GATE 2021 Question Paper XE A Engineering Sciences - Engineering Mathematics

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

Q.1 Gauri said that she can play the keyboard __________ her sister.

(A) as well as

(B) as better as

(C) as nicest as

(D) as worse as

Q.2

A transparent square sheet shown above is folded along the dotted line. The
folded sheet will look like ________.

(A)

(B)

(C)

(D)

XE(A) - Copyright © GATE 2021 Page 1 of 11

Q.3 If  is the angle, in degrees, between the longest diagonal of the cube and
any one of the edges of the cube, then, cos  =

(A) 1
2
(B) 1
√3

(C) 1
√2

(D) √3
2

Q.4 2 2
 1  3
If  x −  −  x −  = x + 2 , then the value of x is:
 2  2

(A) 2

(B) 4

(C) 6

(D) 8

Q.5 Pen : Write :: Knife : _________
Which one of the following options maintains a similar logical relation in the
above?

(A) Vegetables

(B) Sharp

(C) Cut

(D) Blunt

XE(A) - Copyright © GATE 2021 Page 2 of 11

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

Q.6 Listening to music during exercise improves exercise performance and
reduces discomfort. Scientists researched whether listening to music while
studying can help students learn better and the results were inconclusive.
Students who needed external stimulation for studying fared worse while
students who did not need any external stimulation benefited from music.
Which one of the following statements is the CORRECT inference of the
above passage?

(A) Listening to music has no effect on learning and a positive effect on physical
exercise.

(B) Listening to music has a clear positive effect both on physical exercise and on
learning.

(C) Listening to music has a clear positive effect on physical exercise. Music has a
positive effect on learning only in some students.

(D) Listening to music has a clear positive effect on learning in all students. Music
has a positive effect only in some students who exercise.

XE(A) - Copyright © GATE 2021 Page 3 of 11

Q.7

A jigsaw puzzle has 2 pieces. One of the pieces is shown above. Which one
of the given options for the missing piece when assembled will form a
rectangle? The piece can be moved, rotated or flipped to assemble with the
above piece.

(A)

(B)

(C)

(D)

XE(A) - Copyright © GATE 2021 Page 4 of 11

Q.8 The number of students in three classes is in the ratio 3:13:6. If 18 students
are added to each class, the ratio changes to 15:35:21.
The total number of students in all the three classes in the beginning was:

(A) 22

(B) 66

(C) 88

(D) 110

Q.9 350
296 300
300
240
250
200 210
200

150
100
100

50

0
Year 1 Year 2 Year 3
Number of units (`)
Net Profit (INR)

The number of units of a product sold in three different years and the respective
net profits are presented in the figure above. The cost/unit in Year 3 was ` 1, which
was half the cost/unit in Year 2. The cost/unit in Year 3 was one-third of the
cost/unit in Year 1. Taxes were paid on the selling price at 10%, 13% and 15%
respectively for the three years. Net profit is calculated as the difference between
the selling price and the sum of cost and taxes paid in that year.
The ratio of the selling price in Year 2 to the selling price in Year 3 is ________.

A) 4:3

(B) 1:1

(C) 3:4

(D) 1:2

XE(A) - Copyright © GATE 2021 Page 5 of 11

Q.10 Six students P, Q, R, S, T and U, with distinct heights, compare their
heights and make the following observations.
Observation I: S is taller than R.
Observation II: Q is the shortest of all.
Observation III: U is taller than only one student.
Observation IV: T is taller than S but is not the tallest.
The number of students that are taller than R is the same as the number of
students shorter than ______.

(A) T

(B) R

(C) S

(D) P

XE(A) - Copyright © GATE 2021 Page 6 of 11

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

Q.1 Let
𝟐 −𝟒 𝒙𝟏
𝑺 = { 𝑨𝑿 ∶ 𝑨 = [𝟏 𝟏 ] 𝐚𝐧𝐝 𝑿 = [𝒙 ] } .
𝟐
𝟏 −𝟏
−𝟏
If [ 𝜶] ∈ 𝑺, then the value of 𝜶 is
𝟏
(A) −4

(B) −2

(C) 2

(D) 4

Q.2 Let 𝑪 be the boundary of the region 𝑹 ∶ 𝟎 ≤ 𝒙 ≤ 𝝅, 𝟎 ≤ 𝒚 ≤ 𝐬𝐢𝐧 𝒙 in the
𝒙𝒚-plane and 𝜶 be the area of the region 𝑹. If 𝑪 traverses once in the counter
clockwise direction, then the value of the line integral ∮𝑪 (𝟐𝒚 𝒅𝒙 + 𝟓𝒙 𝒅𝒚)
is equal to

(A) 𝛼

(B) 2𝛼

(C) 3𝛼

(D) 4𝛼

XE(A) - Copyright © GATE 2021 Page 7 of 11

Q.3 Given that 𝒊 = √−𝟏. The value of
𝒛𝟑 + 𝟏
𝐥𝐢𝐦𝝅𝒊 𝟒
𝒛 + 𝒛𝟐 + 𝟏
𝒛→ 𝒆𝟑

is

(A) 3
+𝑖 4
√3
4

(B) 3
−𝑖 4
√3
4

(C) −3
+𝑖 4
√3
4

(D) −3 √3
−𝑖 4
4

XE(A) - Copyright © GATE 2021 Page 8 of 11

Q.4 – Q.7 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.4 Let 𝒇(𝒙) be a non-negative continuous function of real variable 𝒙. If the area
𝒂𝟐 𝒂 𝝅 𝝅
under the curve 𝒚 = 𝒇(𝒙) from 𝒙 = 𝟎 to 𝒙 = 𝒂 is 𝟐 + 𝟐 𝐬𝐢𝐧 𝒂 + 𝟐 𝐜𝐨𝐬 𝒂 − 𝟐 ,
𝝅
then the value of 𝒇 ( 𝟐 ) is __________________ (round off to one decimal
place).

