ (round off to 2 decimal places).

A balanced rigid disc mounted on a rigid rotor has four identical point masses, each of $10$ grams, attached to four points on the $100\ \text{mm}$ radius circle shown in the figure.

The rotor is driven by a motor at uniform angular speed of $10\ \text{rad/s}$. If one of the masses gets detached then the magnitude of the resultant unbalance force on the rotor is ___________ $\text{N}$ (round off to 2 decimal places).

A sheet metal with a stock hardness of $250\ \text{HRC}$ has to be sheared using a punch and a die having a clearance of $1\ \text{mm}$ between them. If the stock hardness of the sheet metal increases to $400\ \text{HRC}$, the clearance between the punch and the die should be _________ $\text{mm}$.

A company is hiring to fill four managerial vacancies. The candidates are five men and three women. If every candidate is equally likely to be chosen then the probability that at least one woman will be selected is _______(round off to 2 decimal places).

The compressor of a gas turbine plant, operating on an ideal intercooled Brayton cycle, accomplishes an overall compression ratio of $6$ in a two-stage compression process. Intercooling is used to cool the air coming out from the first stage to the inlet temperature of the first stage, before its entry to the second stage. Air enters the compressor at $300\ \mathrm{K}$ and $100\ \mathrm{kPa}$. If the properties of gas are constant, the intercooling pressure for minimum compressor work is __________ $\mathrm{kPa}$ (round off to $2$ decimal places).

In a concentric tube counter-flow heat exchanger, hot oil enters at $102^\circ\mathrm{C}$ and leaves at $65^\circ\mathrm{C}$. Cold water enters at $25^\circ\mathrm{C}$ and leaves at $42^\circ\mathrm{C}$. The log mean temperature difference (LMTD) is ______________ $^\circ\mathrm{C}$ (round off to one decimal place).

The evaluation of the definite integral $\int_{-1}^{1.4}x|x|\,\mathrm{d}x$ by using Simpson’s $1/3^{\mathrm{rd}}$ (one-third) rule with step size $h=0.6$ yields

A vector field is defined as $\vec{f}(x,y,z)=\frac{x}{[x^2+y^2+z^2]^{3/2}}\hat{i}+\frac{y}{[x^2+y^2+z^2]^{3/2}}\hat{j}+\frac{z}{[x^2+y^2+z^2]^{3/2}}\hat{k}$ where, $\hat{i},\hat{j},\hat{k}$ are unit vectors along the axes of a right-handed rectangular /Cartesian coordinate system. The surface integral $\iint\vec{f}\cdot\mathrm{d}\vec{S}$ (where $\mathrm{d}\vec{S}$ is an elemental surface area vector) evaluated over the inner and outer surfaces of a spherical shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is

Bars of square and circular cross-section with $0.5\ \mathrm{m}$ length are made of a material with shear strength of $20\ \mathrm{MPa}$. The square bar cross-section dimension is $4\ \mathrm{cm}\times4\ \mathrm{cm}$ and the cylindrical bar cross-section diameter is $4\ \mathrm{cm}$. The specimens are loaded as shown in the figure.

Which specimen(s) will fail due to the applied load as per maximum shear stress theory?

The $2\ \mathrm{kg}$ block shown in figure (top view) rests on a smooth horizontal surface and is attached to a massless elastic cord that has a stiffness $5\ \mathrm{N/m}$.

The cord hinged at $O$ is initially unstretched and always remains elastic. The block is given a velocity $v$ of $1.5\ \mathrm{m/s}$ perpendicular to the cord. The magnitude of velocity in $\mathrm{m/s}$ of the block at the instant the cord is stretched by $0.4\ \mathrm{m}$ is

The truss shown in the figure has four members of length $l$ and flexural rigidity $EI$, and one member of length $l\sqrt{2}$ and flexural rigidity $4EI$. The truss is loaded by a pair of forces of magnitude $P$, as shown in the figure.

