GATE 2021 Question Paper PI Production and Industrial Engineering

General Aptitude (GA)
Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 The current population of a city is 11,02,500. If it has been increasing at the
rate of 5% per annum, what was its population 2 years ago?

(A) 9,92,500

(B) 9,95,006

(C) 10,00,000

(D) 12,51,506

Q.2 p q
p and q are positive integers and + = 3,
q p

p2 q2
then, + =
q2 p2

(A) 3

(B) 7

(C) 9

(D) 11

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Q.3

The least number of squares that must be added so that the line P-Q
becomes the line of symmetry is ________

(A) 4

(B) 3

(C) 6

(D) 7

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Q.4 Nostalgia is to anticipation as _______ is to ________
Which one of the following options maintains a similar logical relation in the
above sentence?

(A) Present, past

(B) Future, past

(C) Past, future

(D) Future, present

Q.5 Consider the following sentences:
(i) I woke up from sleep.
(ii) I woked up from sleep.
(iii) I was woken up from sleep.
(iv) I was wokened up from sleep.
Which of the above sentences are grammatically CORRECT?

(A) (i) and (ii)

(B) (i) and (iii)

(C) (ii) and (iii)

(D) (i) and (iv)

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Q. 6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q.6 Given below are two statements and two conclusions.
Statement 1: All purple are green.
Statement 2: All black are green.
Conclusion I: Some black are purple.
Conclusion II: No black is purple.
Based on the above statements and conclusions, which one of the following
options is logically CORRECT?

(A) Only conclusion I is correct.

(B) Only conclusion II is correct.

(C) Either conclusion I or II is correct.

(D) Both conclusion I and II are correct.

Q.7 Computers are ubiquitous. They are used to improve efficiency in almost all
fields from agriculture to space exploration. Artificial intelligence (AI) is
currently a hot topic. AI enables computers to learn, given enough training
data. For humans, sitting in front of a computer for long hours can lead to
health issues.
Which of the following can be deduced from the above passage?
(i) Nowadays, computers are present in almost all places.
(ii) Computers cannot be used for solving problems in engineering.
(iii) For humans, there are both positive and negative effects of using
computers.
(iv) Artificial intelligence can be done without data.

(A) (ii) and (iii)

(B) (ii) and (iv)

(C) (i), (iii) and (iv)

(D) (i) and (iii)

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Q.8 Consider a square sheet of side 1 unit. In the first step, it is cut along the main
diagonal to get two triangles. In the next step, one of the cut triangles is
revolved about its short edge to form a solid cone. The volume of the resulting
cone, in cubic units, is ________

(A) 𝜋
3
(B) 2𝜋
3
(C) 3𝜋
2
(D) 3π

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Q.9
65
Sunday 55

50
Saturday 60

35
Friday 20

55
Thursday 60

50
Wednesday 60

65
Tuesday 55

70
Monday 45

0 10 20 30 40 50 60 70 80
Y X

The number of minutes spent by two students, X and Y, exercising every day
in a given week are shown in the bar chart above.
The number of days in the given week in which one of the students spent a
minimum of 10% more than the other student, on a given day, is

(A) 4

(B) 5

(C) 6

(D) 7

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Q.10

Corners are cut from an equilateral triangle to produce a regular convex
hexagon as shown in the figure above.
The ratio of the area of the regular convex hexagon to the area of the original
equilateral triangle is

(A) 2:3

(B) 3:4

(C) 4:5

(D) 5:6

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Q.1 – Q.17 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 A product has an exponential time-to-failure distribution with a constant
failure rate of 0.00006 per hour. The reliability of the product after 4000
hours of operation is

(A) 0.5866

(B) 0.6866

(C) 0.7866

(D) 0.8866

Q.2 In a typical product development process under concurrent engineering
approach, all elements of product life cycle from conception to disposal are
considered at

(A) Product design stage

(B) Process design stage

(C) Manufacturing stage

(D) Disposal stage

Q.3 When acceptance number of a single sampling plan under attribute category
is zero with sample size less than or equal to 10, the Operating Characteristic
(OC) curve is

(A) A horizontal line

(B) A vertical line

(C) A convex function

(D) An inverted S-shaped curve

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Q.4 Which one of the following is an improvement type heuristic algorithm for
computerized layout design technique?

