Coachings - Vidhyarthi Darpan

GATE 2021 Statistics Previous Year Paper

GATE 2021 Statistics Previous Year Paper

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 and q are positive integers and pq+qp=3then, P²/q²+q²/P²
(A)	3
(B)	7
(C)	9
(D)	11

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 woke up from sleep.I woked up from sleep.I was woken up from sleep.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)

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?Nowadays, computers are present in almost all places.Computers cannot be used for solving problems in engineering.For humans, there are both positive and negative effects of using computers.Artificial intelligence can be done without data.
(A)	(ii) and (iii)
(B)	(ii) and (iv)
(C)	(i), (iii) and (iv)
(D)	(i) and (iii)

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

Statistics

Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.2	Let {(𝒕)}_𝒕≥𝟎 be a standard Brownian motion. Then the variance of𝑾(𝟏)𝑾(𝟐) equals
(A)	1
(B)	2
(C)	3
(D)	4

Q.4	Let {𝒙_𝟏, 𝒙_𝟐, … , 𝒙_𝒏} be a realization of a random sample of size 𝒏 (≥ 𝟐) from a 𝑵(𝝁, 𝝈^𝟐) distribution, where −∞ < 𝝁 < ∞ and 𝝈 > 𝟎. Which of the following statements is/are true? P: 95% confidence interval of 𝜇 based on {𝑥₁, 𝑥₂, … , 𝑥_𝑛} is unique when𝜎 is known Q.: 95% confidence interval of 𝜇 based on {𝑥₁, 𝑥₂, … , 𝑥_𝑛} is NOT unique when 𝜎 is unknown.
(A)	P only
(B)	Q only
(C)	Both P and Q
(D)	Neither P nor Q

Q.10 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.15

A fair die is rolled twice independently. Let 𝑿 and 𝒀 denote the outcomes of the first and second roll, respectively. Then 𝑬(𝑿 + 𝒀 | (𝑿 − 𝒀)^𝟐 = 𝟏) equals

Q.19

Let {𝑿_𝒏}_≥𝟏 be a sequence of independent and identically distributed random variables each having uniform distribution on [𝟎, 𝟐]. For 𝒏 ≥ 𝟏, letZn=-logei = 1n(2-Xi)1nThen, as 𝒏 → ∞, the sequence {𝒁_𝒏}_𝒏≥𝟏 converges almost surely to ___________(round off to 𝟐 decimal places).

Q.26 – Q.43 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).

Q.27	Let 𝒇: [𝟎, ∞) → ℝ be a function, where ℝ denotes the set of all real numbers. Then which one of the following statements is true?
(A)	If 𝑓 is bounded and continuous, then 𝑓 is uniformly continuous
(B)	If 𝑓 is uniformly continuous, then 𝑥∞ 𝑓(𝑥) exists
(C)	If 𝑓 is uniformly continuous, then the function 𝑔(𝑥) = 𝑓(𝑥) sin 𝑥 is also uniformly continuous
(D)	If 𝑓 is continuous and 𝑥∞ 𝑓(𝑥) is finite, then 𝑓 is uniformly continuous

Q.28	Let 𝒇: ℝ → ℝ be a differentiable function such that 𝒇(𝟎) = 𝟎 and 𝒇^′(𝒙) + 𝟐𝒇(𝒙) > 𝟎 for all 𝒙 ∈ ℝ, where 𝒇^′ denotes the derivative of 𝒇 and ℝ denotes the set of all real numbers. Then which one of the following statements is true?
(A)	𝑓(𝑥) > 0, for all 𝑥 > 0 and 𝑓(𝑥) < 0, for all 𝑥 < 0
(B)	𝑓(𝑥) < 0, for all 𝑥 ≠ 0
(C)	𝑓(𝑥) > 0, for all 𝑥 ≠ 0
(D)	𝑓(𝑥) < 0, for all 𝑥 > 0 and 𝑓(𝑥) > 0, for all 𝑥 < 0

Q.30	Let 𝑨 be a 𝟑 × 𝟑 real matrix such that 𝑰_𝟑 + 𝑨 is invertible and let 𝑩 = (𝑰_𝟑 + 𝑨)^−𝟏(𝑰_𝟑 − 𝑨), where 𝑰_𝟑 denotes the 𝟑 × 𝟑 identity matrix. Then which one of the following statements is true?
(A)	If 𝐵 is orthogonal, then 𝐴 is invertible
(B)	If 𝐵 is orthogonal, then all the eigenvalues of 𝐴 are real
(C)	If 𝐵 is skew-symmetric, then 𝐴 is orthogonal
(D)	If 𝐵 is skew-symmetric, then the determinant of 𝐴 equals −1

Q.31	Let 𝑿 be a random variable having Poisson distribution such that 𝑬(𝑿^𝟐) = 𝟏𝟏𝟎. Then which one of the following statements is NOT true?
(A)	𝐸(𝑋^𝑛) = 10 𝐸[(𝑋 + 1)^𝑛−1], for all 𝑛 = 1, 2, 3, …
(B)	𝑃(𝑋 is even) = ¼ (1 + 𝑒⁻²⁰)
(C)	𝑃(𝑋 = 𝑘) < 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 0, 1, … , 8
(D)	𝑃(𝑋 = 𝑘) > 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 10, 11, …

Q.32	Let 𝑿 be a random variable having uniform distribution on –2, 2 Then which one of the following statements is NOT true?
(A)	𝑌 = cot 𝑋 follows standard Cauchy distribution
(B)	𝑌 = tan 𝑋 follows standard Cauchy distribution
(C)
(D)

Q.33	Let 𝛀 = {𝟏, 𝟐, 𝟑, … } represent the collection of all possible outcomes of a random experiment with probabilities 𝑷({𝒏}) = 𝑎_𝒏 for 𝒏 ∈ 𝛀. Then which one of the following statements is NOT true?
(A)	nan=0
(B)
(C)
(D)

Q.35	Let 𝑿_𝟏, 𝑿_𝟐 and 𝑿_𝟑 be three uncorrelated random variables with common variance 𝝈^𝟐< ∞. Let 𝒀_𝟏 = 𝟐𝑿_𝟏 + 𝑿_𝟐 + 𝑿_𝟑, 𝒀_𝟐 = 𝑿_𝟏 + 𝟐𝑿_𝟐 + 𝑿_𝟑 and 𝒀_𝟑 = 𝑿_𝟏 + 𝑿_𝟐 + 𝟐𝑿_𝟑. Then which of the following statements is/are true?P : The sum of eigenvalues of the variance covariance matrix of (𝒀_𝟏, 𝒀_𝟐, 𝒀_𝟑) is 𝟏𝟖𝝈^𝟐.Q : The correlation coefficient between 𝒀_𝟏 𝐚𝐧𝐝 𝒀_𝟐 equals that between 𝒀_𝟐 𝐚𝐧𝐝 𝒀_𝟑.
(A)	P only
(B)	Q only
(C)	Both P and Q
(D)	Neither P nor Q

Q.36	Let {𝑿_𝒏}_≥𝟎 be a time-homogeneous discrete time Markov chain with either finite or countable state space 𝑺. Then which one of the following statements is true?
(A)	There is at least one recurrent state
(B)	If there is an absorbing state, then there exists at least one stationary distribution
(C)	If all the states are positive recurrent, then there exists a unique stationary distribution
(D)	If {𝑋_𝑛}_𝑛≥0 is irreducible, 𝑆 = {1, 2} and [𝜋₁ 𝜋₂] is a stationary distribution, then lim 𝑃(𝑋_𝑛 = 𝑖 \| 𝑋₀ = 𝑖) = 𝜋_𝑖 for 𝑖 = 1, 2𝑛→∞

Q.37	Let customers arrive at a departmental store according to a Poisson process with rate 𝟏𝟎. Further, suppose that each arriving customer is either a male or a female with probability ½ each, independent of all other arrivals. Let (𝒕) denote the total number of customers who have arrived by time 𝒕.Then which one of the following statements is NOT true?
(A)	If 𝑆₂ denotes the time of arrival of the second female customer, then𝑃(𝑆 ≤ 1) = 25 ∫¹𝑠𝑒^−5𝑠𝑑𝑠2 0
(B)	If 𝑀(𝑡) denotes the number of male customers who have arrived by time 𝑡,then 𝑃 (𝑀 (⅓) = 0 \| 𝑀(1) = 1) = ⅓
(C)	𝐸 [(𝑁(𝑡))²] = 100𝑡² + 10𝑡
(D)	𝐸[𝑁(𝑡)𝑁(2𝑡)] = 200𝑡² + 10𝑡

Q.38	Let 𝑿_(𝟏) < 𝑿_(𝟐) < 𝑿_(𝟑) < 𝑿_(𝟒) < 𝑿_(𝟓) be the order statistics corresponding to a random sample of size 𝟓 from a uniform distribution on [𝟎, 𝜽], where𝜽 ∈ (𝟎, ∞). Then which of the following statements is/are true?P : 𝟑𝑿_(𝟐) is an unbiased estimator of 𝜽.Q : The variance of 𝑬[𝟐𝑿_(𝟑) \| 𝑿_(𝟓)] is less than or equal to the variance of 𝟐𝑿_(𝟑).
(A)	P only
(B)	Q only
(C)	Both P and Q
(D)	Neither P nor Q

Q.40	Let 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏 be a random sample of size 𝒏 (≥ 𝟐) from a uniform distribution on [−𝜽, 𝜽], where 𝜽 ∈ (𝟎, ∞). Let 𝑿_(𝟏) = 𝐦𝐢𝐧{ 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏} and 𝑿_(𝒏) = 𝐦𝐚𝐱{ 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏}. Then which of the following statementsis/are true?P : (𝑿_(𝟏), 𝑿_(𝒏)) is a complete statistic. Q : 𝑿_(𝒏) − 𝑿_(𝟏) is an ancillary statistic.
(A)	P only
(B)	Q only
(C)	Both P and Q
(D)	Neither P nor Q

