GATE 2025 Question Paper Production and Industrial Engineering (PI)

GATE
2025
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.

General Aptitude
Q.1 – Q.5 Carry ONE mark Each

Q.1 Courage : Bravery :: Yearning : __________

Select the most appropriate option to complete the analogy.

(A) Longing

(B) Yelling

(C) Yawning

(D) Glaring

Q.2 We __________ tennis in the lawn when it suddenly started to rain.

Select the most appropriate option to complete the above sentence.

(A) have been playing

(B) had been playing

(C) would have been playing

(D) could be playing

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Q.3 A 4 × 4 digital image has pixel intensities (𝑈) as shown in the figure. The number of
pixels with 𝑈 ≤ 4 is:

(A) 3

(B) 8

(C) 11

(D) 9

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Q.4 In the given figure, the numbers associated with the rectangle, triangle, and ellipse
are 1, 2, and 3, respectively. Which one among the given options is the most
appropriate combination of P, Q, and R ?

2

R

4
Q
1 P

3

(A) P = 6; Q = 5; R = 3

(B) P = 5; Q = 6; R = 3

(C) P = 3; Q = 6; R = 6

(D) P = 5; Q = 3; R = 6

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Q.5 A rectangle has a length L and a width W, where L > W. If the width, W, is increased
by 10%, which one of the following statements is correct for all values of L and W?

(A) Perimeter increases by 10%.

(B) Length of the diagonals increases by 10%.

(C) Area increases by 10%.

(D) The rectangle becomes a square.

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Q.6 – Q.10 Carry TWO marks Each

Q.6 Column-I has statements made by Shanthala; and, Column-II has responses given
by Kanishk.

Column-I Column-II

P. This house is in a mess. 1. Alright, I won’t bring it up during
our conversations.

Q. I am not happy with the marks 2. Well, you can easily look it up.
given to me.

R. Politics is a subject I avoid talking 3. No problem, let me clear it up for
about. you.

S. I don’t know what this word 4. Don’t worry, I will take it up with
means. your teacher.

Identify the option that has the correct match between Column-I and Column-II.

(A) P – 2; Q – 3; R – 1; S – 4

(B) P – 3; Q – 4; R – 1; S – 2

(C) P – 4; Q – 1; R – 2; S – 3

(D) P – 1; Q – 2; R – 4; S – 3

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Q.7 Weight of a person can be expressed as a function of their age. The function usually
varies from person to person. Suppose this function is identical for two brothers,
and it monotonically increases till the age of 50 years and then it monotonically
decreases. Let 𝑎1 and 𝑎2 (in years) denote the ages of the brothers and 𝑎1 < 𝑎2 .

Which one of the following statements is correct about their age on the day when
they attain the same weight?

(A) 𝑎1 < 𝑎2 < 50

(B) 𝑎1 < 50 < 𝑎2

(C) 50 < 𝑎1 < 𝑎2

(D) Either 𝑎1 = 50 or 𝑎2 = 50

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Q.8 A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius r
cm as shown in the figure. The side of the dodecagon is d cm. All the triangles
(numbered 1 to 12) in the figure are used to form squares of side r cm and each
numbered triangle is used only once to form a square.

The number of squares that can be formed and the number of triangles required to
form each square, respectively, are:

Note: The figure shown is representative.

(A) 3; 4

(B) 4; 3

(C) 3; 3

(D) 3; 2

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Q.9 2
𝑥2 𝑥 2𝑥
If a real variable 𝑥 satisfies 3 = 27 × 9 , then the value of 𝑥 2 is:
(2 )

(A) 2−1

(B) 20

(C) 23

(D) 215

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Q.10 The number of patients per shift (𝑋) consulting Dr. Gita in her past 100 shifts is
shown in the figure. If the amount she earns is ₹ 1000(𝑋 − 0.2), what is the average
amount (in ₹) she has earned per shift in the past 100 shifts?

Note: The figure shown is representative.