(A) Systematic layout planning (SLP)

(B) Computerized relative allocation of facilities technique (CRAFT)

(C) Computerized relationship layout planning (CORELAP)

(D) Plant layout analysis and evaluation technique (PLANET)

Q.5 Which one of the following is NOT a measure of forecast error?

(A) Mean absolute deviation (MAD)

(B) Mean squared error (MSE)

(C) Mean absolute percent error (MAPE)

(D) Mean sum product error (MSPE)

Q.6 Pearlite microstructure in an eutectoid steel consists of alternating layers of
two phases, namely α ferrite and

(A) Martensite

(B) Austenite

(C) Cementite

(D) Bainite

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Q.7 Which one of the following defects is NOT associated with welding
processes?

(A) Angular distortion

(B) Hot tear

(C) Hydrogen embrittlement

(D) Earring

Q.8 Match the component with the corresponding manufacturing process in the
table below.

Component Manufacturing process

P Aluminum alloy piston for 1 Blow molding
IC engine

Q Low carbon steel oil pan 2 Powder metallurgy

R Tungsten carbide cutting 3 Sand casting
tool insert

S Plastic bottle 4 Deep drawing

(A) P-3, Q-2, R-1, S-4

(B) P-3, Q-4, R-2, S-1

(C) P-2, Q-3, R-4, S-1

(D) P-1, Q-3, R-2, S-4

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Q.9 In a turning operation, doubling the cutting speed (𝑽) reduces the tool life
(𝑻) to 𝟏⁄𝟖th of the original tool life. The exponent 𝒏 in the Taylor’s tool life
equation, 𝑽𝑻𝒏 = 𝑪 is

(A) 1
2
(B) 1
3
(C) 1
4
(D) 1
8

Q.10 Which one among the following mechanisms is NOT used for transforming
rotation to translation in machine tools?

(A) Screw-nut system

(B) 4-bevel gear type differential mechanism

(C) Cam and cam follower system

(D) Whitworth mechanism

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Q.11 Match the measuring feature with the corresponding measuring instrument
in the table below.

Measuring feature Measuring instrument

P Flatness error of a surface 1 Auto collimator
plate

Q Profile of a cam 2 Tool maker’s microscope

R Alignment error of a 3 Dividing head and dial gauge
machine tool slide way

S Pitch and angle errors of 4 Optical interferometer
screw thread

(A) P-4, Q-1, R-2, S-3

(B) P-1, Q-3, R-4, S-2

(C) P-2, Q-4, R-3, S-1

(D) P-4, Q-3, R-1, S-2

Q.12 The frequency of pulsing in a die-sinking electric discharge machine (EDM)
is 10 kHz. The pulse off-time is set at 40 micro-seconds. The duty factor at
this setting is

(A) 0.40

(B) 0.60

(C) 0.67

(D) 2.50

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Q.13 A cantilever beam of length 0.3 m is subjected to a uniformly distributed
load C = 10 kN/m, as shown in the figure. The bending (flexural) rigidity of
the beam is 5000 Nm2. Neglecting the self-weight of the beam, the magnitude
of beam curvature in m-1 at the fixed end is

0.3 m

C = 10 kN/m

x

(A) 1.10

(B) 0.02

(C) 0.09

(D) 0.05

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Q.14 A circular rod of length l = 2 m is subjected to a compressive load P, as shown
in the figure. The bending (flexural) rigidity of the rod is 2000 Nm2. If both
ends are pinned, then the critical load Pcr in N (rounded to the nearest
integer) at which the rod buckles elastically is
P

l = 2m

(A) 4935

(B) 2000

(C) 5167

(D) 1238

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Q.15 Two cylindrical parts of equal length l, as shown in the figure, made of steel
having Young’s modulus E = 200 GPa and Poisson’s ratio  = 0.33 are press
fitted upon one another. If radial interference  = 0.05 mm, and radii R = 25
mm and R0 = 40 mm, then the contact pressure P in MPa at the interface
upon press fit is