Q.41	Let {𝑿_𝒏}_≥𝟏 be a sequence of independent and identically distributed random variables having common distribution function 𝑭(⋅). Let 𝒂 < 𝒃 be two real numbers such that 𝑭(𝒙) = 𝟎 for all 𝒙 ≤ 𝒂, 𝟎 < 𝑭(𝒙) < 𝟏 for all𝒂 < 𝒙 < 𝒃 and (𝒙) = 𝟏 for all 𝒙 ≥ 𝒃. Let 𝑺(𝒙) be the empirical distribution function at 𝒙 based on 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏, 𝒏 ≥ 𝟏. Then which one of the following statements is NOT true?
(A)
(B)
(C)
(D)

Q.43	Let 𝒀 follow 𝑵(𝟎, 𝑰_𝟖) distribution, where 𝑰_𝟖 is the 𝟖 × 𝟖 identity matrix. Let 𝒀^𝑻𝚺_𝟏𝒀 and 𝒀^𝑻𝚺_𝟐𝒀 be independent and follow central chi-square distributions with 𝟑 and 𝟒 degrees of freedom, respectively, where 𝚺_𝟏 and𝚺_𝟐 are 𝟖 × 𝟖 matrices and 𝒀^𝑻 denotes transpose of 𝒀. Then which of the following statements is/are true?P : 𝚺_𝟏 and 𝚺_𝟐 are idempotent.Q : 𝚺_𝟏𝚺_𝟐 = 𝟎, where 𝟎 is the 𝟖 × 𝟖 zero matrix.
(A)	P only
(B)	Q only
(C)	Both P and Q
(D)	Neither P nor Q

Q.44 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.44	Let (𝑿, 𝒀) have a bivariate normal distribution with the joint probability density functionfX,Y(x,y)=1e (3/2xy – 25/32x² – 2y²), -<x,y<Then 𝟖 𝑬(𝑿𝒀) equals _________.

Q.45	Let 𝒇: ℝ × ℝ → ℝ be defined by (𝒙, 𝒚) = 𝟖𝒙^𝟐 − 𝟐𝒚, where ℝ denotes the set of all real numbers. If 𝑴 and 𝒎 denote the maximum and minimum values of 𝒇, respectively, on the set {(𝒙, 𝒚) ∈ ℝ × ℝ ∶ 𝒙^𝟐 + 𝒚^𝟐 = 𝟏}, then𝑴 − 𝒎 equals_________(round off to 𝟐 decimal places).

Q.46	Let 𝑨 = [𝒂 𝒖_𝟏 𝒖_𝟐 𝒖_𝟑], 𝑩 = [𝒃 𝒖_𝟏 𝒖_𝟐 𝒖_𝟑] and 𝑪 = [𝒖_𝟐 𝒖_𝟑 𝒖_𝟏 𝒂 + 𝒃] be three 𝟒 × 𝟒 real matrices, where 𝒂, 𝒃, 𝒖_𝟏, 𝒖_𝟐 𝐚𝐧𝐝 𝒖_𝟑 are 𝟒 × 𝟏 real column vectors. Let 𝐝𝐞(𝑨), 𝐝𝐞𝐭(𝑩) and 𝐝𝐞𝐭(𝑪) denote the determinants of the matrices 𝑨, 𝑩 and 𝑪, respectively. If 𝐝𝐞𝐭(𝑨) = 𝟔 and 𝐝𝐞𝐭(𝑩) = 𝟐, then𝐝𝐞𝐭(𝑨 + 𝑩) − 𝐝𝐞𝐭(𝑪) equals ________

Q.47	Let 𝑿 be a random variable having the moment generating function Then (𝑿 > 𝟏) equals_____(round off to 𝟐 decimal places).

Q.48	Let {𝑿_𝒏}_≥𝟏 be a sequence of independent and identically distributed random variables each having uniform distribution on [𝟎, 𝟑]. Let 𝒀 be a random variable, independent of {𝑿_𝒏}_𝒏≥𝟏, having probability mass function Then 𝑷(𝐦𝐚𝐱{𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒀} ≤ 𝟏) equals (round off to decimal places).

Q.49

Let {𝑿_𝒏}_𝒏≥𝟏 be a sequence of independent and identically distributed random variables each having probability density function

Let _(𝒏) = 𝐦𝐚𝐱{𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏} for 𝒏 ≥ 𝟏. If 𝒁 is the random variable to which {𝑿_(𝒏) − 𝐥𝐨𝐠_𝒆 𝒏}_𝒏≥𝟏converges in distribution, as 𝒏 → ∞, then the median of 𝒁 equals ______________(round off to 𝟐 decimal places).

Q.50

Consider an amusement park where visitors are arriving according to a Poisson process with rate 𝟏. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function

If at a given time point, there are 𝟏𝟎 visitors in the park and 𝒑 is the probability that there will be exactly two more arrivals before the next departure, then 1/p equals ________.

Q.52

Let a random sample of size 𝟏𝟎𝟎 from a normal population with unknown mean 𝝁 and variance 𝟗 give the sample mean 𝟓. 𝟔𝟎𝟖. Let (⋅) denote the distribution function of the standard normal random variable. If (𝟏. 𝟗𝟔) = 𝟎. 𝟗𝟕𝟓, 𝚽(𝟏. 𝟔𝟒) = 𝟎. 𝟗𝟓 and the uniformly most powerful unbiased test based on sample mean is used to test 𝑯_𝟎: 𝝁 = 𝟓. 𝟎𝟐 against 𝑯_𝟏: 𝝁 ≠ 𝟓. 𝟎𝟐, then the 𝒑-value equals____(round off to 𝟑 decimal places).

Q.52

Q.54

Let 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝟏𝟎 be a random sample from a probability density function𝒇_𝜽(𝒙) = 𝒇(𝒙 − 𝜽), −∞ < 𝒙 < ∞,where −∞ < 𝜽 < ∞ and 𝒇(−𝒙) = 𝒇(𝒙) for −∞ < 𝒙 < ∞. For testing 𝑯_𝟎: 𝜽 = 𝟏. 𝟐 against 𝑯_𝟏: 𝜽 ≠ 𝟏. , let 𝑻⁺ denote the Wilcoxson Signed- rank test statistic. If 𝜼 denotes the probability of the event {𝑻⁺ < 𝟓𝟎} under 𝑯_𝟎, then 𝟑𝟐 𝜼 equals____(round off to 2 decimal places).

Q.55

Consider the multiple linear regression model𝒀_𝒊 = 𝖰_𝟎 + 𝖰_𝟏𝒙_𝟏, + 𝖰_𝟐𝒙_𝟐,𝒊 + ⋯ + 𝖰_𝟐𝟐𝒙_{𝟐𝟐,𝒊} + 𝝐_𝒊, 𝒊 = 𝟏, 𝟐, … , 𝟏𝟐𝟑, where, for 𝒋 = 𝟎, 𝟏, 𝟐, … , 𝟐𝟐, 𝖰_𝒋’s are unknown parameters and 𝝐_𝒊’s are independent and identically distributed 𝑵(𝟎, 𝝈^𝟐), 𝝈 > 𝟎,random variables.If the sum of squares due to regression is 𝟑𝟑𝟖. 𝟗𝟐, the total sum of squares is 𝟓𝟐𝟐. 𝟑𝟎 and 𝑹^𝟐 denotes the value of adjusted 𝑹^𝟐, then 𝟏𝟎𝟎 𝑹^𝟐 equal _____(round off to 𝟐 decimal places).

Answer Key

Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans
1	C	1	A	11	33.50 to 34.50	21	0.98 to 1.00	31	B	41	C	51	0.50 to 0.50
2	B	2	C	12	2 to 2	22	8.60 to 8.75	32	D	42	D	52	0.045 to 0.055
3	C	3	A	13	1.70 to 1.75	23	0.14 to 0.18	33	B	43	C	53	0.20 to 0.22
4	C	4	B	14	0.32 to 0.35	24	1.00 to 1.00	34	C	44	3 to 3	54	31.60 to 31.80
5	B	5	A	15	7 to 7	25	6.31 to 6.35	35	C	45	10.10 to 10.15	55	57.00 to 57.40
6	C	6	A	16	0.32 to 0.35	26	C	36	B	46	72 to 72
7	D	7	A	17	7 to 7	27	D	37	B	47	0.60 to 0.66
8	A	8	B	18	0.50 to 0.50	28	A	38	C	48	0.20 to 0.26
9	C	9	A	19	0.27 to 0.35	29	B	39	A	49	0.32 to 0.42
10	A	10	2 to 2	20	50 to 50	30	C	40	D	50	143 to 143

GATE 2021 Production and Industrial Engineering Previous Year Paper

GATE 2021 Production and Industrial Engineering Previous Year Paper

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 and q are positive integers and
(A)	3
(B)	7
(C)	9
(D)	11

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 woke up from sleep.I woked up from sleep.I was woken up from sleep.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)

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?Nowadays, computers are present in almost all places.Computers cannot be used for solving problems in engineering.For humans, there are both positive and negative effects of using computers.Artificial intelligence can be done without data.
(A)	(ii) and (iii)
(B)	(ii) and (iv)
(C)	(i), (iii) and (iv)
(D)	(i) and (iii)

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

Production & Industrial Engineering (PI)

– 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

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

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.9	In a turning operation, doubling the cutting speed (𝑽) reduces the tool life (𝑻) to ^𝟏⁄_𝟖^thof the original tool life. The exponent 𝒏 in the Taylor’s tool life equation, 𝑽𝑻^𝒏 = 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

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

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

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 σ₁= σ, σ₂= 0, σ₃= −σ with σ₁≥ σ₂≥ σ₃, then the stress σ 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 ΔF = 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*]


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 by𝒇(𝒙, 𝒚, 𝒛) = 𝒙^𝟐 + 𝟓𝒚^𝟐 + 𝟓𝒛^𝟐 − 𝟒𝒙 + 𝟒𝟎𝒚 − 𝟒𝟎𝒛 + 𝟑𝟎𝟎is ________. [*in integer*]

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

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

Q.31	The eigenvalues of matrix A = are 5 and 10. For matrix B = A + α I,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.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 30_{𝟎. 𝟎𝟎𝟎}^{𝟎. 𝟎𝟒𝟎}mm. The matingshaft 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. 𝟎𝟏𝟎}^{+𝟎. 𝟎𝟕𝟎}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