50
40
40
Number of shifts

30
30
20
20
10
10

0
5 6 7 8
Number of patients per shift (X)

(A) 6,100

(B) 6,300

(C) 6,000

(D) 6,500

Organizing Institute: IIT Roorkee Page 9 of 35

Q.11 – Q.35 Carry ONE mark Each

0 −1
Q.11 The eigenvalues of the matrix   are
1 0 

(A) − −1 and −1

(B) −1 and 1

(C) 1 + −1 and 1 − −1

(D) −1 + −1 and −1 − −1

Q.12 If i , j and k are the orthogonal unit vectors in Cartesian x-y-z coordinate system,
the curl of the vector −2y i + x j is

(A) 3k

(B) −3k

(C) −k

(D) k

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Q.13 If F ( s ) denotes the Laplace transform of some function f ( t ) , then the Laplace
transform of e f ( t ) , where 𝑏 is a real constant, is
bt

(A) F ( s − b)

(B) F ( s + b)

(C) −F ( s )

(D) F (b − s )

Q.14 Which one of the following equations is a linear differential equation?

dy
(A) + 2x = y2
dx

dy
(B) x3 + xy = x 2
dx

d2 y dy
(C) x 2
+ 2y =0
dx dx

2
 dy 
(D)   + 2x = y
 dx 

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Q.15 A bag contains 5 red, 7 green and 3 blue balls. Two balls are drawn at random from
the bag one-by-one. The probability of the second drawn ball being red is

1
(A)
5

1
(B)
3

2
(C)
5

2
(D)
3

Q.16 Exit-hole occurrence is common in

(A) Electron Beam Welding

(B) Submerged Arc Welding

(C) Friction Welding

(D) Friction Stir Welding

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Q.17 An aircraft has two engines, each having a reliability R. The aircraft will crash only
when both engines stop working. The reliability of the aircraft flying without crash
is

(A) R2

(B) 2R

(C) 2R− R2

(D) R2−2R

Q.18 The proper sequence of design of a product is

(A) Conceptual design, Embodiment design, Detailed design

(B) Conceptual design, Detailed design, Embodiment design

(C) Embodiment design, Conceptual design, Detailed design

(D) Embodiment design, Detailed design, Conceptual design

Organizing Institute: IIT Roorkee Page 13 of 35

Q.19 In a work sampling, out of n observations, a worker was sitting idle in x
observations. The standard deviation of the mean proportion of idle time is given
by

(A) x2
n3

(B) x (n − x)
n3

(C) x (n − x)
n2

(D) x
n

Q.20 Atomic packing factor of a body centered cubic structure is closest to

(A) 0.34

(B) 0.52

(C) 0.68

(D) 0.74

Organizing Institute: IIT Roorkee Page 14 of 35

Q.21 Which one of the following statements is FALSE with respect to the injection
molding of polymer composite?

(A) Molten polymer along with the reinforcements is injected into a closed mold cavity.

(B) Material deforms plastically to adopt the shape of the mold cavity.

(C) Melt temperature, injection speed and screw speed are some important process
parameters.

(D) Commonly used reinforcements are particles, whiskers and short fibers.

Q.22 A through hole of 8 mm diameter is to be drilled in a 30 mm thick mild steel plate.
Which one of the following processes is the most appropriate to achieve high
dimensional accuracy with less processing time?

(A) Conventional drilling using a carbide drill bit

(B) Die sinking EDM using a copper electrode

(C) ECM using a copper electrode

(D) Plasma arc machining

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Q.23 Which one of the following casting defects is caused due to the supply of the molten
metal through two gates?

(A) Cold shut

(B) Shift

(C) Pin hole

(D) Rat tail

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Q.24 Match the following with reference to the CNC machine and its minimum number
of axes available in the machine.

Type of CNC Machine Minimum number of axes

P Turning center (CNC Lathe) i 3

Q Machining center (CNC Vertical milling) ii 2

iii 4

(A) P – i, Q – ii

(B) P – ii, Q – i

(C) P – i, Q – iii

(D) P – ii, Q – iii

Organizing Institute: IIT Roorkee Page 17 of 35

Q.25 A simply supported beam AB of span L is shown in the figure. A moment M is
applied at point C. The magnitude of the reaction force at point A is

(A) M
L

(B) M
a

(C) M
L−a

(D) M
L+a

Organizing Institute: IIT Roorkee Page 18 of 35

Q.26 The relationship between the hoop stress  1 and the longitudinal stress  2 of a
closed cylindrical thin-walled pressure vessel is

(A)  1 = 2 2

(B) 1 =  2

(C) 2
1 =
2

(D) 1
1 =  2
3

Organizing Institute: IIT Roorkee Page 19 of 35

Q.27 The starting simplex table of a linear programming problem is given below, where
S1 , S 2 , S3 and S 4 are the slack variables. The objective of the problem is

