GATE 2026 Question Paper Production and Industrial Engineering (PI)

FOR GATE EXAM PREPARATION

GATE 2026
Question Paper ·
Production Industrial
Engineering (PI)
EXAM YEAR TYPE SUBJECT

GATE 2026 Question Paper Production Industrial Engineering (PI)

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Production & Industrial Engineering (PI)

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General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each

Q.1 ‘The shopkeeper sells lemons.’

In this sentence, the word ‘lemons’ is the ________.
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(A)
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(B) subject

(C) predicate

(D) verb
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Production & Industrial Engineering (PI)

Q.2 The figure below is supposed to show three non-overlapping shapes – one oval
and two triangles. Which one of the following figures P, Q, R, or S fits the
missing portion indicated by ‘?’ and completes the oval and the two triangles?

P Q

R S

(A) P

(B) Q

(C) R

(D) S

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Production & Industrial Engineering (PI)

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Q.3 At how many points will the curves 𝑦 = 𝑥 2 and 𝑦 = −𝑥 2 − 2𝑥 − 1 intersect in
the real (𝑥, 𝑦) plane?

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(A) 0

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(B)
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(C) a 2

(D) 3

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Production & Industrial Engineering (PI)

Q.4 ‘If Anish had scored hundred runs in today’s match, he would have been made the
captain of his team. He would have then become the youngest captain in his
team’s history. Unfortunately, he got out without scoring any runs. Hence, there
won’t be any change in the captaincy for now.’

Based on the paragraph above, which one of the following statements is true?

(A) Anish made hundred runs but was denied captaincy.

(B) Anish was the captain of his team before the game today.

(C) The current captain is older than Anish.

(D) Anish is the youngest player in his team.

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Q.5 Which one of the following figures P, Q, R, or S, correctly shows the 45°
clockwise-rotated version of figure (I)?

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(I)
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P Q

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R S

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(A) P
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c. o(B) s e m
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(D) S

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Q.6 – Q.10 Carry TWO marks Each

Q.6 Match the words in Column I with their synonyms in Column II.

Column I Column II
(i) Lonely (p) Verbatim
(ii) Literal (q) Solitary
(iii) Lousy (r) Deadly
(iv) Lethal (s) Terrible

(A) (i)-(q); (ii)-(p); (iii)-(s); (iv)-(r)

(B) (i)-(q); (ii)-(s); (iii)-(r); (iv)-(p)

(C) (i)-(s); (ii)-(p); (iii)-(q); (iv)-(r)

(D) (i)-(r); (ii)-(s); (iii)-(p); (iv)-(q)

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Q.7 ̅̅̅̅ is the diameter of a circle with center 𝑂. Two points 𝑅
In the given figure, 𝑃𝑄
and 𝑆 are chosen on the circle such that ∠𝑅𝑂𝑆 = 80°. When ̅̅̅̅ 𝑃𝑅 and ̅̅̅̅
𝑄𝑆 are
extended, they meet at 𝑇. The value of ∠𝑅𝑇𝑆 is _________________

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(A) 40°

(B) 50°

(C) 60°

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(D) 80°
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Q.8 Based on the relationship between each polygon and the number inside it, the
value of ‘X’ is ______

(A) 720

(B) 596

(C) 24

(D) 240

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Q.9 Consider a linear arrangement of seven bulbs, each of which can be in the ON or
OFF states. The initial configuration of the bulbs is shown in the figure. In every
Step, the states of the bulbs are changed based on the following rules:

 Any OFF bulb with exactly one ON neighbor at the end of the

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previous Step is turned ON.

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
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Any ON bulb with both neighbors ON at the end of the previous

 . c o Step is turned OFF.
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The state of any bulb not meeting the conditions above is left
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l as unchanged.
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agThe state of bulbs at the end of Step 1 and Step 2 are also shown in the figure.
The number of bulbs which are ON at the end of Step 8 is ______

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(A) 5

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(B) 4

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Q.10 𝑃 and 𝑄 are two positive integers such that 𝑃2 = 𝑄 2 + 13.

The product of the numbers 𝑃 and 𝑄 is _________

(A) 13

(B) 26

(C) 39

(D) 42

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Q.11 – Q.35 Carry ONE mark Each

Q.11 Which ONE of the following steps is NOT a part of product concept selection?