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

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

Q.37

The probability mass function (𝒙) of a discrete random variable 𝑿 is given by 𝑷(𝒙) =1/𝟐^𝒙, 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.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 7^thyear is ___________. [in integer]

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 (𝝈_𝒅) in MPa for the above process is given by 𝝈 = 𝝈̅ 𝐥𝐧 , where 𝝈̅ 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 mm²is 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 casting process is estimated by Chvorinov’s equation 𝒕 = 𝑩 [ ^𝑽/A ]² , where 𝑩𝑨 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*]

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 = 3×10⁵J/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*]

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]

Answer Key

Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans
1	C	1	C	11	B OR D	21	9545.0 to 9555.0	31	B	41	335000 to 335000	51	-5773.7 to -5773.4
2	B	2	A	12	B	22	2.4 to 2.6	32	B	42	5 to 5	52	5.0 to 5.2
3	C	3	C	13	C	23	52.2 to 52.5	33	A	43	16.80 to 16.84	53	33.4 to 33.6
4	C	4	B	14	A	24	2.4 to 2.6	34	C	44	8.8 to 9.2	54	20943.8 to 20944.1
5	B	5	D	15	C	25	136 to 136	35	A	45	74.5 to 75.5	55	1.85 to 1.95
6	C	6	C	16	D	26	A	36	A	46	28.0 to 30.0
7	D	7	D	17	MTA	27	C	37	2 to 2	47	89.0 to 91.0
8	A	8	B	18	404.00 to 405.00	28	B	38	0.365 to 0.370	48	0.193 to 0.203
9	C	9	B	19	0.17 to 0.19	29	C	39	4 to 4	49	1999.0 to 2001.0
10	A	10	B	20	2100 to 2100	30	A	40	90 to 90	50	147.00 to 147.30

^*MTA means Marks to All

GATE 2021 Physics Previous Year Paper

General Aptitude (GA)

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

Q.1	Arun and Aparna are here.Arun and Aparna is here.Arun’s families is here.Arun’s family is here.Which of the above sentences are grammatically CORRECT?
(A)	(i) and (ii)
(B)	(i) and (iv)
(C)	(ii) and (iv)
(D)	(iii) and (iv)

Q.3	Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
(A)	1/36
(B)	1/12
(C)	1/8
(D)	1/4

Q.4	⊕ and ⊙ are two operators on numbers p and q such that𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p ⊕ q = p× qThen, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=
(A)	40
(B)	-26
(C)	-33
(D)	-40

Q.5	Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A)	6
(B)	9
(C)	18
(D)	24

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

Q.6	On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.The distance of point Q to point O, in the same units, should be____________
(A)	3
(B)	4
(C)	5
(D)	6

Q.7	The author said, “Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk on to the stage and start speaking. In my opinion, it is no less important for public speakers to rehearse their talks.”Based on the above passage, which one of the following is TRUE?
(A)	The author is of the opinion that rehearsing is important for musicians, actors and public speakers.
(B)	The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors.
(C)	The author is of the opinion that rehearsing is more important only for musicians than public speakers.
(D)	The author is of the opinion that rehearsal is more important for actors than musicians.

Q.8	Some football players play cricket.All cricket players play hockey.Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
(A)	No football player plays hockey.
(B)	Some football players play hockey.
(C)	All football players play hockey.
(D)	All hockey players play football.

Q.10	In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum area of the triangle PQR possible, in cm², is
(A)	18
(B)	24
(C)	48√3
(D)	144√3

Physics (PH)

– Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.2	A matter wave is represented by the wave function_{𝚿(𝒙, 𝒚, 𝒛, 𝒕) = 𝑨𝒆}𝒊 (𝟒𝒙+𝟑𝒚+𝟓𝒛−𝟏𝟎𝝅𝒕)where 𝑨 is a constant. The unit vector representing the direction of propagation of this matter wave is
(A)
(B)
(C)
(D)

Q.4	As shown in the figure, two metal-semiconductor junctions are formed between an n-type semiconductor 𝑺 and metal 𝑴. The work functions of 𝑺 and 𝑴 are 𝝋_𝑺 𝐚𝐧𝐝 𝝋_𝑴, respectively with 𝝋_𝑴 > 𝝋_𝑺. The 𝑰 − 𝑽 characteristics (on linear scale) of the junctions is best represented by
(A)
(B)
(C)
(D)

Q.5	Consider a tiny current loop driven by a sinusoidal alternating current. If the surface integral of its time-averaged Poynting vector is constant, then the magnitude of the time-averaged magnetic field intensity, at any arbitrary position, ⃗𝒓→, is proportional to
(A)	1/r³
(B)	1/r2
(C)	1/𝑟
(D)	r

Q.7	Assume that ¹³N (𝒁 = 𝟕) undergoes first forbidden β⁺decay from its ground state with spin-parity 𝑱_i^𝝅, to a final state 𝑱_𝒇^𝝅. The possible values for 𝑱_i^𝝅 and𝑱_f^𝝅, respectively, are
(A)
(B)
(C)
(D)

Q.8	In an experiment, it is seen that an electric-dipole (𝑬𝟏) transition can connect an initial nuclear state of spin-parity 𝑱_i^𝝅 = 𝟐⁺ to a final state 𝑱_𝒇^𝝅. All possible values of 𝑱_f^𝝅 are
(A)	1⁺, 2⁺
(B)	1⁺, 2⁺, 3⁺
(C)	1⁻, 2⁻
(D)	1⁻, 2⁻, 3⁻

Q.9	Choose the correct statement from the following.
(A)	Silicon is a direct band gap semiconductor.
(B)	Conductivity of metals decreases with increase in temperature.
(C)	Conductivity of semiconductors decreases with increase in temperature.
(D)	Gallium Arsenide is an indirect band gap semiconductor.

Q.10 – Q.16 Multiple Select Question (MSQ), carry ONE mark each (no negative marks).

Q.12	A material is placed in a magnetic field intensity 𝑯. As a result, bound current density 𝑱_𝒃 is induced and magnetization of the material is 𝑴. The magnetic flux density is 𝑩. Choose the correct option(s) valid at the *surface* of the material.
(A)	𝛁. 𝑴 = 0
(B)	𝛁. 𝑩 = 0
(C)	𝛁. 𝑯 = 0
(D)	𝛁. 𝑱_𝑏 = 0

Q.13	For a finite system of Fermions where the density of states increases with energy, the chemical potential
(A)	decreases with temperature
(B)	increases with temperature
(C)	does not vary with temperature
(D)	corresponds to the energy where the occupation probability is 0.5

Q.14	Among the term symbols^𝟒𝑺_𝟏, ^𝟐𝑫_𝟕/𝟐, ^𝟑𝑺_𝟏 and ^𝟐𝑫_𝟓/𝟐choose the option(s) possible in the 𝑳𝑺 coupling notation.
(A)	⁴S₁
(B)	²𝐷_7/2
(C)	³𝑆₁
(D)	²𝐷_5/2

Q.15	To sustain lasing action in a three-level laser as shown in the figure, necessary condition(s) is(are)
(A)	lifetime of the energy level 1 should be greater than that of energy level 2
(B)	population of the particles in level 1 should be greater than that of level 0
(C)	lifetime of the energy level 2 should be greater than that of energy level 0
(D)	population of the particles in level 2 should be greater than that of level 1

Q.17 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.17	The donor concentration in a sample of n-type silicon is increased by a factor of 100. Assuming the sample to be non-degenerate, the shift in the Fermi level (in meV) at 300 K (rounded off to the nearest integer) is ________.(*Given:* 𝒌_𝐁𝑻= 25 meV at 300 K)

Q.18	Two observers O and Oʹ observe two events P and Q. The observers have a constant relative speed of 0.5c. In the units, where the speed of light, c, is taken as unity, the observer O obtained the following coordinates:Event P: 𝒙 = 5, 𝒚 = 3, 𝒛 = 5, 𝒕 = 3 Event Q: 𝒙 = 5,𝒚 = 1, 𝒛 = 3, 𝒕 = 5The length of the space-time interval between these two events, as measured by Oʹ, is 𝑳. The value of \|𝑳\| (in integer) is ________________.

Q.19	A light source having its intensity peak at the wavelength 289.8 nm is calibrated as 10,000 K which is the temperature of an equivalent black body radiation. Considering the same calibration, the temperature of light source (in K) having its intensity peak at the wavelength 579.6 nm (rounded off to the nearest integer) is ______.

Q.26 – Q.41 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).

Q.27	In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobility of the hole to electron is
(A)	4:9
(B)	4:11
(C)	9:4
(D)	11:4

Q.28	Consider a spin 𝑺 = ℏ⁄𝟐 particle in the state \|𝝓⟩ = . The probability that a measurement finds the state with 𝑺_𝒙 = +ℏ⁄𝟐 is
(A)	5/18
(B)	11/18
(C)	15/18
(D)	17/18

Q.33	Consider a system of three distinguishable particles, each having spin 𝑺=𝟏/𝟐 such that 𝑺_𝒛 = ±𝟏/𝟐 with corresponding magnetic moments 𝝁_𝒛 = ±𝝁. When the system is placed in an external magnetic field 𝑯 pointing along the𝒛-axis, the total energy of the system is 𝝁𝑯. Let 𝒙 be the state where the first spin has 𝑺_𝒛 = 𝟏/𝟐. The probability of having the state 𝒙 and the mean magnetic moment (in the +𝒛 direction) of the system in state 𝒙 are
(A)
(B)
(C)
(D)

Q.35	Consider a state described by 𝝍(𝒙, 𝒕) = 𝝍_𝟐(𝒙, 𝒕) + 𝝍_𝟒(𝒙, 𝒕), where 𝝍_𝟐(𝒙, 𝒕) and 𝝍_𝟒(𝒙, 𝒕) are respectively the second and fourth normalized harmonic oscillator wave functions and 𝑚 is the angular frequency of the harmonic oscillator. The wave function 𝝍(𝒙, 𝒕 = 𝟎) will be orthogonal to 𝝍(𝒙, 𝒕) at time 𝒕 equal to
(A)	𝜋/2𝜔
(B)	𝜋/𝜔
(C)	𝜋/4𝜔
(D)	𝜋/6𝜔

Q.40	Consider the potential 𝑼(𝒓) defined aswhere 𝑎 and 𝑼_𝟎 are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with 𝑼(𝒓) for a (wave-vector) momentum transfer 𝒒 and 𝑎 → 𝟎, is proportional to(*Useful integral:*
(A)	q⁻²
(B)	q⁻¹
(C)	𝑞
(D)	q²

Q.42 – Q.46 Multiple Select Question (MSQ), carry TWO mark each (no negative marks).