The smallest value of $P$, at which any of the truss members will buckle is

A rigid mass-less rod of length $L$ is connected to a disc (pulley) of mass $m$ and radius $r=L/4$ through a friction-less revolute joint. The other end of that rod is attached to a wall through a friction-less hinge. A spring of stiffness $2k$ is attached to the rod at its mid-span. An inextensible rope passes over half the disc periphery and is securely tied to a spring of stiffness $k$ at point C as shown in the figure. There is no slip between the rope and the pulley. The system is in static equilibrium in the configuration shown in the figure and the rope is always taut.

Neglecting the influence of gravity, the natural frequency of the system for small amplitude vibration is

A strip of thickness $40\ \mathrm{mm}$ is to be rolled to a thickness of $20\ \mathrm{mm}$ using a two-high mill having rolls of diameter $200\ \mathrm{mm}$. Coefficient of friction and arc length in $\mathrm{mm}$, respectively are

For an assembly line, the production rate was $4$ pieces per hour and the average processing time was $60$ minutes. The WIP inventory was calculated. Now, the production rate is kept the same, and the average processing time is brought down by $30$ percent. As a result of this change in the processing time, the WIP inventory

A small metal bead (radius $0.5\ \mathrm{mm}$), initially at $100^\circ\mathrm{C}$, when placed in a stream of fluid at $20^\circ\mathrm{C}$, attains a temperature of $28^\circ\mathrm{C}$ in $4.35$ seconds. The density and specific heat of the metal are $8500\ \mathrm{kg/m^3}$ and $400\ \mathrm{J/kg.K}$, respectively. If the bead is considered as lumped system, the convective heat transfer coefficient (in $\mathrm{W/m^2.K}$) between the metal bead and the fluid stream is

Consider two exponentially distributed random variables $X$ and $Y$, both having a mean of $0.50$. Let $Z=X+Y$ and $r$ be the correlation coefficient between $X$ and $Y$. If variance of $Z$ equals $0$, then the value of $r$ is ______________ (round off to $2$ decimal places).

An analytic function of a complex variable $z=x+iy$ ($i=\sqrt{-1}$) is defined as $f(z)=x^2-y^2+i\,\psi(x,y),$ where $\psi(x,y)$ is a real function. The value of the imaginary part of $f(z)$ at $z=(1+i)$ is ____________ (round off to $2$ decimal places).

In a disc-type axial clutch, the frictional contact takes place within an annular region with outer and inner diameters $250\ \mathrm{mm}$ and $50\ \mathrm{mm}$, respectively. An axial force $F_1$ is needed to transmit a torque by a new clutch. However, to transmit the same torque, one needs an axial force $F_2$ when the clutch wears out. If contact pressure remains uniform during operation of a new clutch while the wear is assumed to be uniform for an old clutch, and the coefficient of friction does not change, then the ratio $F_1/F_2$ is ______________ (round off to $2$ decimal places).

A cam with a translating flat-face follower is desired to have the follower motion $y(\theta)=4\cdot[2\pi\theta-\theta^2],\qquad 0\leq\theta\leq2\pi.$ Contact stress considerations dictate that the radius of curvature of the cam profile should not be less than $40\ \mathrm{mm}$ anywhere. The minimum permissible base circle radius is __________ $\mathrm{mm}$ (round off to one decimal place).

A rectangular steel bar of length $500\ \mathrm{mm}$, width $100\ \mathrm{mm}$ and thickness $15\ \mathrm{mm}$ is cantilevered to a $200\ \mathrm{mm}$ steel channel using $4$ bolts, as shown.

For an external load of $10\ \mathrm{kN}$ applied at the tip of the steel bar, the resultant shear load on the bolt at B, is ______________ $\mathrm{kN}$ (round off to one decimal place).

The barrier shown between two water tanks of unit width ($1\ \mathrm{m}$) into the plane of the screen is modeled as a cantilever.