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(A)
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Preparation of the selection matrix
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(B) a Ranking of the concepts

(C) Combination and improvisation of the concept

(D) Development of a cost model of the product

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Q.12 Which ONE of the following is FALSE with respect to filler material used in
brazing?
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(A) Melting temperature of the filler material must be less than the base metal.

(B) Wettability of the base metal by the filler material must be low.
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c. o(C) Viscosity of the filler material must be low for penetration into them
s e interface.

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(D) Chemical reactivity between the filler material and the base
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Q13. An arc welding process is being carried out with a power source of 6 kW to weld
two similar metals. The total energy loss is 50%, the area of cross-section of the
weld is 10 mm2, and the specific energy needed to melt the metal is 15 J/mm3.

The speed of welding is ______ mm/s.

(A) 10

(B) 20

(C) 30

(D) 40

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Q.14 Which ONE of the following options represents correctly ALL the constraints
resulting in the feasible region for the given linear programming problem as shown
in the figure?

Note: Figure not to scale

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(A) 𝑥1 ≥ 0; 𝑥2 ≥ 0; 𝑥2 ≥ 3; 𝑥1 − 𝑥2 ≤ 1; 𝑥1 − 0.5𝑥2 ≥ 1

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(B) 𝑥 ≥ 0; 𝑥 ≥ 0; 𝑥 ≤ 3; 𝑥 − 𝑥 ≤ 1; 𝑥 + 0.5𝑥 ≥ 1
1 2 2 1 2 1 2

(C) 𝑥1 ≥ 0; 𝑥2 ≥ 0; 𝑥2 ≤ 4; 𝑥1 − 𝑥2 ≤ 1; 𝑥1 + 0.5𝑥2 ≥ 1

(D) 𝑥1 ≥ 0; 𝑥2 ≥ 0; 𝑥2 ≤ 3; 𝑥1 + 𝑥2 ≤ 1; 𝑥1 + 0.5𝑥2 ≥ 1

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Q. 15 Which ONE of the following options is the closest approximation to the value of
the given definite integral?
1
2
∫ 𝑒 −𝑥 𝑑𝑥
0

Note: Use numerical integration

(A) 1

(B) 1
𝑒

(C) 𝑒+1
2𝑒

(D) 1+𝑒
2

Q. 16 1
Which ONE of the following is the value of 𝑛 ?
𝑖

where 𝑖 = √−1 , and 𝑛 is an even positive integer.

(A) +1 or −1

(B) +𝑖 or −𝑖

(C) only +1

(D) only −𝑖

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Q.17 Which ONE of the following CORRECTLY matches the metrology instruments
and their respective most common applications?

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Metrology Instrument Application
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s e P 1 Surface roughness of milled planar
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Q Profilometer

R Vernier calliper 3 Checking shaft diameter tolerance
without direct measurement

Ring GO gauge 4
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S Length of any side of a cuboid

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

(B) P-4; Q-3; R-2; S-1
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P-4; Q-1; R-2; S-3
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Q.18 A plain carbon steel sample of eutectoid composition is heat treated from
austenizing temperature to achieve a microstructure consisting of ONLY bainite.

Which ONE of the following is the corresponding heat treatment method?

(A) Normalizing

(B) Annealing

(C) Martempering

(D) Austempering

Q.19 An Iron-Carbon alloy at room temperature has pro-eutectoid ferrite as a constituent
in the microstructure, that formed above the eutectoid temperature. The alloy also
has pearlite as another constituent that has formed below eutectoid temperature.

Which ONE of the following best describes this Iron-Carbon alloy?

(A) Hyper-eutectic steel

(B) Hyper-eutectoid steel

(C) Hypo-eutectoid steel

(D) Eutectoid steel

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Q.20 An engineering student is fabricating a setup and needs a key for connecting a shaft
to the hub of a rotating machine element. The student uses a sunk key which is
tapered and prevents relative axial motion.

Which ONE of the following keys did the student most likely use?

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(A) Saddle key

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(B) a Woodruff key

(C) Feather key

(D) Gib-head key

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Q.21
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The ratio of hydrostatic stress to volumetric strain is known as ______.
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(A) Bulk modulus

(B) Compressibility

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Poisson’s ratio
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Young’s modulus
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(D)

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Q.22 In the given weighted graph, the weights represent distance between the
corresponding vertices. The shortest distance between vertex A and vertex E is d.
The number of paths with distance d is p.

The value of d/p is ______.