Q.42	A function (𝒕) is defined only for 𝒕 ≥ 𝟎. The Laplace transform of 𝒇(𝒕) is 𝐿(𝒇; 𝒔) = ∫^∞_𝟎 𝒆^−𝒔𝒕 𝒇(𝒕) 𝒅𝒕 whereas the Fourier transform of 𝒇(𝒕) is 𝒇^̃(𝑚) = ∫^∞_𝟎 𝒇(𝒕) 𝒆^{−𝒊𝑚𝒕} 𝒅𝒕 . The correct statement(s) is(are)
(A)	The variable 𝑠 is always real.
(B)	The variable 𝑠 can be complex.
(C)	(𝑓; 𝑠) and 𝑓^̃(𝜔) can never be made connected.
(D)	(𝑓; 𝑠) and 𝑓^̃(𝜔) can be made connected.

Q.43	𝑷 and 𝑸 are two Hermitian matrices and there exists a matrix 𝑹, which diagonalizes both of them, such that 𝑹𝑷𝑹^−𝟏 = 𝑺_𝟏 and 𝑹𝑸𝑹^−𝟏 = 𝑺_𝟐 , where 𝑺_𝟏 and 𝑺_𝟐 are diagonal matrices. The correct statement(s) is(are)
(A)	All the elements of both matrices 𝑆₁ and 𝑆₂ are real.
(B)	The matrix 𝑃𝑄 can have complex eigenvalues.
(C)	The matrix 𝑄𝑃 can have complex eigenvalues.
(D)	The matrices 𝑃 and 𝑄 commute.

Q.45	The Gell-Mann – Okuba mass formula defines the mass of baryons as 𝑴 = 𝑴 + 𝒂𝒀 + 𝒃 [𝑰(𝑰 + 𝟏) − 1/4 𝒀^𝟐], where 𝑴 , 𝒂 and 𝒃 are constants, 𝑰 represents the isospin and 𝒀 represents the hypercharge. If the mass of 𝚺 hyperons is same as that of 𝚲 hyperons, then the correct option(s) is(are)
(A)	𝑀 ∝ 𝐼(𝐼 + 1)
(B)	𝑀 ∝ 𝑌
(C)	𝑀 does not depend on 𝐼
(D)	𝑀 does not depend on 𝑌

Q.46	The time derivative of a differentiable function 𝒈(𝒒_𝒊, 𝒕) is added to a Lagrangian 𝑳(𝒒_𝒊, 𝒒̇_𝒊, 𝒕) such that, where 𝒒_𝒊, 𝒒̇ _𝒊, 𝒕 are the generalized coordinates, generalized velocities and time, respectively. Let 𝒑_𝒊 be the generalized momentum and 𝑯 the Hamiltonian associated with (𝒒_𝒊, 𝒒̇ _𝒊, 𝒕). If 𝒑^′ and 𝑯^′ are those associated with 𝑳^′, then the correct option(s) is(are)
(A)	Both 𝐿 and 𝐿^′ satisfy the Euler-Lagrange’s equations of motion
(B)
(C)	If 𝑝_𝑖 is conserved, then 𝑝^′ is necessarily conserved
(D)

Q.47 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).


Q.47	A linear charged particle accelerator is driven by an alternating voltage source operating at 10 MHz. Assume that it is used to accelerate electrons. After a few drift-tubes, the electrons attain a velocity 𝟐. 𝟗 × 𝟏𝟎^𝟖 m s^–1. The minimum length of each drift-tube, in m, to accelerate the electrons further (rounded off to one decimal place) is _______________.

Q.48	The Coulomb energy component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is _____________. (*Given:e²/𝟒𝝅E_𝟎= 𝟏. 𝟒𝟒 MeV fm)*

Q.49	For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices 𝝈_𝟏, 𝝈_𝟐 for particles 1 and 2, respectively. The value of (𝝈_𝟏 ⋅ 𝝈_𝟐) (in integer) is ________.

Q.50

A contour integral is defined as where 𝒏 is a positive integer and 𝑪 is the closed contour, as shown in the figure, consisting of the line from −𝟏𝟎𝟎 to 𝟏𝟎𝟎 and the semicircle traversed in the counter-clockwise sense.

The value of 𝑰_𝒏(in integer) is _________.

Q.51	The normalized radial wave function of the second excited state of hydrogen atom is where 𝒂 is the Bohr radius and 𝒓 is the distance from the center of the atom.The distance at which the electron is most likely to be found is 𝒚 × 𝒂. The value of 𝒚 (in integer) is ____________.

Q.52	Consider an atomic gas with number density 𝒏 = 𝟏𝟎^𝟐𝟎 𝐦^−𝟑, in the ground state at 300 K. The valence electronic configuration of atoms is 𝒇^𝟕. The paramagnetic susceptibility of the gas 𝝌 = 𝒎 × 𝟏𝟎^−𝟏𝟏. The value of 𝒎 (rounded off to two decimal places) is ________.(*Given*: 𝐌𝐚𝐠𝐧𝐞𝐭𝐢𝐜 𝐩𝐞𝐫𝐦𝐞𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐟𝐫𝐞𝐞 𝐬𝐩𝐚𝐜𝐞 𝝁_𝟎 = 𝟒𝝅 × 𝟏𝟎^−𝟕 𝐇 𝐦^−𝟏𝐁𝐨𝐡𝐫 𝐦𝐚𝐠𝐧𝐞𝐭𝐨𝐧 𝝁_𝐁 = 𝟗. 𝟐𝟕𝟒 × 𝟏𝟎^−𝟐𝟒𝐀 𝐦^𝟐𝐁𝐨𝐥𝐭𝐳𝐦𝐚𝐧𝐧 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝒌_𝐁 = 𝟏. 𝟑𝟖𝟎𝟕 × 𝟏𝟎^−𝟐𝟑 𝐉 𝐊^−𝟏 )

Answer Key

Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans	Q. No.	Ans
1	B	1	A	11	B	21	5 to 6	31	A	41	A	51	4 to 4
2	B	2	A	12	B; D	22	17367 to 17371	32	D	42	B; D	52	5.40 to 5.50
3	D	3	B	13	A; D	23	6 to 6	33	A	43	A; D	53	0.24 to 0.26
4	D	4	A	14	C; D	24	2 to 2	34	C	44	A; C	54	27 to 93
5	C	5	C	15	A; B	25	-1 to -1	35	A	45	B; C	55	3 to 5
6	C	6	C	16	B; C	26	B	36	B	46	A; B
7	A	7	A	17	115 to 116	27	D	37	C	47	14.0 to 15.0
8	B	8	D	18	2 to 2	28	D	38	B	48	8 to 8 OR 9 to 9
9	C	9	B	19	5000 to 5000	29	A	39	A	49	-3 to -3
10	D	10	B; C; D	20	2 to 2	30	D	40	A	50	5 to 5

GATE 2021 Petroleum Engineering Previous Year Paper

GATE 2021 Petroleum Engineering Previous Year Paper

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.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 woke up from sleep.I woked up from sleep.I was woken up from sleep.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)

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?Nowadays, computers are present in almost all places.Computers cannot be used for solving problems in engineering.For humans, there are both positive and negative effects of using computers.Artificial intelligence can be done without data.
(A)	(ii) and (iii)
(B)	(ii) and (iv)
(C)	(i), (iii) and (iv)
(D)	(i) and (iii)

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)	3𝜋2
(D)	3π

Petroleum Engineering (PE)

Q.1 – Q.13 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	MOPU (in the context of offshore drilling and production systems) stands for
(A)	Mobile Offshore Process Unit
(B)	Mobile Offshore Piping Unit
(C)	Mobile Offshore Production Unit
(D)	Mobile Oil Production Unit

Q.2	Which ONE of the following statements is INCORRECT?
(A)	Conductor is the outer casing of a well
(B)	Riser is used for transporting fluid
(C)	Conductor and riser have the same functions
(D)	Conductor is used for shielding the well flow lines from external forces

Q.3	Which ONE of the following offshore installations uses Dynamic Positioning System (DPS) for station keeping?
(A)	Jacket
(B)	Jack-up
(C)	Semi-submersible
(D)	Tension Leg Platform

Q.4	Which ONE of the following is NOT a primary safety system for offshore installations?
(A)	Emergency Shut Down
(B)	Isolation
(C)	Fire Protection
(D)	Blowdown

Q.5	The primary function of the thruster in the Dynamic Positioning System (DPS) of an offshore installation is
(A)	To apply thrust in the direction opposite to the resultant environmental force
(B)	To apply thrust in the same direction as the resultant environmental force
(C)	To apply thrust in the direction opposite to the motion
(D)	To apply thrust in the same direction as the motion

Q.6	Select the CORRECT firefighting system for electrical switchgear room in an offshore facility.
(A)	Wet chemical
(B)	Halon system
(C)	Foam
(D)	Water sprinklers

Q.7	Which ONE of the following options can be used to quantify secondary porosity?
(A)	Sonic and Gamma Ray Logs
(B)	Sonic and Neutron Logs
(C)	Sonic and Caliper Logs
(D)	Density and Neutron Logs

Q.8	Among the options given below, what is the typical temperature range for significant oil generation in a source rock associated with conventional crude-oil reservoirs?
(A)	10 ^oC – 40 ^oC
(B)	60 ^oC – 175 ^oC
(C)	225 ^oC – 325 ^oC
(D)	350 ^oC – 425 ^oC