(A) 10.7

(B) 60.9

(C) 121.9

(D) 1005.3

Q.16 The dimensionless number defined by the ratio of inertial force to viscous
force is called

(A) Mach number

(B) Froude number

(C) Weber number

(D) Reynolds number

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Q.17 A small capillary tube of 3 mm inner diameter is inserted into a fluid having
density 900 kg/m3, surface tension 0.1 N/m, and contact angle 30⁰. The rise
in the height of fluid in the capillary tube due to surface tension is

(A) 111.4 mm

(B) 128.3 mm

(C) 89.1 mm

(D) 154.1 mm

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Q.18 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.18 A given steel has identical yield strength of 700 MPa in uni-axial tension and
uni-axial compression. If the steel is subjected to pure shear stress such that
the three principal stresses are  =   =   = − with  ≥  ≥ , then
the stress  in MPa for the initiation of plastic yielding in the steel as per
von Mises yield criterion is _____________. [round off to 2 decimal places]

Q.19 A cylindrical mild steel tensile test specimen of gauge length 50 mm and
diameter 10 mm is extended in two stages at a deformation speed of 4
mm/min. The specimen is extended from 50 mm to 55 mm in the first stage,
and from 55 mm to 60 mm in the second stage. Neglecting elastic
deformation, the total longitudinal true strain is ____________. [round off to
2 decimal places]

Q.20 A M30 bolt needs to be subjected to pretension Fi = 350 kN. If the torque
coefficient K of the bolt is 0.2, then the torque in Nm needed to achieve this
pretension is _________. [in integer]

Q.21 A 150 mm wide polyamide flat belt is transmitting 15 kW power through a
belt-pulley system. The driving pulley of 150 mm pitch diameter is rotating
at 200 RPM. If F1 is the belt tension on high tension side, and F2 is the belt
tension on low tension side, then the difference in belt tensions F = F1 – F2
in N is ___________. [round off to one decimal place]

Q.22 Heat is being removed from a refrigerator at a rate of 300 kJ/min to maintain
its inside temperature at 2 ⁰C. If the input power to the refrigerator is 2 kW,
the coefficient of performance of the refrigerator is __________.
[round off to one decimal place]

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Q.23 In an ideal Otto cycle, 800 kJ/kg is transferred to air during the constant
volume heat addition process and 381 kJ/kg is removed during the constant
volume heat rejection process. The thermal efficiency in % of the cycle is
__________.
[round off to one decimal place]

Q.24 If (3i +1)x + (4i + 4)y + 5 = 0 with x, y being real and i = √−𝟏, then x =
________. [correct up to one decimal place]

Q.25 The minimum value of function f defined by
𝒇(𝒙, 𝒚, 𝒛) = 𝒙𝟐 + 𝟓𝒚𝟐 + 𝟓𝒛𝟐 − 𝟒𝒙 + 𝟒𝟎𝒚 − 𝟒𝟎𝒛 + 𝟑𝟎𝟎
is ____________. [in integer]

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Q.26 – Q.36 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong
answer: – 2/3).

Q.26 For a given process control chart, there are four rules for determining out-
of-control state of the process which are being used simultaneously. The
probability of Type-I error for the four rules are 0.005, 0.02, 0.03, and 0.05.
Assuming independence of the rules, the probability of overall Type-I error
when all the four rules are used simultaneously is

(A) 0.101

(B) 0.201

(C) 0.001

(D) 0.301

Q.27 An in-control process has an estimated standard deviation of 2 mm. The
specification limits of the component being processed are 120 ± 8 mm. When
the process mean shifts to 118 mm, the values of the process capability
indices, Cp and Cpk , respectively are,

(A) 1.000, 1.667

(B) 1.333, 1.667

(C) 1.333, 1.000

(D) 1.000, 1.000

Q.28 There are a number of identical components in a parallel system. When the
system reliability is 0.97 and the reliability of each individual component is
0.68, the number of identical components in the system is (if actual value is
a fraction, it may be rounded up to the next higher integer).