(A) 5

(B) 3

(C) 2

(D) 8

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Q.23 Which ONE of the following is NOT an assumption of the economic order quantity
(EOQ) model?

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(A)
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Demand is known and deterministic.
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(B) Lead time is zero.

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(C) a Stockouts are permissible.

(D) Holding cost for the quantity in stock is linearly increasing with the order
quantity.

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Q.24
m.the free project.
Painting is one of the activities in a construction
that takes 6 days, both the total floateand
For the painting activity

l a s float are zero.

Which ONE of the followinggis TRUE for the painting activity?
a

(A) Painting is a dummy activity.

(B) Painting is a non-critical activity.
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The start of painting activity can be delayed without causing delay in completion
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a (D) The start of painting activity cannot be delayed without causing delay in
completion of the construction project.

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Which ONE or MORE among the following tube cross-sections shown can be
Q.25
produced by true centrifugal casting method?

Note: The cross-sections shown are perpendicular to the axis of rotation. Figures
are not to scale.

(A)

(B)

(C)

(D)

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Q.26 Which ONE or MORE among the following options CORRECTLY match(es) the
type of defects with the corresponding metal working processes in which they are
likely to occur?

Defect Process
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P 1 Deep drawing
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R Zipper cracks 3 Extrusion

S Cold shut 4 Closed die forging

5 Casting
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(A) P-3; Q-1; R-2; S-4

(B) P-5; Q-2; R-3; S-4

(C) P-1; Q-2; R-4; S-5

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(D) P-3; Q-1; R-2; S-5
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Q.27 Which ONE or MORE among the options given is/are well established format(s) of
data transfer used in geometrical modelling related to computer aided design
(CAD)?

(A) IGES

(B) STEP

(C) NDC

(D) STL

Q.28 In statistical analysis of numerical data, which ONE or MORE among the options
is/are possible value(s) of correlation coefficient?

(A) 0

(B) 10.3

(C) − 0.9

(D) 2

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Q.29 Which ONE or MORE among the following processes of the Carnot cycle is/are
isentropic?

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Note: Figure is not to scale

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(A) 1-2
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(B) 2-3

(C) 3-4

(D) 4-1

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a Q.30 A square of side length 50 mm is to be blanked from a strip of 2 mm thickness. The
sheet metal has a shear strength of 200 MPa.

To enable a single step blanking operation, the theoretical minimum blanking force
required is ______ × 103 N (in integer).

Note: Neglect the effect of clearances and friction.

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Q.31 A box in a machine shop consists of 5 coated and 10 uncoated cutting inserts which
are of otherwise similar characteristics. The inserts got mixed up randomly in the
box. An operator has taken 4 of them at once without noticing the differences to
mount on a 4-tooth face milling cutter that uses inserts.

The probability of the milling cutter having all uncoated inserts is ______ (rounded
off to two decimal places).

Q.32 A customer has made a lumpsum investment of INR 2,00,000 with a bank. The
discount rate is 15% per year. The investment enables the customer to receive a
payout of INR 1,00,000 yearly for next three consecutive years.

The net present value (NPV) of his/her investment is INR ______ (rounded off to
the nearest integer).

Q.33 A modified Taylor tool life equation is given as follows:

𝑉𝑇 𝑛 𝑓 𝑚 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

where 𝑉 is the cutting speed (m/s), 𝑇 is the tool life in minutes and 𝑓 is the feed in
mm/rev, 𝑛 = 0.25 and 𝑚 = 0.5. Under two different cutting conditions (𝑉1 , 𝑓1 ) and
(𝑉2 , 𝑓2 ) the tool life (𝑇1 , 𝑇2 ) was found to be the same.

If the ratio of the cutting speeds (𝑉1 /𝑉2) used is 2/3, then the ratio of corresponding
feeds (𝑓1 /𝑓2 ) must be ______ (rounded off to two decimal places).

Q. 34 Consider the line integral

∫ (2𝑥 𝑑𝑥 + 2𝑦 𝑑𝑦 + 2𝑧 𝑑𝑧)
𝐶

where 𝐶 is a semi-circle in the 𝑧 = 0 plane with start point at (0, 0, 0) and end point
at (1, 0, 0).

The value of the line integral is ______ (in integer).

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Q.35 Consider the aggregate planning problem for a toy car manufacturing organization
whose demand at a plant is as follows:

Month September October November December
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Demand (in units) 8000 6500 10500 9500

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The organization has ONLY the following options available for aggregate planning
meet the demand: (1) in-house production, (2) subcontracting, and (3) inventory
Assume that the quantity subcontracted in a given month is available in the
same month.