Q.9	In the original Darcy’s law as proposed by Henry Darcy, which of the following drives the fluid flow through a fully saturated sand column?
(A)	Pressure-gradient or hydraulic-gradient
(B)	Viscous force per unit volume
(C)	Capillary force per unit volume
(D)	Inertial force per unit volume

Q.10	At which ONE of the following scales is Darcy’s law for fluid flow through a porous medium applicable?
(A)	Nano-scale
(B)	Molecular-scale
(C)	Microscopic-scale
(D)	Macroscopic-scale

Q.11	Pressure – Temperature phase diagram of CO2 is shown below. Identify the correct phases from the given options.
(A)	I = Solid Phase, II = Liquid Phase, III = Gas Phase, IV = Supercritical Phase
(B)	I = Gas Phase, II = Supercritical Phase, III = Solid Phase, IV = Liquid Phase
(C)	I = Supercritical Phase, II = Liquid Phase, III = Solid Phase, IV = Gas Phase
(D)	I = Gas Phase, II = Solid Phase, III = Liquid Phase, IV = Supercritical Phase

Q.12	A measure of the potential of crude oil to form surfactants for Enhanced Oil Recovery (EOR) is given by the Total Acid Number (TAN). TAN is the mass of _________(in milligrams) that is required to neutralize one gram of crude oil.
(A)	Ca(OH)2
(B)	NaCl
(C)	KOH
(D)	NaOH

Q.13	In Water-Alternating-Gas (WAG) injection, the purpose of the injection is to_____I _____ the “relative permeability” of gas and to _____ II _____ the “mobility” of the gas.
(A)	I = reduce, II = enhance
(B)	I = reduce, II = reduce
(C)	I = enhance, II = reduce
(D)	I = enhance, II = enhance

Q.14 – Q.19 Multiple Select Question (MSQ), carry ONE mark each (no negative marks

Q.14	Solids that may possibly form in the offshore pipelines during the production of oil and gas from deep-water reservoirs are
(A)	Wax
(B)	Char
(C)	Hydrates
(D)	Asphaltenes

Q.15	Oil and gas pipelines, which are at an elevated pressure (about 3 MPa) and sub-ambient temperature (below 298 K), may get blocked by the formation of solid hydrates. One of the strategies adopted to inhibit the formation of hydrates is the injection of Thermodynamic Hydrate Inhibitors (THIs) into the reservoir fluid.Identify all suitable chemicals that are commonly used as THIs.
(A)	Sodium Chloride
(B)	Methanol
(C)	Polyvinylpyrrolidone
(D)	Sodium Dodecyl Sulphate

Q.17	Heavy oil recovered from reservoirs can be represented by CXH1.5X. Suitable processes to reduce the density of heavy oil are
(A)	Carbon Rejection
(B)	Pyrolysis
(C)	Hydrogenation
(D)	Filtration

Q.18	Identify the CORRECT statements for a 𝒏 × 𝒏 matrix.
(A)	Under elementary row operations, the rank of the matrix remains invariant
(B)	Under elementary row operations, the eigenvalues of the matrix remain the same
(C)	If the elements in a row can be written as a linear combination of two or more rows, then the matrix is singular.
(D)	The rank of the matrix is equal to 𝑛 if the determinant of the matrix is zero.

Q.19	During a drilling operation, kick occurs if
(A)	the shear ram in the Blow Out Preventer (BOP) does not work.
(B)	the formation pressure is equal to the drilling fluid pressure.
(C)	the volume of the mud used to fill the hole is less than that of the pipe being pulled out.
(D)	the formation pressure is more than the drilling fluid pressure.

Q.20 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).


Q.20	The value of is_________(correct up to one decimal place).

Q.21	Given two complex numbers, 𝒛_𝟏 = 𝟒 + 𝟑𝒊 𝐚𝐧𝐝 𝒛_𝟐 = 𝟐 − 𝟓𝒊, the real part of (𝒛_𝟏𝒛̅̅_𝟐̅) is .

Q.22	The number of ‘three-digit numbers’ that can be formed using the digits from 1 to 9 without the repetition of each digit is .

Q.23	The estimate for the root of the function 𝒇(𝒙) = 𝒆^𝟐𝒙 + 𝟐𝒙 after one iteration with an initial guess of 𝒙_𝟎 = 𝟎, using the Newton-Raphson method is__________(correct up to two decimal places).

Q.24	A saturated oil reservoir has an average reservoir pressure of 3000 psia, tested for flowing bottom-hole pressure (BHP) of 2000 psia and production rate of 500 STB/day. The maximum reservoir deliverability based on Vogel’s equation for two-phase flow is_____________STB/day.

Q.25	If the specific heat ratio of natural gas is 1.28, the critical pressure ratio (ratio of outlet pressure to upstream pressure) through a choke is_____(round off to two decimal places).

Q.26 – Q.37 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3)

Q.32	Select the INCORRECT statement related to Enhanced Oil Recovery (EOR) techniques from the following options.
(A)	Alkaline flooding recovers crude oil by reduction of interfacial tension (IFT) and reversal of wettability of rocks.
(B)	In-situ combustion recovers crude oil by the application of heat, thus lowering the viscosity of the crude oil.
(C)	Nitrogen flue gas flooding recovers crude oil by vaporizing the lighter components of the crude oil.
(D)	Polymer flooding recovers crude oil by reducing the viscosity and increasing the mobility of water.

Q.34	An Ideal Pressure Buildup Test yields a single straight line for all times, where shut-in Bottom-Hole Pressure (Pws) is plotted against 𝐥𝐨𝐠_𝟏𝟎. Here 𝒕_𝒑 is the well production time and 𝚫𝒕 is the time elapsed since shut-in. However, in an actual Pressure Buildup Test, a non-linear curve is obtained which can be logically divided into distinct regions.Choose the INCORRECT option from the following.
(A)	A late-time region, in which the radius of investigation has reached the well’s drainage boundaries.
(B)	A middle-time region during which the pressure transient has moved away from the wellbore and into the bulk formation.
(C)	An early-time region during which a pressure transient is moving through the formation nearest the wellbore.
(D)	An early-time region during which a pressure transient is moving away from the drainage boundary.

Q.35	When two immiscible fluid phases are placed in contact with a solid surface, one phase usually is attracted to solid more strongly than the other phase.The more strongly attracted phase is called the ‘wetting phase’.The inter-molecular interaction of the non-wetting phase with the solid is_____I than its intra-molecular interaction. Due to this, the non-wetting phase tends to occupy the II of the reservoir.
(A)	I = stronger, II = smaller pores
(B)	I = stronger, II = larger pores
(C)	I = weaker, II = smaller pores
(D)	I = weaker, II = larger pores

Q.36	Coal bed methane is methane gas adsorbed in coal seams. To desorb the methane from the coal seam it should be exposed to CO₂ and (or) N₂. Which ONE of the following is an appropriate reason to enhance the desorption process?
(A)	N2 is used because it has low kinetic energy compared to CH4
(B)	CO2 is used as it has high kinetic energy compared to CH4
(C)	CO2 is used as it strongly binds with coal compared to CH4
(D)	N2 is used as it strongly binds with coal compared to CH4

Q.38 – Q.39 Multiple Select Question (MSQ), carry TWO mark each (no negative marks).

Q.38	Select the CORRECT statements from the following. Well testing operations on a typical crude oil reservoir
(A)	do not measure rock and fluid properties of the reservoir when the well is flowing or shut-in.
(B)	measure variation in pressure response of the reservoir with time when the well is flowing or shut-in.
(C)	measure productivity index and partial well completion.
(D)	do not measure length and conductivity of hydraulic fractures.

Q.39	Crude oil from oil sands contains bitumen and asphaltene and this crude is heavy and viscous at room temperature. Assume that one such crude oil is represented by CXH1.2XOY. For easier transportation through pipelines it should be processed further. Identify the processes which help in transportation of this crude oil.
(A)	Drying
(B)	Vis-breaking
(C)	Coking process
(D)	Hydro-treating

Q.40 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.40	Given matrix 𝑨 =. The eigenvalue corresponding to the eigenvectoris ______________ .

Q.41	The maximum value of the function 𝒇(𝒙) = 𝒙^𝟒 − 𝟖𝒙^𝟐 + 𝟐 for −𝟐 ≤ 𝒙 ≤ 𝟐 is_______.

Q.42	Given the second order ordinary differential equation: 𝒚^′′ + 𝟑𝒚^′ − 𝟒𝒚 = 𝟎 with the initial conditions 𝒚(𝟎) = 𝟑, 𝐚𝐧𝐝 𝒚^′(𝟎) = −𝟕, the value of y(1) is________(round off to two decimal places).

Q.43	The directional derivative of 𝒇(𝒙, 𝒚, 𝒛) = 𝒙^𝟐 + 𝟑𝒚^𝟐 + 𝒛^𝟐, at point (2,1,0) along the unit vector in x-direction, 𝒊̂, is __.

Q.44

A productivity test conducted on a crude oil well indicates a stabilized flow rate of 150 STB/day (water-free oil production) at a bottom-hole flowing pressure of 935 psig. After shutting the well for 24 hours, the bottom-hole pressure reached a static value of 1250 psig.The Absolute Open Flow (AOF) potential of the well is__________STB/day.