(A) 2

(B) 4

(C) 6

(D) 8

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Q.29 A retail chain company has identified four sites A, B, C and D to open a new
retail store. The company has selected four factors as the basis for evaluation
of these sites. The factors, their weights, and the score for each site are given
in the following table.

Score for site (out of 100)
Factor
Factor weight A B C D

Average community income 0.4 60 70 80 50

Demand growth potential 0.1 30 80 50 40

Proximity to existing store 0.3 50 10 40 60

Availability of public transport 0.2 40 30 40 20

The site that should be selected to open the new retail store is

(A) Site A

(B) Site B

(C) Site C

(D) Site D

Q.30 In the classical economic order quantity (EOQ) model, let 𝑸 and 𝑪 denote
the optimal order quantity and the corresponding minimum total annual
cost (the sum of the inventory holding and ordering costs). If the order
quantity is estimated incorrectly as 𝑸′ = 𝟐𝑸, then the corresponding total
annual cost 𝑪′ is

(A) 𝐶 ′ = 1.25𝐶

(B) 𝐶 ′ = 1.5𝐶

(C) 𝐶 ′ = 1.75𝐶

(D) 𝐶 ′ = 2𝐶

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Q.31 𝟖 𝟑
The eigenvalues of matrix A = [ ] are 5 and 10. For matrix B = A + α I,
𝟐 𝟕
where α is a constant and I is 2×2 identity matrix, its eigenvalues are

(A) 5, 10

(B) 5 + α, 10 + α

(C) 5 - α, 10 - α

(D) 5α, 10α

Q.32 A company manufactures two products P and Q with unit profit of 4 and 5,
respectively. The production requires manpower and two kinds of raw
materials R1 and R2. The following table summarizes the requirement and
availability of resources.

Resource Resource usage per unit Amount of
of production resource available

P Q

manpower 1 1 10

R1 1 2 18

R2 2 1 18

The maximum profit the company can make is

(A) 45

(B) 48

(C) 42

(D) 54

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Q.33 A tool of an NC machine has to move along a circular arc from (20,20) to
(10,10), while performing an operation. The center of the arc is at (20,10).
Which one of the following NC tool commands performs the above
mentioned operation?

(A) N020 G03 X20 Y20 X10 Y10 R10

(B) N020 G02 X20 Y20 X10 Y10 R10

(C) N020 G02 X10 Y10 X20 Y20 R10

(D) N020 G01 X20 Y20 X10 Y10 R10

Q.34 In a shaft-hole assembly, the hole is specified as 30𝟎. 𝟎𝟒𝟎 mm. The mating
𝟎. 𝟎𝟎𝟎
shaft has a clearance fit with minimum clearance of 0.01 mm. The tolerance
on the shaft is 0.03 mm. The maximum clearance in mm between the hole
and the shaft is

(A) 0.04

(B) 0.05

(C) 0.08

(D) 0.10

Q.35 ‘GO’ and ‘NO GO’ snap gauges are to be designed for a shaft 36.000+𝟎. 𝟎𝟕𝟎
+𝟎. 𝟎𝟏𝟎
mm. Gauge tolerance can be taken as 5% of the hole tolerance. Following
the ISO system of gauge design, the respective sizes of ‘GO’ and ‘NO GO’
gauges are

(A) 36.013 mm and 36.067 mm

(B) 36.015 mm and 36.065 mm

(C) 36.018 mm and 36.062 mm

(D) 36.020 mm and 36.060 mm

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Q.36 A circular tank of 4 m diameter is filled up to a height of 3 m. Assuming
almost steady flow and neglecting losses, the time taken in seconds to empty
the tank through a 5 cm diameter hole located at the center of the tank
bottom (take acceleration due to gravity g = 9.81 m/s2) is [round off to the
nearest integer]