A part of their optimal solution is as follows:

Inventory at the end of October = 1500 units

Units produced in-house in November = 8000 units
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Units subcontracted in November = 1000 units
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The inventory at the end of November is ______ units (in integer).

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Q.36 – Q.65 Carry TWO marks Each

Q.36 In a polymer extrusion process, some cross-sectional shapes of the extruded
polymers are shown. Cross-sections of available dies are also shown.

Which ONE of the following CORRECTLY matches the extruded cross section
with the die opening that most likely generated it?

Note: The cross-sections are in a plane orthogonal to the extrusion direction.
Figures are not to scale.

(A) P-1; Q-4

(B) P-1; Q-2

(C) P-3; Q-2

(D) P-3; Q-4

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Q.37 Iron powder produced by gas atomization has a tap density and an apparent density
of 3.26 g/cm3 and 2.66 g/cm3, respectively. The density of solid iron is 7.85 g/cm3.

Which ONE of the following options is the closest approximation to the degree of
densification (in percentage) of this iron powder?

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(A) 41.52

g l a
(B) a 81.59

(C) 33.89

(D) 75.41

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Q.38
Which ONE of the following options CORRECTLY matches the matrices to their
properties?

Matrix Property

−30 17 5
P [ 17 0 −12] 1 Singular
5 −12 4

0 1 9
Q [7 0 2] 2 Triangular
12 3 0

2/3 1/3 2/3
R [ 0 2/3 −1/3] 3 Symmetric
0 4/3 −2/3

1 2 3
S [0 1 4 ] 4 Trace free
0 0 1

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

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

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

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

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Q.39 A sheet metal drawing is considered feasible if the following conditions are met:

 Reduction ratio < 0.5
 Thickness to diameter ratio > 1%

A sheet metal drawing process is being planned on a sheet metal with a thickness
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of 2 mm with a punch of diameter 100 mm with various blanks of diameter 𝐷𝑏 .
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Which ONE of the following blank diameters 𝐷𝑏 (in mm) will result in feasible
m s e m
e
drawing?
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(A) 150

(B) 250

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(C) 350

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(D) 450
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Q. 40 A student is trying to determine the flatness and parallelism of the top surface of a
part whose cross-section, assumed uniform in the Z direction, is shown in the figure.

Which ONE of the following options is CORRECT?

(A) 𝑎 is flatness while 𝑏 is parallelism.

(B) 𝑎 is parallelism while 𝑏 is flatness.

(C) 𝑎 + 𝑏 is parallelism while 𝑏 − 𝑎 is flatness.

(D) 𝑎 + 𝑏 is flatness while 𝑏 − 𝑎 is parallelism.

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Q.41. The function, 𝑟(𝑡), signifies the likelihood that a component has survived up to time
𝑡 and fails in the next instant of time.

𝑟(𝑡) = 𝑓(𝑡)/(1 − 𝐹(𝑡))

where 𝑓(𝑡) denotes the probability density function for time to failure with
m
.co
corresponding cumulative distribution function given by,
m
.co e m
𝑏
𝐹(𝑡) = 1 − 𝑒 −𝑎𝑡 , 𝑎 > 0, 𝑏 > 0
e m l as
as
Which ONE of the following represents 𝑟(𝑡)?
l ag
ag

𝑏
(A) 𝑎 × 𝑏 × 𝑡 𝑏−1 × 𝑒 −𝑎𝑡

(B) 𝑎 × 𝑏 × 𝑡 𝑏−1

m
(C) 1 − (𝑎 × 𝑏 × 𝑡 𝑏−1 )
m .co
s e
1 − (𝑎 × 𝑏 × 𝑡 𝑏−1 × 𝑒 −𝑎𝑡
𝑏
) gl
a
a
(D)

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.42 Which ONE of the following options CORRECTLY matches the machining
process and associated predominant material removal mechanism?

Predominant material removal
Advanced machining process
mechanism

P Laser beam machining 1 Impact based erosion

Q Abrasive waterjet machining 2 Sputtering

R Ion beam machining 3 Melting and vaporization

S Mechanical milling 4 Shearing

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

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

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

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

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Q. 43 A glass tube of 2.5 mm diameter is immersed vertically in a fluid.