Q.46

A tubing with an inner diameter of 2.259 inch delivers oil from a well at the rate of 1000 bbl/day. The API gravity and viscosity of the oil are 40^oand 1.2 cP, respectively. The tubing makes an angle of 15^owith the vertical. Assuming a fanning friction factor of 0.006, the pressure-drop over a length of 1000 ft tubing is ______ psi (round off to nearest integer).[1 bbl = 5.615 ft³]

Q.47

A cylindrical crude oil reservoir with a radius of 3000 ft is under water influx from a cylindrical aquifer with an estimated radius of 9000 ft. The reservoir has the following properties.Aquifer thickness, h = 40 ft Porosity, ϕ = 15%Formation compressibility, Cf = 4.5 × 10^-6 psi^-1 Water compressibility, Cw = 4.0 × 10^-6 psi^-1Assuming a pot reservoir model with fractional encroachment angle as unity, the water influx into the reservoir for a pressure drop of 700 psi is ________MMbbl (million barrels) (round off to two decimal places).(π = 3.14, 1 bbl = 5.615 ft³)

Q.48

A heavy oil reservoir with an initial oil recovery of 10% has the following properties.Confined area A = 1.5 acres, thickness of the reservoir h = 15 ft, effective porosity ϕ = 15%, irreducible water saturation Swr = 25%, oil formation volume factor Bo = 1.10 bbl/STB.An in-situ combustion test was conducted in the above reservoir. Oil recovery due to the combustion process at the well is observed to be 12000 bbl.The total (overall) oil recovery at the end of the in-situ combustion process is__________% (round off to nearest integer) of the original oil in place.(1 acre = 43560 ft², 1 bbl = 5.615 ft³)

Q.49

A double acting duplex pump with a rod diameter of 2.5 inch and a stroke of 20 inch is to be operated at 60 strokes per minute for drilling down to 10000 ft. The flow rate is 600 gpm. If the volumetric efficiency of the pump is 80%, the liner size is _________inch (round off to one decimal place).[1 gallon = 231 inch³]

Q.50

The fluid flow through an under-saturated oil reservoir is driven by solution gas drive mechanism. The reservoir parameters are as given below.Compressibility of water, 𝒄_𝒘 = 𝟏 × 𝟏𝟎^−𝟔 psi^-1Compressibility of formation, 𝒄_𝒇 = 𝟏 × 𝟏𝟎^−𝟓 psi^-1Connate water saturation, 𝑺_𝐰𝐜 = 𝟎. 𝟐Initial reservoir pressure, 𝒑_𝒊 = 𝟒𝟎𝟎𝟎 psiReservoir pressure at bubble-point, 𝒑_𝐛 = 𝟑𝟎𝟎𝟎 psi Oil formation volume factor, 𝑩_𝐨𝐢 = 𝟏. 𝟐𝟒 rb/STBFormation volume factor at bubble point pressure, 𝑩_𝐨𝐛 = 𝟏. 𝟐𝟔 rb/STB.The percentage of oil recovered as a fraction of the Original Oil in Place (OOIP) is ___________% (round off to one decimal place).

Q.51	During drilling, a well is damaged out to a radial distance of 5 ft from the periphery of the wellbore so that the permeability within the damaged zone is reduced to 1/50^thof the undamaged effective permeability. After completion, the well is stimulated so that the permeability out to a radial distance of 15 ft from the periphery of the wellbore is increased to twenty times the permeability of the undamaged zone.The radial inflow equation for stabilized flow conditions under semi-steady state conditions is given by where 𝒑_𝐞 is effective pressure, 𝒑_𝐰𝐟 is flowing bottom-hole pressure, 𝒒 is flow- rate, 𝝁 is viscosity, 𝒌_𝐞 is average effective permeability, 𝒉 is reservoir thickness, 𝒓_𝐞 is drainage radius, 𝒓_𝐰 is wellbore radius and 𝑺 is skin factor.If 𝒓_𝐰 = 𝟎. 𝟓 ft and 𝒓_𝐞 = 𝟓𝟎𝟎 ft, then the increase in Productivity Index ratio is (round off to one decimal place).

Q.52	A depleted and shut-in oil reservoir originally contained 𝟐𝟓 × 𝟏𝟎^𝟔 STB of oil with a formation volume factor of 1.35 res bbl/STB and a connate water saturation of 0.25. Cumulative oil production to date has been 𝟐. 𝟓 × 𝟏𝟎^𝟔 STB of oil. The oil formation volume factor is now 1.25 res bbl/STB. Assuming no water influx, the gas saturation in the reservoir is _______% (round off to one decimal place).

Q.54	Miscible displacement process is one of the EOR techniques. The performance of this process depends on fluid physical properties that affect flow behavior in a reservoir. Two of the important properties are density and viscosity.Consider the use of CO2 for one such process. The density of CO2 at the reservoir condition is lb/ft³(round off to one decimal place).Relevant data for this calculation are given below. Reservoir temperature = 300 ^oF (422 K) Reservoir pressure = 1470 psig (100 atm)Compressibility factor (z) at the reservoir condition = 0.5Values of Universal Gas Constant (R) in different units are listed below. Universal Gas Constant (R) = 8.314 m³.Pa.K^-1mol^-1= 10.731 psi. ft³.lb. mol^{-1 o}R^-1= 0.082 L. atm. K^-1mol^-1

Q.55	In a counter current heat exchanger, the hot fluid enters at 175 ^oF and exits at 100 ^oF. The cold fluid enters at 75 ^oF and exits at 85 ^oF. For the calculation of heat transfer rate, consider the tube surface area (per unit length) to be 0.26 ft²/ft and a tube length of 40 ft. The overall heat transfer coefficient of the exchanger is 100 BTU/hr-ft². The minimum number of tubes required in the exchanger for a heat duty of 𝟏𝟓 × 𝟏𝟎^𝟓 BTU/hr is _______(round off to nearest integer).

Answer Key

Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans
1	C	1	C	11	D	21	-7 to -7	31	A	41	2 to 2	51	32 to 42
2	B	2	C	12	C	22	504 to 504	32	D	42	2.71 to 2.79	52	11 to 14
3	C	3	C	13	B	23	-0.25 to -0.25	33	B	43	4 to 4	53	61 to 65
4	C	4	C	14	A; C; D	24	960 to 990	34	D	44	580 to 610	54	15.7 to 16.1
5	B	5	A OR C	15	A; B	25	0.52 to 0.57	35	D	45	5 to 6	55	MTA
6	C	6	B	16	B; D	26	D	36	C	46	340 to 360
7	D	7	B	17	A; C OR A; B; C	27	C	37	A	47	1.36 to 1.50
8	A	8	B	18	A; C	28	C	38	A; B; C	48	68 to 78
9	C	9	A	19	C; D OR D	29	B	39	B; C; D	49	6.5 to 7.5
10	A	10	D	20	0.5 to 0.5	30	C	40	3 to 3	50	2.6 to 3.0

*MTA means Marks to All

GATE 2021 Metallurgical Engineering Previous Year Paper

General Aptitude (GA)

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

Q.1	Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P should not be seated adjacent to S. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A)	2
(B)	3
(C)	4
(D)	5

Q.2	Consider the following sentences:The number of candidates who appear for the GATE examination is staggering.A number of candidates from my class are appearing for the GATE examination.The number of candidates who appear for the GATE examination are staggering.A number of candidates from my class is appearing for the GATE examination.Which of the above sentences are grammatically CORRECT?
(A)	(i) and (ii)
(B)	(i) and (iii)
(C)	(ii) and (iii)
(D)	(ii) and (iv)

Q.3	A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM.The immediate next time that they will beep together is__________
(A)	10.08 AM
(B)	10.42 AM
(C)	11.00 AM
(D)	10.00 PM

Q.4	If ⊕÷⊙= 2 ; ⊕÷ Δ = 3; ⊙ +Δ = 5; Δ ×⊗ =10,Then, the value of (⊗−⊕)², is:
(A)	0
(B)	1
(C)	4
(D)	16

Q.5	The front door of Mr. X’s house faces East. Mr. X leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another 50 m and stops. The direction of the point Mr. X is now located at with respect to the starting point is_______
(A)	South-East
(B)	North-East
(C)	West
(D)	North-West

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 1 and 2, and two conclusions I and II. Statement 1: All entrepreneurs are wealthy.Statement 2: All wealthy are risk seekers. Conclusion I: All risk seekers are wealthy.Conclusion II: Only some entrepreneurs are risk seekers.Based on the above statements and conclusions, which one of the following options is CORRECT?
(A)	Only conclusion I is correct
(B)	Only conclusion II is correct
(C)	Neither conclusion I nor II is correct
(D)	Both conclusions I and II are correct

Q.7	A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is
(A)	3/16
(B)	45/236
(C)	1/4
(D)	3/4

Q.8	The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is______
(A)	1/8
(B)	1/6
(C)	1/4
(D)	1/2

Q.9	Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to________
(A)	1/4
(B)	1/8
(C)	1/16
(D)	1/32

Q.10	The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to weak demand.In relation to the first sentence above, what does the second sentence do?
(A)	Restates an idea from the first sentence.
(B)	Second sentence entirely contradicts the first sentence.
(C)	The two statements are unrelated.
(D)	States an effect of the first sentence.

Metallurgical Engineering (MT)

Q.1– Q.13 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	For the matrix given below, the eigenvalues are:
(A)	0, 2, 2
(B)	1, 1, 2
(C)	0, 1, 2
(D)	0, 1, 3

Q.2	Which one of the following is a homogeneous function of degree three?
(A)	𝑥³ + 2𝑥²𝑦²
(B)	𝑦²𝑥 + 2𝑦𝑥²
(C)	𝑦³ + 2𝑥²
(D)	𝑥𝑦² + 3𝑥𝑦

Q.3	The divergence of a vector field ^⃗𝑽^→(𝒙, 𝒚, 𝒛), where its three components (𝑽_𝒙 ,𝑽_𝒚, 𝑽_𝒛) are functions of x,y,z, is:
(A)
(B)
(C)
(D)

Q.5	Single crystal turbine blades of nickel-based superalloys for aero-engines are manufactured using:
(A)	Investment casting
(B)	Die casting
(C)	Squeeze casting
(D)	Directional solidification

Q.6	Elements A and B have the same crystal structure. For a dilute solution of B in A, which one of the following is true?(Given: Δ𝑯_𝒎𝒊𝒙 – Mixing enthalpy, a_B– Activity of B and X_B– Mole fraction of B)
(A)	𝐼𝑓 Δ𝐻_𝑚𝑖𝑥 = 0, then 𝑎_𝐵 < 𝑋_𝐵
(B)	𝐼𝑓 Δ𝐻_𝑚𝑖𝑥 = 0, then 𝑎_𝐵 > 𝑋_𝐵
(C)	𝐼𝑓 Δ𝐻_𝑚𝑖𝑥 > 0, then 𝑎_𝐵 < 𝑋_𝐵
(D)	𝐼𝑓 Δ𝐻_𝑚𝑖𝑥 < 0, then 𝑎_𝐵 < 𝑋_𝐵

Q.7	For uniaxial tensile stress-strain behaviour of polycrystalline aluminium, which one of the following statements is FALSE?
(A)	True stress is always higher than the engineering stress.
(B)	At the ultimate tensile stress point on the true stress – strain curve, 𝑑σ/𝑑ε = 0
(C)	Resilience is the area under the elastic region of the engineering stress – strain curve.
(D)	Maximum true stress does not correspond to the maximum load.