Maximize z = 6 x1 + 4 x2

The leaving variable among the basic variables is

x1 x2 S1 S2 S3 S4 Solution
Basic

S1 6 3 1 0 0 0 36

S2 2 1 0 1 0 0 40

S3 −1 3 0 0 1 0 2

S4 0 3 0 0 0 1 3

(A) S1

(B) S2

(C) S3

(D) S4

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Q.28 For an ideal Diesel cycle, the heat addition is an

(A) isobaric process

(B) isothermal process

(C) isochoric process

(D) isentropic process

Q.29 Three principal stresses at a point in a material are 300 MPa, 250 MPa and 100
MPa. If the yielding just starts at that point, the yield strength (in MPa) of the
material as per Tresca criterion is________. (Answer in integer)

Q.30 In an orthogonal straight turning process, the feed is 0.1 mm/rev and the depth of
cut is 0.5 mm. In ASA system, the side cutting edge angle of the cutting tool is 0.
The width (in mm) of the chip is________. (Rounded off to one decimal place)

Q.31 The pitch of the single-start lead screw of a lathe is 6 mm. It is used to cut double
start thread of 3 mm pitch on a cylindrical work piece. During the thread cutting,
the spindle rotates at 400 revolutions per minute (RPM). The speed (in RPM) of the
lead screw is________. (Answer in integer)

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Q.32 Two options are available to meet the annual demand of batteries in a toy company.
In option 1, batteries are manufactured in the plant having fixed cost of Rupees
2,00,000 and a variable cost of Rupees 70 per unit. Option 2 consists of buying
batteries from the market at a price of Rupees 90 per unit. The annual demand (in
number of batteries) at which the company should switch from buying to making
the batteries in the plant is________. (Answer in integer)

Q.33 A company estimates the demand of 2000 bulbs for the next year. The ordering cost
is Rupees 300 per order and the annual carrying cost per bulb is Rupees 30. The
economic order quantity (number of bulbs) is________. (Answer in integer)

Q.34 While inspecting final assembly of automobile-gear-boxes, 15 features were
considered critical-to-quality (CTQ). During last quarter, 40000 gear boxes were
produced among which 1500 defects were found of the CTQ features. The defects
per million opportunities (DPMO) is________. (Answer in integer)

Q.35 The hole and the shaft dimensions (in mm) are given as

Hole dimension = 30+0.04
−0.02

Shaft dimension = 30 +0.06
+0.03

The maximum possible clearance (in mm) is________. (Rounded off to two decimal
places)

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Q.36 – Q.65 Carry TWO marks Each

Q.36 The solution of the linear differential equation
dy
+ y = ex ,
dx
when y ( 0 ) = 0 , is

(A) 1 1
y = e x − e− x
2 2

(B) 1 1
y = e x + e− x
2 2

(C) y = e x − e− x

(D) y = e x + e− x

Q.37 Which one of the following functions is analytic, given i = −1 ?

(A) e x ( cos y + i sin y )

(B) e x ( cos y − i sin y )

(C) e x ( − cos y + i sin y )

(D) e− x ( cos y + i sin y )

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Q.38 Match the following with reference to the machining process and its feature.

Process Feature

P EDM 1 Loss of dimensional accuracy due to under cutting

Q LBM 2 Cutting edible items

R CHM 3 Machining of a deep square blind hole on a mild steel
plate

S WJM 4 High speed profile cutting on a thin mild steel plate

(A) P – 3, Q – 4, R – 1, S – 2

(B) P – 4, Q – 3, R – 2, S – 1

(C) P – 4, Q – 2, R – 1, S – 3

(D) P – 1, Q – 2, R – 3, S – 4

Organizing Institute: IIT Roorkee Page 24 of 35

Q.39 Match the operation/phenomenon in a grinding process with the corresponding
definition listed in the table.