(A) 5005

(B) 1807

(C) 8097

(D) 3154

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Q.37 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.37 The probability mass function 𝑷(𝒙) of a discrete random variable 𝑿 is given
𝟏
by 𝑷(𝒙) = 𝟐𝒙 , where 𝒙 = 𝟏, 𝟐, ⋯ , ∞. The expected value of X is __________.
[in integer]

Q.38 The time to pass through a security screening at an airport follows an
exponential distribution. The mean time to pass through the security
screening is 15 minutes. To catch the flight, a passenger must clear the
security screening within 15 minutes. The probability that the passenger will
miss the flight is __________. [round off to 3 decimal places]

Q.39 A machine shop has received four jobs A, B, C and D for processing on a
single CNC machine. All jobs are available for processing on the first day of
the production schedule calendar, and processing times and due dates as
applicable on the first day are given below. Using earliest due date rule, the
average tardiness (in days) is ________. [in integer]

Processing time Due date
Job (in days) (day)

A 8 14

B 5 10

C 7 12

D 9 19

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Q.40 A time study is carried out for a spot welding operation which is being
performed by an operator. The time taken (in seconds) for five observations
are recorded as 40, 35, 45, 37 and 43, respectively. If the standard time and
the allowance for this operation are 45 seconds and 9 seconds, respectively,
then the performance rating (in percentage) of the operator is ________. [in
integer]

Q.41 The initial cost of a machine is INR 10,00,000 and its salvage value after 10
years of use is INR 50,000. Using the straight line depreciation method, the
book value in INR of the machine at the end of 7th year is ________. [in
integer]

Q.42 A project consists of eight activities. The time required for each activity and
its immediate predecessor(s) are given in the table below.

Activity Activity time (in days) Immediate predecessor(s)

A 2 -

B 3 -

C 2 A

D 4 A, B

E 4 C

F 3 C

G X D, E

H 2 F, G

If the project completion time using critical path method (CPM) is 15 days, then
the value of X (in days) is ________. [in integer]

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Q.43 A wire of 5 mm diameter is drawn into a wire of 4 mm diameter through a
conical die at a constant pulling speed of 5 m/s. Neglecting the coefficient of
friction and redundant work, the drawing stress (𝝈𝒅 ) in MPa for the above
𝟏
process is given by 𝝈𝒅 = 𝝈 ̅ 𝐥𝐧 [𝟏−𝒓], where 𝝈 ̅ is the mean flow strength of
wire material in MPa, and 𝒓 is the ratio of decrease in area of cross-section
to initial area of cross-section of the wire. If the mean flow strength of wire
material is 600 MPa, then the power required in kW in the above wire
drawing process is ________.
[round off to 2 decimal places]

Q.44 In an arc welding process, the DC power source characteristic is linear with
an open circuit voltage of 60 V and short circuit current of 600 A. The heat
required for melting a metal during the welding is 10 J/mm3, and the heat
transfer and melting efficiencies are 80% and 25%, respectively. If the weld
cross-sectional area of 20 mm2 is made using the maximum arc power, then
the required welding speed in mm/s is ________. [round off to one decimal
place]

Q.45 A company is producing a disc-shaped product of 50 mm thickness and 1.0
m diameter using sand casting process. The solidification time of the above
𝑽 𝟐
casting process is estimated by Chvorinov’s equation 𝒕 = 𝑩 [ 𝑨 ] , where 𝑩
is the mold constant, and 𝑽 and 𝑨 are the volume and surface area of the
casting, respectively. It is decided to modify both the thickness and diameter
of the disc to 25 mm and 0.5 m, respectively, maintaining the same casting
condition. The percentage reduction in solidification time of the modified
disc as compared to that of the bigger disc is _________.[round off to one
decimal place]

Q.46 A single point cutting tool with 15⁰ orthogonal rake angle is used to machine
a mild steel plate under orthogonal machining condition. The depth of cut
(uncut thickness) is set at 0.9 mm. If the chip thickness is 1.8 mm, then the
shear angle in degree is __________. [round off to one decimal place]