Q.8	Which one of the following is FALSE for creep deformation?
(A)	The minimum creep rate is obtained in the primary stage (stage I).
(B)	Creep resistance decreases with decrease in grain size.
(C)	Coble creep occurs via grain boundary diffusion.
(D)	Nabarro-Herring creep occurs via lattice diffusion.

Q.9	Which one of the following elements alloyed with iron is a ferrite stabilizer?
(A)	Nickel
(B)	Manganese
(C)	Carbon
(D)	Silicon

Q.10	Which one of the following is the correct decreasing sequence of Quenching Power for quenchants used in heat treatment of steels?
(A)	Oil > Water > Brine > Air
(B)	Brine > Oil > Water > Air
(C)	Brine > Water > Oil > Air
(D)	Water > Brine > Oil > Air

Q.11	For a zeroth order chemical reaction, which one of the following is FALSE?
(A)	Concentration versus time plot is a straight line.
(B)	Increase in concentration of reacting species increases the rate of reaction.
(C)	Half-life depends on the initial concentration and zero-order rate constant.
(D)	Rate of reaction depends on temperature.

Q.12	Which one of the following elements oxidizes first in basic oxygen steel making process?
(A)	Silicon
(B)	Carbon
(C)	Manganese
(D)	Phosphorus

Q.13	Which one of the following is a hydrometallurgical operation?
(A)	Roasting
(B)	Leaching
(C)	Zone refining
(D)	Smelting

Q.14 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.14

Q.15

The grain size (X) of annealed specimens follows a symmetric distribution with a mean (µ) of 5 µm and a standard deviation (σ) of 0.5 µm. The percentage of specimens with grain size in the range 5 to 6 µm is expected to be: ________(round off to nearest integer).Given: For the symmetric distribution: Probability (𝐗 ≤ µ + 𝟐σ) = 𝟎. 𝟗𝟖

Q.16

If 𝑬^𝒐_{𝑵𝒊𝟐+/𝑵𝒊} − 0.25 V, the value of µ^𝒐_𝑵𝒊 _𝟐+ (in J mol^-1) at 298 K is:______(round off to nearest integer).Given: F = 96500 C mol^-1

Q.17

Melting point of Cu is 1358 K and its enthalpy of melting is 13400 J mol^-1. The value of free energy change (in J mol^-1) for liquid to solid transformation at 1058 K is: ________(round off to nearest integer).Assume: 𝑪_𝑷^{𝒍𝒊𝒒𝒖𝒊𝒅}= 𝑪_𝑷^{𝒔𝒐𝒍𝒊𝒅}

Q.18

A body is subjected to a state of stress given by the following stress tensor:

If yielding is predicted by the Tresca Criterion, the uniaxial tensile yield stress (in MPa) of the body should be less than or equal to:_______(round off to nearest integer).

Q.19

Consider homogeneous nucleation of a spherical solid in liquid. For a given undercooling, if surface energy of a nucleus increases by 20 %, the corresponding increase (in percent) in the critical radius of the nucleus is:_______(round off to nearest integer).

Q.20

If saturation magnetization of iron at room temperature is 1700 kA m^-1, the magnetic moment (in A m²) per iron atom in the crystal is: _____ × 10^-23(round off to 1 decimal place).(Given: Lattice parameter of iron at room temperature = 0.287 nm)

Q.21

In the X-ray diffraction pattern of a FCC crystal, the first reflection occurs at a Bragg angle (θ) of 30°. The Bragg angle (in degree) for the second reflection will be: ____(round off to 1 decimal place).

Q.22

A 0.6 wt.% C steel sample is slowly cooled from 900 °C to room temperature. The fraction of proeutectoid ferrite in the microstructure is:______-(round off to 2 decimal places). Given: Eutectoid composition: 0.8 wt.% CMaximum solubility of carbon in α-Fe: 0.025 wt.% C

Q.23

If the degree of polymerization of polyethylene is 30000, the average molecular weight (in g mol^-1) is: (round off to nearest integer).(Given: Atomic weights of carbon and hydrogen are 12 and 1, respectively)

Q.25

The vacancy concentration in a crystal doubles upon increasing the temperature from 27 °C to 127 °C. The enthalpy (in kJ mol^-1) of vacancy formation is: ______ (round off to 2 decimal places).Given: R = 8.314 J mol^-1K^-1

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

Q.26	The minimum value of y for the equation 𝒚 = 𝒙^𝟐 − 𝟐𝒙 + 𝟒 is .
(A)	3
(B)	1
(C)	4
(D)	6

Q.29	Number of degrees of freedom for the following reacting system is: M (s) + CO2 (g) = MO (s) + CO (g)
(A)	0
(B)	1
(C)	2
(D)	3

Q.31	In the absence of any external stress, which one of the following statements related to the interaction of point defect and a dislocation is FALSE:
(A)	An oversized solute atom would preferentially migrate below the slip plane of an edge dislocation.
(B)	A spherically symmetric point defect can interact with both the hydrostatic and shear stress fields of a dislocation.
(C)	A point defect can locally modify the elastic modulus and thereby can change the interaction energy.
(D)	Vacancies are attracted towards the compressive region of dislocation.

Q.32	A single crystal aluminium sample is subjected to uniaxial tension along[112] direction. If the applied tensile stress is 100 MPa and the critical resolved shear stress (CRSS) is 25 MPa, which one of the following slip systems will be activated?
(A)	[1^̅01](111)
(B)	[1^̅10](111)
(C)	[101](111^̅)
(D)	[011](111^̅)

Q.34	For a fully developed 1-D flow of a Newtonian fluid through a horizontal pipe of radius R (see figure), the axial velocity (𝒗_𝒛) is given by: where, Δ𝑷 is the pressure difference (𝑷𝟏 – 𝑷𝟐), μ is the viscosity, 𝒓 is the radial distance from the axis and L is the length of the tube. The shear stress exerted by the fluid on the tube wall is:
(A)	∆𝑃𝑅 /2𝐿
(B)	∆𝑃𝑅/𝐿
(C)	3∆𝑃𝑅/2𝐿
(D)	2∆𝑃𝑅/𝐿

Q.36	A blast furnace uses hematite ore with 80% Fe2O3 and 20% gangue materials. It uses 600 kg coke per ton of hot metal. The coke contains 85% C and 15% ash. The composition of hot metal is 95.5% Fe and 4.5% C.The weight of iron ore used and slag produced per ton of hot metal respectively, are:Given: Atomic weight: O = 16, C = 12, N = 14, Fe = 56 All the compositions are in wt.%.1 ton = 1000 kgAssume that the gangue materials of the ore and ash content of coke form slag while Fe2O3 in the ore is consumed in making hot metal.
(A)	1705 kg, 431 kg
(B)	2131 kg, 546 kg
(C)	1705 kg, 331 kg
(D)	1500 kg, 431 kg

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

Q.37

Consider the function (𝒙) = 𝒙 − 𝐜𝐨𝐬 𝒙. Using Newton-Raphson method, the estimated root of (𝒙) after the first iteration is: (round off to 3 decimal places).Assume: Initial guess of the root = 0.5 radians.

Q.38

The work done by a force ^⃗𝑭^→= 𝟐𝒙𝒊 + 𝟑𝒚𝒋 along a straight line from point (0, 0) to (1, 2) is: ________(round off to nearest integer).

Q.39

A coin is tossed three times. Given that there are more heads than tails, the probability of getting exactly one tail is: __________(round off to 2 decimal places).

Q.40

A continuous fillet weld is made using a 3000 W welding machine. At a travel speed of 6 mm s^-1, the cross-sectional area (in mm²) of the weld is:______(round off to nearest integer).Given: The unit energy required to melt the metal is 6 J mm^-3.Heat transfer factor = 0.6 Melting factor = 0.5

Q.41

Liquid iron is cast into a spherical sand mold (6 cm radius) and a cubical sand mold (12 cm edge length). If solidification time is 60 minutes in the spherical casting, the time (in minutes) required to solidify in the cubical casting is: _________(round off to nearest integer).

Q. 42

True strain for 60% height reduction of a sample subjected to hot forging is: ______(round off to 2 decimal places).

Q.43

For the equilibrium reaction: 𝟐𝑪𝒖 (𝒔) + 𝑺𝑶_𝟐(𝒈) = 𝑪𝒖_𝟐𝑺 (𝒔) + 𝑶_𝟐 (𝒈), thevalue of 𝒍𝒏

at 973 K is:_____(round off to 2 decimal places).Given: 𝟐𝑪𝒖 (𝒔) + 𝟎. 𝟓𝑺_𝟐(𝒈) = 𝑪𝒖_𝟐𝑺 (𝒔) ΔG^oat 973 K = -100 kJ𝑺𝑶_𝟐(𝒈) = 𝟎. 𝟓𝑺_𝟐(𝒈) + 𝑶_𝟐(𝒈) ΔG^oat 973 K = 292 kJ R = 8.314 J mol^-1K^-1Assume: Cu and Cu2S are pure solids.

Q.44

One mole of an ideal gas at 10 atm. and 300 K undergoes reversible adiabatic expansion to a pressure of one atm. The work done (in Joule) by the gas is: ___________(round off to nearest integer).Given: R = 8.314 J mol^-1K^-1; 1 atm. = 101325 Pa; Cp = 2.5R

Q.47

In a material, a shear stress of 100 MPa is required to bow a dislocation line between precipitates with a spacing of 0.2 µm. If the spacing between the precipitates is increased to 0.5 µm, the shear stress (in MPa) to bow the dislocation would be: ____________(round off to nearest integer).