Operation/phenomenon Definition

P Loading 1 Regenerating the sharpness of the
grinding wheel

Q Glazing 2 Filling of grinding chips in the space
between the abrasive grits

R Dressing 3 Restoring the geometry/shape of the
grinding wheel

S Truing 4 Condition of dull grinding wheel with
worn-out grains

(A) P – 4, Q – 1, R – 3, S – 2

(B) P – 2, Q – 3, R – 1, S – 4

(C) P – 2, Q – 4, R – 1, S – 3

(D) P – 4, Q – 3, R – 1, S – 2

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Q.40 A CNC vertical milling is used for cutting a straight line slot in x-y plane. Cutter is
 dy 
located at point P. The slope   of the straight line created by the cutter is 1.25.
 dx 
Feed rate of x-axis is 120 mm/min. The new position of the cutter after 20 seconds
is

(A) (50, 55)

(B) (60, 50)

(C) (60, 48)

(D) (100, 80)

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Q.41 The network diagram of eight activities (A to H) along with their time durations (in
days, given in bracket) of a project is shown in the figure. The critical path of the
project is

(A) 1−2−3−6

(B) 1−4−3−6

(C) 1−5−6

(D) 1−4−5−6

Q.42 The benefit(s) of product standardization is/are

(A) Need of less number of drawings

(B) Reduction in unit cost

(C) Reduction in inventory cost

(D) Greater product variety

Organizing Institute: IIT Roorkee Page 27 of 35

3
Q.43
The value of the integral  ( x 2 − 2 x ) dx obtained by using Simpson’s 1/3 rule with
1
n
4 subintervals is equal to . The value of n is________. (Answer in integer)
3

Q.44 If i , j and k are the orthogonal unit vectors in Cartesian x-y-z coordinate system,
the rate of change of the function f ( x, y, z ) = x 2 + 2 y 2 + z at point (1, 1, 1) in the
direction of 3 i + 4 k is________. (Answer in integer)

Q.45 In a wire drawing of a perfectly-plastic material with flow stress of 300 MPa, the
back tension is zero and front tension is 200 MPa. Assuming ideal deformation with
zero friction, the percentage reduction of the cross-sectional area of the wire
is________. (Rounded off to one decimal place)

Q.46 In a cold rolling process without front and back tensions, the required minimum
coefficient of friction is 0.04. Assume large rolls. If the draft is doubled and roll
diameters are halved, then the required minimum coefficient of friction is________.
(Rounded off to two decimal places)

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Q.47 In a direct current arc welding, the voltage V (in volt) is related to the arc length
l (in cm) as

V = ( 30 + 30 l ) .

The open circuit voltage is 80 volts. The maximum possible arc length (in cm)
is________. (Rounded off to two decimal places)

Q.48 A worker is allowed half an hour personal time in a normal 8-hour shift. If the
normal time for manufacturing a product is 5 minutes, the standard time (in second)
is________. (Answer in integer)

Q.49 A product has to be manufactured in a single-line layout by carrying out the six
tasks in a sequence. The time (in minute) of the six sequential tasks are 37, 8, 19,
34, 36 and 17. These tasks cannot be further sub-divided. For minimizing the cycle
time, the number of stations to be used is________. (Answer in integer)

Q.50 A through hole of 10 mm diameter is to be drilled in a mild steel plate of 30 mm
thickness. The selected spindle speed and feed for drilling hole are 600 revolutions
per minute (RPM) and 0.3 mm/rev, respectively. Take initial approach and
breakthrough distances as 3 mm each. The total time (in minute) for drilling one
hole is________. (Rounded off to two decimal places)

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Q.51 In the iron-carbon equilibrium phase diagram, the eutectoid reaction occurs at
723 C with the eutectoid composition of 0.83 weight % carbon. Ferrite and
cementite phases are considered to contain 0.022 weight % carbon and 6.67 weight
% carbon, respectively. If a steel specimen with 0.7 weight % carbon is cooled from
950 C to below 723 C, the fraction of eutectoid ferrite is________. (Rounded off
to two decimal places)

Q.52 During orthogonal cutting with a tool of 10 rake angle, the cutting and thrust forces
are 900 N and 275 N, respectively. The coefficient of friction on the rake surface of
the tool is________. (Rounded off to two decimal places)

Q.53 In casting a cube of 80 mm side, the volumetric shrinkages due to solidification and
solid contraction are 4.5% and 2%, respectively. Assume uniform cooling in all
directions. The side (in mm) of the cubical pattern for getting the required size
casting is________. (Rounded off to two decimal places)

Q.54 The solidification of a casting starts at 10 AM. However, the solidification of the
molten metal at the center-line of the mold starts at 10:03 AM and ends at
10:10 AM. The casting is considered solidified completely when the solidification
is completed at the center-line of the mold. The center-line feeding resistance (CFR)
in percentage is________. (Answer in integer)