Assume contact angle is zero. The surface tension of the fluid is 0.1 N/m, density
of the fluid is 1000 kg/m3 and acceleration due to gravity is 10 m/s2.

The approximate capillary rise is ______ mm.
m
m .co
m .co s e m
e l a
2 as ag
(A)
g l
a
(B) 8

(C) 16

(D) 32
m
.co
s em
lacustomers arrive at a rate of 3 per minute, and the
Q. 44 In an M/M/1 queuing system,gthe
a in the system is 𝑛 at steady state. Assume that the
average number of customers
server utilization is less than 1.

Which ONE of the following is the average time a customer spends in the system
(in minutes) at steady state?

m
m
c. o(A) 3𝑛
m.co
m s e
s e g l a
la
𝑛⁄
g
(B) 3 a
a
(C) 𝑛⁄
6

(D) 3⁄
𝑛

m .
.co s e m
s em l a
la
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Q.45 A steel cube of side 10 cm is made by the sand-casting process. A cylindrical side-
riser with diameter, 𝑑, and height, ℎ, needs to be used.

Assume:

 No surface sharing between the riser and casting.
 𝑑 = ℎ.
 The connecting link between the riser and casting does not freeze before the
casting.
 All the surfaces of the riser and casting are subjected to identical cooling
conditions.

In this situation, which ONE or MORE among the following values of d (in cm)
can theoretically fully compensate for shrinkage during casting?

(A) 5

(B) 8

(C) 15

(D) 20

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Q.46 In a rolling operation, a 200 mm wide strip of 23 mm thickness needs to be reduced
to 20 mm in a single pass. The roll diameter is 200 mm. Four lubricants P, Q, R,
and S with coefficient of friction 0.05, 0.1, 0.2, and 0.25, respectively, are available
for use at the roll-strip interface. Assume that the strip width remains constant
throughout the process.

m
.co
Which ONE or MORE among the following lubricants will enable the rolling
m
.co
process to achieve the desired final thickness of the strip in a single pass?
e m
e m l as
l as ag
g
aP
(A)

(B) Q

(C) R

m
(D) S
m .co
s e
g la
a

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q. 47 In the theory of orthogonal machining, assuming that the shear plane will take up
an angle that minimizes energy, the shear angle can be shown to be

𝛼 𝛽
𝜙=𝐶− +
2 2
where 𝜙 is the shear angle, 𝛼 the rake angle, and 𝛽 the friction angle.

Which ONE or MORE among the following is/are TRUE?

(A) 𝐶 is constant if the shear stress on the shear plane is independent of the normal
stress.

(B) 𝐶 is constant if the shear stress on the shear plane is dependent on the normal
stress.

(C) 𝐶 depends on material properties if the shear stress on the shear plane is
independent of the normal stress.

(D) 𝐶 depends on material properties if the shear stress on the shear plane is linearly
dependent on the normal stress.

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Q.48 Which ONE or MORE among the following homogeneous representations in
robotics involve(s) rotation about Y-axis by an angle 𝜃?

m
m .co
.co
(A) cos 𝜃 − sin 𝜃 0 0
sin 𝜃 cos 𝜃 0 0
e m
em
(
0 0 1 0
)
l as
las 0 0 0 1
ag
g
a 1 0
(B) 0 0
0 cos 𝜃 − sin 𝜃 0
( )
0 sin 𝜃 cos 𝜃 0
0 0 0 1

(C) cos 𝜃 0 sin 𝜃 0
0 1 0 0
( )
m
.co
− sin 𝜃 0 cos 𝜃 0
0 0 0 1
e m
las
(D)
(
cos 𝜃
0
0 sin 𝜃 3
1 0 4 a
)
g
− sin 𝜃 0 cos 𝜃 2
0 0 0 1

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.49 Which ONE or MORE among the following options is/are TRUE for the ideal
operating characteristics (OC) curve given in the figure?

Note: 𝑝0 represents the acceptable quality level (AQL) and 𝑝1 represents the lot
tolerance percent defective (LTPD).

(A) Producer’s risk is 100%.

(B) Consumer’s risk is 100%.

(C) Producer’s risk is zero.

(D) Consumer’s risk is zero.

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Q.50 The operation chart of a method of assembling nuts and bolts for a left-handed
operator has been illustrated in the figure.