Taking the density of water as $1000\ \mathrm{kg/m^3}$, and the acceleration due to gravity as $10\ \mathrm{m/s^2}$, the maximum absolute bending moment developed in the cantilever is __________ $\mathrm{kN\cdot m}$ (round off to the nearest integer).

The magnitude of reaction force at joint C of the hinge-beam shown in the figure is ______________ $\mathrm{kN}$ (round off to $2$ decimal places).

A slot of $25\ \mathrm{mm}\times25\ \mathrm{mm}$ is to be milled in a workpiece of $300\ \mathrm{mm}$ length using a side and face milling cutter of diameter $100\ \mathrm{mm}$, width $25\ \mathrm{mm}$ and having $20$ teeth.

For a depth of cut of $5\ \mathrm{mm}$, feed per tooth $0.1\ \mathrm{mm}$, cutting speed $35\ \mathrm{m/min}$ and approach and over travel distance of $5\ \mathrm{mm}$ each, the time required for milling the slot is ______ minutes (round off to one decimal place).

The following data applies to basic shaft system:

tolerance for hole $=0.002\ \mathrm{mm}$,

tolerance for shaft $=0.001\ \mathrm{mm}$,

allowance $=0.003\ \mathrm{mm}$,

basic size $=50\ \mathrm{mm}$.

The maximum hole size is ____ $\mathrm{mm}$ (round off to $3$ decimal places).

A steel part with surface area of $125\ \mathrm{cm^2}$ is to be chrome coated through an electroplating process using chromium acid sulphate as an electrolyte. An increasing current is applied to the part according to the following current time relation: $I=12+0.2t$ where, $I=$ current ($\mathrm{A}$) and $t=$ time (minutes). The part is submerged in the plating solution for a duration of $20$ minutes for plating purpose. Assuming the cathode efficiency of chromium to be $15\%$ and the plating constant of chromium acid sulphate to be $2.50\times10^{-2}\ \mathrm{mm^3/A\cdot s}$, the resulting coating thickness on the part surface is ______________ $\mathrm{\mu m}$ (round off to one decimal place).

In a turning process using orthogonal tool geometry, a chip length of $100\ \mathrm{mm}$ is obtained for an uncut chip length of $250\ \mathrm{mm}$.

The cutting conditions are: cutting speed $=30\ \mathrm{m/min}$, rake angle $=20^\circ$.

The shear plane angle is _______ degrees (round off to one decimal place).

The thickness of a steel plate with material strength coefficient of $210\ \mathrm{MPa}$, has to be reduced from $20\ \mathrm{mm}$ to $15\ \mathrm{mm}$ in a single pass in a two-high rolling mill with a roll radius of $450\ \mathrm{mm}$ and rolling velocity of $28\ \mathrm{m/min}$. If the plate has a width of $200\ \mathrm{mm}$ and its strain hardening exponent, $n$ is $0.25$, the rolling force required for the operation is __________ $\mathrm{kN}$ (round off to $2$ decimal places).

Note: $\text{Average Flow Stress}=\text{Material Strength Coefficient}\times\dfrac{(\text{True Strain})^n}{(1+n)}$

Two business owners Shveta and Ashok run their businesses in two different states. Each of them, independent of the other, produces two products A and B, sells them at Rs. $2,000$ per kg and Rs. $3,000$ per kg, respectively, and uses Linear Programming to determine the optimal quantity of A and B to maximize their respective daily revenue. Their constraints are as follows: i) for each business owner, the production process is such that the daily production of A has to be at least as much as B, and the upper limit for production of B is $10\ \mathrm{kg}$ per day, and ii) the respective state regulations restrict Shveta’s production of A to less than $20\ \mathrm{kg}$ per day, and Ashok’s production of A to less than $15\ \mathrm{kg}$ per day. The demand of both A and B in both the states is very high and everything produced is sold.