Q.5 𝟒
If the numerical approximation of the value of the integral ∫𝟎 𝟐𝜶𝒙 𝒅𝒙 using
the Trapezoidal rule with two subintervals is 𝟗, then the value of the real
constant 𝜶 is __________________ (round off to one decimal place).

Q.6 Let the transformation 𝒚(𝒙) = 𝒆𝒙 𝒗(𝒙) reduce the ordinary differential
equation
𝒅𝟐 𝒚 𝒅𝒚
𝒙 𝟐 + 𝟐(𝟏 − 𝒙) + (𝒙 − 𝟐) 𝒚 = 𝟎; 𝒙 > 𝟎
𝒅𝒙 𝒅𝒙
to
𝒅𝟐 𝒗 𝒅𝒗
𝜶𝒙 𝟐
+ 𝟐𝜷 + 𝟑𝜸𝒗 = 𝟎,
𝒅𝒙 𝒅𝒙
where 𝜶, 𝜷, 𝜸 are real constants. Then, the arithmetic mean of 𝜶, 𝜷, 𝜸
is ______________ (round off to three decimal places).

Q.7 A person, who speaks the truth 𝟑 out of 𝟒 times, throws a fair dice with six
faces and informs that the outcome is 𝟓. The probability that the outcome is
really 5 is __________________ (round off to three decimal places).

XE(A) - Copyright © GATE 2021 Page 9 of 11

Q. 8 – Q. 9 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q. 8 Let 𝒇(𝒙, 𝒚) = 𝒙𝟒 + 𝒚𝟒 − 𝟐𝒙𝟐 + 𝟒𝒙𝒚 − 𝟐𝒚𝟐 + 𝜶 be a real valued function.
Then, which one of the following statements is TRUE for all 𝜶 ?

(A) (0, 0) is not a stationary point of 𝑓

(B) 𝑓 has a local maxima at (0, 0)

(C) 𝑓 has a local minima at (0, 0)

(D) 𝑓 has a saddle point at (0, 0)

Q. 9 Let 𝒖(𝒙, 𝒚) = (𝒙𝟐 − 𝒚𝟐 )𝒗(𝒙, 𝒚) be such that both 𝒖(𝒙, 𝒚) and 𝒗(𝒙, 𝒚) satisfy
the Laplace equation in a domain 𝛀 of the 𝒙𝒚-plane. Then, which one of the
following is TRUE in 𝛀 ?

(A) 𝑥 𝜕𝑣 − 𝑦 𝜕𝑣 = 0
𝜕𝑥 𝜕𝑦

(B) 𝑥 𝜕𝑣 + 𝑦 𝜕𝑣 = 0
𝜕𝑥 𝜕𝑦

(C) 𝑥 𝜕𝑣 − 𝑦 𝜕𝑣 = 0
𝜕𝑦 𝜕𝑥

(D) 𝑥 𝜕𝑣 + 𝑦 𝜕𝑣 = 0
𝜕𝑦 𝜕𝑥

XE(A) - Copyright © GATE 2021 Page 10 of 11

Q. 10 – Q. 11 Numerical Answer Type (NAT), carry TWO marks each (no negative
marks).

Q. 10 Let 𝑰 denote the identity matrix of order 𝟕, and 𝑨 be a 𝟕 × 𝟕 real matrix
having characteristic polynomial 𝑪𝑨 (𝝀) = 𝝀𝟐 (𝝀 − 𝟏)𝜶 (𝝀 + 𝟐)𝜷 , where 𝜶
and 𝜷 are positive integers. If 𝑨 is diagonalizable and 𝒓𝒂𝒏𝒌(𝑨) = 𝒓𝒂𝒏𝒌(𝑨 +
𝟐𝑰), then 𝒓𝒂𝒏𝒌(𝑨 − 𝑰) is ____________ (in integer).

Q. 11 𝟒 𝟑
Let 𝑪𝟏 be the line segment from (𝟎, 𝟏) to (𝟓 , 𝟓), and let 𝑪𝟐 be the arc of the
𝟒 𝟑
circle 𝒙𝟐 + 𝒚𝟐 = 𝟏 from (𝟎, 𝟏) to (𝟓 , 𝟓). If

𝟐𝒙 𝟏 − 𝒙𝟐 𝟐𝒙 𝟏 − 𝒙𝟐
𝜶= ∫( 𝒊̂ + 𝒋̂ ) ⋅ ⃗
𝒅𝒓 𝐚𝐧𝐝 𝜷 = ∫ ( 𝒊̂ + ⃗ ,
𝒋̂) ⋅ 𝒅𝒓
𝒚 𝒚𝟐 𝒚 𝒚𝟐
𝑪𝟏 𝑪𝟐

where 𝒓⃗ = 𝒙 𝒊̂ + 𝒚 𝒋̂, then the value of 𝜶𝟐 + 𝜷𝟐 is __________________
(round off to two decimal places).

END OF THE QUESTION PAPER

XE(A) - Copyright © GATE 2021 Page 11 of 11