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Q.47 The top layer of a flat 750 mm × 300 mm rectangular mild steel plate is to be
machined with a single depth of cut using a shaping machine. The plate has
been fixed by keeping 750 mm side along the tool travel direction. If the
approach and the over-travel are 25 mm each, average cutting speed is 10
m/min, feed rate is 0.4 mm/stroke, and the ratio of return time to cutting
time of the tool is 1:2, the time (in minutes) required to complete the
machining operation is _______. [round off to one decimal place]

Q.48 A 3 mm thick steel sheet, kept at room temperature of 30 ⁰C, is cut by a fiber
laser beam. The laser spot diameter on the top surface of the sheet is 0.2 mm.
The laser absorptivity of the sheet is 50%. The properties of steel are density
= 8000 kg/m3, specific heat = 500 J/kg.⁰C, melting temperature = 1530 ⁰C,
and latent heat of fusion = 3×105 J/kg. Assume that melting efficiency is
100% and that the kerf width is equal to the laser spot diameter. The
maximum speed (in m/s) at which the sheet can be fully cut at 2 kW laser
power is __________.
[round off to 3 decimal places]

Q.49 In a point-to-point open-loop NC drive, a stepper motor with 1.8⁰ step angle
is coupled to a leadscrew through a gear reduction of 4:1 (4 rotations of the
motor enables 1 rotation of leadscrew). The single-start leadscrew has a
pitch of 4 mm. The worktable of the system is driven by the leadscrew. If the
table moves at a uniform speed of 10 mm/s, the pulse frequency (in Hz)
required to drive the stepper motor is ________. [round off to one decimal
place]

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Q.50 A 30 kg smooth, solid sphere rests on two frictionless inclines as shown in
the figure. The magnitude of contact force in N acting at the point A is (take
acceleration due to gravity g = 9.81 m/s2 and consider both sphere and
inclines to be rigid) _________. [round off to 2 decimal places]

A
B
600
300

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Q.51 Consider the truss shown in the figure. The members AB, BC, and CA are
all rigid and form an equilateral triangle. The contact between roller and
ground at C is frictionless. If the self-weight of members is neglected, the
force in member BC in N is (negative sign should be used if the force is
compressive and positive if the force in the member is tensile) _________.
[round off to one decimal place]
10 kN

B

A C

Q.52 A fluid with dynamic viscosity  = 1 Pa.s is flowing through a circular pipe
with diameter 1 cm. If the flow rate (discharge) in the pipe is 0.2 liter/s, the
maximum velocity in m/s of the fluid in the pipe is (assume fully developed
flow and take fluid density  = 1000 kg/m3) ________. [round off to one
decimal place]

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Q.53 Values of function y(x) at discrete values of x for 0 ≤ x ≤ 10 are given in
𝟏𝟎
table. Using trapezoidal rule, ∫𝟎 𝒚(𝒙)𝒅𝒙 = _________.
[round off to one decimal place]

x 0 1 2 3 4 5 6 7 8 9 10

y(x) 5 3 0 -5 -10 -6 0 5 11 18 30

Q.54 Temperature field inside a sphere of radius R = 1 m with origin at its center
is T(x, y, z) = 100 – 70x + 51y – 80z – 10x2 – 20y2 – 20z2 . If thermal
conductivity of the sphere material is K = 50 W/m.K and Fourier law of heat
conduction is valid, net heat leaving the sphere per unit time in W is
_________.
[round off to one decimal place]

Q.55 A 3.5 mm thick sheet is rolled using a two high rolling mill to reduce the
thickness under plane strain condition. Both rolls have a diameter of 500 mm
and are rotating at 200 RPM. The coefficient of friction at the sheet and roll
interface is 0.08, and the elastic deflection of the rolls is negligible. If the
mean flow strength of the sheet material is 400 MPa, then the minimum
possible thickness (in mm) of sheet that can be produced in a single pass is
_________.
[round off to 2 decimal places]

END OF THE QUESTION PAPER

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