Which ONE or MORE among the following options is/are CORRECT?

m
m .co
m .co s e m
s e l a
g l a ag
a

m
m .co
s e
g la
a

m
m .co
(A) The work distribution between the two hands is not balanced.

m .co s e m
s e (B) The two hands do not follow opposite motions.
g l a
g la a
a (C) The work is well distributed between the two hands.

(D) The dominant hand of the operator performs majority of the tasks.

m .
.co s e m
s em l a
la
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Q. 51 Which ONE or MORE among the following work materials is/are
NOT commonly machined using diamond cutting tools?

(A) Mild steel

(B) Stainless steel

(C) Silicon

(D) Aluminium alloy

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A linear slot is to be milled in a single pass from the point (0, 0) to
Q.52
(180, 180) on XY plane by a CNC machine. The actual velocity along the Y-axis is
5% less than the intended value. The programmed feed rate along the intended slot
is 150 mm/min.

The magnitude of the positional error along the Y-axis when the X-coordinate
m
.co
reaches 180 mm, is ______ mm (in integer).
m
.co
Note: All coordinates are in mm.
m s e m
In a s e g l a
l a
laser beam machining, the time (𝑡 ) required for the material to attain the melting
𝑚

agexpression,
Q.53
temperature from a room temperature (𝜃 ) of 32 ℃ is expressed
𝑜 by the following

𝜋 (𝜃𝑚 − 𝜃𝑜 ) 𝑘 2
𝑡𝑚 = ( )
𝛼 2𝐻

where 𝛼 is thermal diffusivity, 𝜃𝑚 is melting temperature, 𝑘 is thermal conductivity,
𝐻 is heat flux.

m
If a uniformly distributed 1 kW power laser beam with a beam diameter of 0.1 mm

.co
is used for machining tungsten carbide, and 10% of beam absorption is assumed,
the time 𝑡𝑚 is ______ µs (rounded off to one decimal place).
e m
s
laW/cm- ℃;
Note: Thermal properties of tungsten carbide: melting temperature = 3400 ℃;
ag
thermal conductivity = 2.15 diffusivity = 0.79 cm 𝑠 ; assume 2 −1

𝜋 = 3.14.

Q.54 In the measurement of surface roughness using a 2D stylus profilometer, a surface
profile measured over a length of 0.8 mm was recorded on a graph paper. During
recording, vertical magnification of 10,000 and horizontal magnification of 100
m
.co
were used. The areas in the as-recorded graph above and below the datum line are
m
.co
as follows:
e m
e m Above (mm2): 140, 60, 150, 50
l as
las Below (mm2): 70, 50, 120, 160
ag
ag The average surface roughness (𝑅𝑎 ) of the surface is ______ µm (rounded off to
two decimal places).

m .
.co s e m
s em l a
la
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Q.55 A cylindrical metal component of 10 mm diameter is subjected to uniform uniaxial
tension during operation. It was observed that a force of 14 kN produces a uniform
reduction of 3 × 10−3 mm in diameter. Assume that material behaviour is
homogeneous, isotropic, and linear elastic.

If its Young’s modulus is 150 GPa, the Poisson’s ratio of the material is ______
(rounded off to two decimal places).

Note: Assume 𝜋 = 3.14.

Q.56 A custom pinion, with a width equal to 10 times the module, has 20 full depth teeth
and a pressure angle of 20⁰ is being designed. It should transmit a torque of
95 N-m to a corresponding spur gear.

Considering the safe bending stress to be 180 MPa and form factor to be 0.342, the
module of the pinion is ______ mm (rounded off to one decimal place).

Note: The Lewis equation gives the tangential force as 𝐹𝑡 = 𝜎𝑏𝑌𝑚, where 𝜎 is the
safe bending stress, 𝑏 the width, 𝑌 the form factor, and 𝑚 the module.

Q.57 A company has a fixed cost of INR 3,00,000 and a variable cost of
INR 150 per unit for manufacturing of a product. The company sells 5,000 units of
that product making a profit equivalent to 20 % of the total sales revenue.

The break-even quantity for that product is _____ units (rounded off to the nearest
integer).