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Q.55 A link OA of length 200 mm is rotating counter clockwise about O in x-y plane with
a constant angular velocity of 100 rad/s, as shown in the figure. The absolute value
of the x-component of the linear velocity (in m/s) of point A at the instant shown in
the figure is________. (Rounded off to one decimal place)

Q.56 A force of 1000 N is acting at point A on a bracket fixed at point B as shown in the
figure. The magnitude of the moment of the force about B (in Nm) is________.
(Rounded off to one decimal place)

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Q.57 A steel plate is fastened to a channel using three identical bolts as shown in the
figure. The bolts are made of carbon steel of permissible yield strength in shear as
400 N/mm2. The plate is subjected to a force of 12 kN. Neglect the weight of the
plate. The magnitude of the resultant shear force (in N) on bolt 2 is________.
(Answer in integer)

Q.58 The annual profit of a company depends on its annual marketing expenditure. The
information of preceding 3 years’ annual profit and marketing expenditure is given
in the table. Based on linear regression, the estimated profit (in units) of the 4th year
at a marketing expenditure of 5 units is__________. (Rounded off to two decimal
places)

Year Expenditure for marketing (units) Annual profit (units)

1 3 22

2 4 27

3 6 36

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Q.59 Three plants P1, P2 and P3 produce 6, 1 and 9 thousand liters of fruit juice,
respectively. The produced fruit juice is transported to three distribution centers D1,
D2 and D3 with requirement of 7, 5 and 4 thousand liters of juice, respectively. The
transportation cost (in hundreds of Rupees) from each plant to each distribution
center is given in the table. The total transportation cost (in hundreds of Rupees) in
initial basic feasible solution using Vogel’s approximation method is________.
(Answer in integer)

D1 D2 D3 Supply

P1 2 3 11 6

P2 1 0 6 1

P3 5 8 15 9

Demand 7 5 4

Q.60 A company purchases items in bulk for getting quantity discount in the item’s price.
The price break-up is given in the table. The annual demand for the item is
5000 units. The ordering cost is Rupees 400 per order. The annual inventory
carrying cost is 30 percent of the purchase price per unit. The optimal order size (in
units) is________. (Answer in integer)

Quantity of item ( Q in units) Unit price of item
(Rupees)

0  Q  1200 10

1200  Q  2000 8

2000  Q 7

Organizing Institute: IIT Roorkee Page 33 of 35

Q.61 The zero line of the Vernier scale lies between divisions 20 and 21 of the main scale.
The 4th Vernier scale division exactly coincides with a main scale division. The
5 divisions of the Vernier scale are equal to 4 divisions of the main scale. If one
main scale division is 1 mm, the measured value (in mm) is __________. (Rounded
off to one decimal place)

Q.62 A broaching machine makes key slots with a mean dimension of 10.56 mm and a
standard deviation of 0.05 mm. The upper control limit for mean of sample size 5
calculated using X-bar ( X ) chart is __________. (Rounded off to two decimal
places)

Q.63 The table shows the data of running a machine for five years. The original machine
cost is Rupees 70,000. In order to minimize the average total cost per year for
running the machine, the machine should be replaced after________ years. (Answer
in integer)

1st year 2nd year 3rd year 4th year 5th year

Resale value 40000 30000 25000 22000 20000
(Rupees)

Maintenance 19100 20300 23500 30500 40000
cost
(Rupees)

Organizing Institute: IIT Roorkee Page 34 of 35

Q.64 Water flows through a smooth circular pipe of diameter 10 cm and length 10 m. The
pressure drop across the length of the pipe is 0.2 Pa. Kinematic viscosity and density
of water are 1×10−6 m2/s and 1000 kg/m3, respectively. Assuming laminar and fully
developed flow throughout the pipe, the velocity of water (in mm/s) at the center of
the pipe is __________. (Rounded off to one decimal place)

Q.65 The left-hand side of a 20 cm thick wall is maintained at 25 C. The right-hand side
of the wall is exposed to hot air at 50 C. There is no heat generation inside the wall
and its thermal conductivity is 100 W/mK. The convective heat transfer coefficient
is 50 W/m2K. Under steady state condition, the temperature (in C) of the right-
hand side surface of the wall is________. (Rounded off to one decimal place)

Organizing Institute: IIT Roorkee Page 35 of 35

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