The absolute value of the difference in daily (optimal) revenue of Shveta and Ashok is ______________ thousand Rupees (round off to $2$ decimal places).

Consider two cases as below.

Case 1: A company buys $1000$ pieces per year of a certain part from vendor ‘X’. The changeover time is $2$ hours and the price is Rs. $10$ per piece. The holding cost rate per part is $10\%$ per year.

Case 2: For the same part, another vendor ‘Y’ offers a design where the changeover time is $6$ minutes, with a price of Rs. $5$ per piece, and a holding cost rate per part of $100\%$ per year. The order size is $800$ pieces per year from ‘X’ and $200$ pieces per year from ‘Y’.

Assume the cost of downtime as Rs. $200$ per hour. The percentage reduction in the annual cost for Case $2$, as compared to Case $1$ is __________ (round off to $2$ decimal places).

Consider steady, viscous, fully developed flow of a fluid through a circular pipe of internal diameter $D$. We know that the velocity profile forms a paraboloid about the pipe centre line, given by: $V=-C\left(r^2-\dfrac{D^2}{4}\right)\ \mathrm{m/s}$, where $C$ is a constant. The rate of kinetic energy (in $\mathrm{J/s}$) at the control surface A-B, as shown in the figure, is proportional to $D^n$. The value of $n$ is ______________.

Air discharges steadily through a horizontal nozzle and impinges on a stationary vertical plate as shown in figure.

The inlet and outlet areas of the nozzle are $0.1\ \mathrm{m^2}$ and $0.02\ \mathrm{m^2}$, respectively. Take air density as constant and equal to $1.2\ \mathrm{kg/m^3}$. If the inlet gauge pressure of air is $0.36\ \mathrm{kPa}$, the gauge pressure at point O on the plate is ______ $\mathrm{kPa}$ (round off to two decimal places).

Air (ideal gas) enters a perfectly insulated compressor at a temperature of $310\ \mathrm{K}$. The pressure ratio of the compressor is $6$. Specific heat at constant pressure for air is $1005\ \mathrm{J/kg.K}$ and ratio of specific heats at constant pressure and constant volume is $1.4$. Assume that specific heats of air are constant. If the isentropic efficiency of the compressor is $85$ percent, the difference in enthalpies of air between the exit and the inlet of the compressor is _______ $\mathrm{kJ/kg}$ (round off to nearest integer).

One kg of air, initially at a temperature of $127^\circ\mathrm{C}$, expands reversibly at a constant pressure until the volume is doubled. If the gas constant of air is $287\ \mathrm{J/kg.K}$, the magnitude of work transfer is _________ $\mathrm{kJ}$ (round off to $2$ decimal places).

For an ideal Rankine cycle operating between pressures of $30\ \mathrm{bar}$ and $0.04\ \mathrm{bar}$, the work output from the turbine is $903\ \mathrm{kJ/kg}$ and the work input to the feed pump is $3\ \mathrm{kJ/kg}$. The specific steam consumption is __________ $\mathrm{kg/kW.h}$ (round off to $2$ decimal places).

For a Kaplan (axial flow) turbine, the outlet blade velocity diagram at a section is shown in figure.

The diameter at this section is $3\ \mathrm{m}$. The hub and tip diameters of the blade are $2\ \mathrm{m}$ and $4\ \mathrm{m}$, respectively. The water volume flow rate is $100\ \mathrm{m^3/s}$. The rotational speed of the turbine is $300\ \mathrm{rpm}$. The blade outlet angle $\beta$ is _____ degrees (round off to one decimal place).

The indicated power developed by an engine with compression ratio of $8$, is calculated using an air-standard Otto cycle (constant properties). The rate of heat addition is $10\ \mathrm{kW}$. The ratio of specific heats at constant pressure and constant volume is $1.4$. The mechanical efficiency of the engine is $80$ percent.

The brake power output of the engine is ______ $\mathrm{kW}$ (round off to one decimal place).