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Q 58 A company must allocate oil produced from their two plants to meet all the demands
of two markets. The cost per litre of allocation from plant 𝑖 ∈ {1,2} to market
𝑗 ∈ {1,2} is denoted by 𝐶𝑖𝑗 . The market demand 𝐷𝑗 , plant production capacity
𝐾𝑖 and 𝐶𝑖𝑗 values are given in the table.

m
.co
Market 1 (𝑗 = 1) Market 2 (𝑗 = 2) Capacity 𝐾𝑖
m
.co m
in litres

s e
s em1 (𝑖 = 1) l a
g la
Plant 𝐶11 = 250 𝐶12 = 280 𝐾 = 500
1
ag
a Plant 2 (𝑖 = 2) 𝐶21 = 150 𝐶22 = 180 𝐾2 = 600

Demand 𝐷𝑗 in litres 𝐷1 = 300 𝐷2 = 400

The company hired an intern to formulate an optimization model to decide on the
quantity (𝑋𝑖𝑗 ) to be allocated from plant 𝑖 to market 𝑗 and the formulation is given
m
.co
below:

Minimize 𝑍 = ∑2𝑖=1 ∑2𝑗=1 𝐶𝑖𝑗 𝑋𝑖𝑗 e m
las
Subject to
ag
∑2𝑗=1 𝑋𝑖𝑗 ≤ 𝐾𝑖 ∀ 𝑖 ∈ {1,2}

∑2𝑖=1 𝑋𝑖𝑗 ≤ 𝐷𝑗 ∀ 𝑗 ∈ {1,2}

𝑋𝑖𝑗 ≥ 0

The optimal value of the objective function of the linear programming problem
m
.co
formulated by the intern is ______ (in integer).
m
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q 59 The computer centre (CC) in an institute manages the central server for which
researchers queue to run their computer programs. CC processes programs only one
at a time. Four computer programs are available at 10:00 am on a given day. The
time required to run each program on the central server and their respective
promised time of completion (i.e., due date) on that same day are given in the table.

If the CC follows shortest processing time (SPT) rule to sequence the computer
programs, then number of tardy jobs is ______.

Note: Ignore program change-over times.

Program 1 2 3 4

Processing time (in minutes) 60 40 90 100

Promised time (i.e., due date) 12 noon 11:30 am 1:00 pm 1:30 pm

Q 60 𝑑𝑦
If = 2𝑦, and the value of 𝑦 at 𝑡 = 0 is 2, then the value of 𝑦 at 𝑡 = 1 is ______
𝑑𝑡
(rounded off to two decimal places).

Q 61 The number of soldering defects that occur in a semiconductor device follows a
discrete Poisson distribution with a probability mass function
𝑒 −𝜆 𝜆𝑥
𝑝(𝑥) = .
𝑥!

The average number of soldering defects per semiconductor device is 3.

The probability that a randomly selected semiconductor device will have at least
two soldering defects is ______ (rounded off to two decimal places).

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Q.62 A time study was carried out for a job broken into two elements. For each element,
three iterations of task were observed, and observed time was recorded. The
observed times (in seconds) and the rating factor for each element are given in the
table.

Observed time
m
.co
Work element Rating factor
m (seconds)
c. o A s e m
e m 20; 25; 21 105%
l a
l as ag
ag B 19; 17; 18 90%

If a further 20% needs to be added for allowances, then the standard output of the
worker is ______ units per hour (in integer).

Q. 63 In a hole-shaft assembly, the following are the dimensions and tolerances (all in
mm) provided for the hole and shaft:
m
Hole: 22+0.018
+0.010
.co
Shaft: 22+0.075 s em
+0.004

g a
lwhen
a
The magnitude of the clearance the shaft is at the minimum material condition
and hole is at the maximum material condition is ______ mm (to three decimal
places).

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.64 Consider an assembly line producing an item through a process involving three
activities in the following order: cutting, welding, and polishing. There are three
workers available: one is assigned only for cutting, one is assigned only for welding,
and one is assigned only for polishing.

The activity time per item is given in the table.

Assuming that the process is running in steady state, the total number of items
coming out of the assembly line in six hours is ______ (in integer).

Note: Assume that workers, items, and machines/tools are available as and when
needed.

Activity name Cutting Welding Polishing

Activity time/item (in minutes) 30 40 25

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Q.65 In the composite filament winding process as shown in the figure, the mandrel
diameter is 700 mm and is rotating at a speed, 𝑁 = 6 rev/min.

If a 45° helical winding angle is needed, the axial velocity, 𝑣𝑐 , of the slider should
be ______ m/s (rounded off to two decimal places).

m
.co
Assume 𝜋 = 22/7. Ignore the thickness of the composite layers already wound on
m
.co
the mandrel.
e m
e m l as
l as ag
ag

m
.co
s em
g la
aNote: Figure is not to scale

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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