Q.48

A metal plate is in a state of plane strain (ε_𝒛𝒛 = 𝟎) with σ_𝒙𝒙 = σ_𝒚𝒚≠ 𝟎 andτ_𝒙𝒚 = τ_𝒙𝒛 = τ_𝒚𝒛 = 𝟎. If the Poisson’s ratio is 0.3, the ratio, σ_𝒛𝒛/σ_𝒙𝒙 is________(round off to 1 decimal place).

Q.49

An infinite metal plate has a central through-thickness crack of length 80/πmm. The maximum applied stress (in MPa) that the plate can sustain in mode I is: ________________(round off to nearest integer). Assume: Linear elastic fracture mechanics is validGiven: Fracture toughness, 𝑲_𝑰𝑪= 20 MPa m^1/2

Q.52

At 25 °C, iron corrodes in a deaerated acid of pH 3 with a corrosion current density of 4 µA cm^-2. The corrosion potential (V) is:_____ (round off to 2 decimal places).Given: βc = 0.1 V per decade of current densityExchange current density of hydrogen on iron surface = 10^-9A cm^-2R = 8.314 J mol^-1K^-1, F = 96500 C mol^-1All potentials are with reference to standard hydrogen electrode.

Q.53

The radius of an interstitial atom which just fits (without distorting the structure) inside an octahedral void of a bcc-iron crystal (in nm) is:______(round off to 3 decimal places).Assume the radius of Fe atom to be 0.124 nm.

Q.54

Nickel undergoes isothermal oxidation at 800 K for a duration of 400 s resulting in a weight gain of 2 mg cm^-2. The weight gain (mg cm^-2) after a duration of 1600 s is: _________(round off to nearest integer).Assume: Weight gain is proportional to square root of time.

Answer Key

Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans
1	B	1	C	11	B	21	34.8 to 36.1	31	B	41	58 to 62	51	0.65 to 0.75
2	A	2	B	12	A	22	0.22 to 0.30	32	A	42	0.90 to 0.94 OR -0.94 to -0.90	52	−0.60 to −0.50
3	A	3	A	13	B	23	840000 to 840000	33	A	43	−24.00 to − 23.50	53	0.017 to 0.023
4	B	4	D	14	25 to 25	24	0.53 to 0.59	34	A	44	2230 to 2270	54	4 to 4
5	D	5	D	15	48 to 48	25	6.85 to 7.00	35	A	45	63.00 to 71.00	55	0.74 to 0.76
6	C	6	D	16	-48251 to – 48240	26	A	36	A	46	0 to 0
7	B	7	B	17	-2962 to -2958	27	B	37	0.745 to 0.770	47	40 to 40
8	C	8	A	18	150 to 150	28	C	38	7 to 7	48	0.6 to 0.6
9	B	9	D	19	20 to 20	29	C	39	0.75 to 0.75	49	98 to 102
10	D	10	C	20	1.7 to 2.3	30	B	40	25 to 25	50	298 to 302

GATE 2021 Mining Engineering Previous Year Paper

General Aptitude (GA)

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

Q.1	Arun and Aparna are here. Arun and Aparna is here. Arun’s families is here.Arun’s family is here. Which of the above sentences are grammatically CORRECT?
(A)	(i) and (ii)
(B)	(i) and (iv)
(C)	(ii) and (iv)
(D)	(iii) and (iv)

Q.3	Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
(A)	1/36
(B)	1/12
(C)	1/8
(D)	1/4

Q.4	⊕ and ⊙ are two operators on numbers p and q such that𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p ⊕ q = px qThen, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=
(A)	40
(B)	-26
(C)	-33
(D)	-40

Q.5	Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A)	6
(B)	9
(C)	18
(D)	24

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

Q.6	On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.The distance of point Q to point O, in the same units, should be
(A)	3
(B)	4
(C)	5
(D)	6

Q.7	The author said, “Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk on to the stage and start speaking. In my opinion, it is no less important for public speakers to rehearse their talks.”Based on the above passage, which one of the following is TRUE?
(A)	The author is of the opinion that rehearsing is important for musicians, actors and public speakers.
(B)	The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors.
(C)	The author is of the opinion that rehearsing is more important only for musicians than public speakers.
(D)	The author is of the opinion that rehearsal is more important for actors than musicians.

Q.8	Some football players play cricket.All cricket players play hockey.Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
(A)	No football player plays hockey.
(B)	Some football players play hockey.
(C)	All football players play hockey.
(D)	All hockey players play football.

Q.10	In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum area of the triangle PQR possible, in cm², is
(A)	18
(B)	24
(C)	48√3
(D)	144√3

Mining Engineering (MN

– Q.19 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	Tricone roller bit is used with
(A)	down-the-hole hammer.
(B)	Jack hammer.
(C)	rotary-percussive drill.
(D)	rotary drill.

Q.2	Resuing method of mining is practiced for
(A)	thick vein deposit.
(B)	massive shallow deposit.
(C)	narrow vein deposit.
(D)	massive deep-seated deposit.

Q.3	The equipment used for both drop cut and terrace cut in surface mining is
(A)	surface miner.
(B)	shovel.
(C)	dragline.
(D)	bucket wheel excavator.

Q.4	Surface miner does NOT have a
(A)	differential gear for turning.
(B)	tensioning arrangement for crawler.
(C)	scraper plate behind the drum.
(D)	pick cooling system.

Q.5	Induced blasting enhances production in
(A)	sublevel stoping.
(B)	block caving.
(C)	cut and fill mining.
(D)	shrinkage stoping.

Q.6	The measures of dispersion of a dataset are
(A)	standard deviation, range and mode.
(B)	standard deviation, range and interquartile range.
(C)	variance, range and median.
(D)	interquartile range, median and mode.

Q.7	NONEL is used as down-the-hole initiator to
(A)	avoid generation of air overpressure.
(B)	provide precise delay.
(C)	avoid deflagration of column charge.
(D)	reduce ground vibration.

Q.8	The vectors ⃗𝒂→ and ^⃗𝒃^→act in a plane as shown below. The magnitude of the vector⃗𝒄→ = ( → +) × (→ −) is
(A)	zero.
(B)	half to the area bounded by the vectors → and.
(C)	equal to the area bounded by the vectors → and.
(D)	twice the area bounded by the vectors → and.

Q.9	As per MMDR Act 1957, for the allocation of lease of minor minerals
(A)	the State Government is authorised to give the permit.
(B)	the Central Government is authorised to give the permit.
(C)	the State Government is authorised to give permit but with the consent of Central Government.
(D)	the Central Government is authorised to give permit but with the consent of State Government.

Q.10	In photogrammetry, the ‘Tilt of a photograph’ refers to the angle between the
(A)	lines joining the opposite fiducial marks of a photograph.
(B)	normal to the plane of photograph and optical axis.
(C)	vertical and the axis of the flight.
(D)	vertical and optical axis of the camera.

Q.11	The hydraulic sand stowing pipeline layout should be such that
(A)	the geometric profile must coincide with the hydraulic gradient line.
(B)	the hydraulic profile should always be below the hydraulic gradient line.
(C)	the hydraulic profile should always be above the hydraulic gradient line.
(D)	the geometric profile should always be above the hydraulic gradient line.

Q.12	For a “positive definite” square matrix, the TRUE statement is
(A)	the matrix is singular.
(B)	all the eigen values of the matrix are greater than zero.
(C)	all the eigen values of the matrix are zero.
(D)	some of the eigen values can be less than zero.

Q.13	The standard normal distribution is a
(A)	non-parametric distribution.
(B)	single parameter distribution.
(C)	two-parameter distribution.
(D)	three-parameter distribution.

Q.14	Variance of the sum of two statistically independent random variables 𝑿and 𝒀, 𝝈^𝟐 _𝑿+𝒀, is
(A)	𝜎²𝑋 + 𝜎²𝑌.
(B)	𝜎²_𝑋 + 𝜎²_𝑌 + 2𝜎_𝑋𝑌.
(C)	𝜎²_𝑋 + 𝜎²_𝑌 + 𝜎_𝑋𝑌.
(D)	𝜎²_𝑋 + 𝜎²_𝑌− 2𝜎_𝑋𝑌.

Q.15	The difference between depreciation and amortization allowances in tax calculation is that
(A)	depreciation is for a tangible asset applicable on its declared life; whereas amortization is for an intangible asset applicable on a specified period.
(B)	depreciation is for an intangible asset applicable on its declared life; whereas amortization is for a tangible asset applicable on a specified period.
(C)	depreciation is for a tangible asset applicable on a specified period; whereas amortization is for an intangible asset applicable on its declared life.
(D)	depreciation is for an intangible asset applicable on a specified period; whereas amortization is for a tangible asset applicable on its declared life.

Q.16	Owning cost of a machine does NOT include
(A)	purchase price.
(B)	insurance.
(C)	interest.
(D)	operating cost.

Q.17	Folds are the structural features resulting from
(A)	ductile deformation of earth crust.
(B)	brittle deformation of earth crust.
(C)	high impact tectonic stresses of earth crust.
(D)	fracturing of earth crust.

Q.20 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).


Q.20	A coal sample having moisture content of 8.0% has unit weight 15.6 kN/m³. The dry unit weight of the sample, in kN/m³is_________ .[*round off to 2 decimal places*]

Q.21	The value of the integral 𝑰 =computed using Simpson’s 1/3 rule with 2 sub intervals is _________.[*round off to 3 decimal places*]

Q.22	In the context of sound frequency analysis, the lower and upper frequencies of a 1/1 octave band are 710 Hz and 1420 Hz respectively. The corresponding centre frequency of the band in Hz, is__________ *. [round off to the nearest integer***]

Q.23	In Battle Environmental Evaluation System (BEES) of Environmental Impact Assessment (EIA), “air pollution” has a Parameter Importance Unit (PIU) value of 52. The Environmental Quality (EQ) score of a project with respect to air pollution was 0.8 before the project implementation and it becomes 0.6 after the project implementation. The difference in the “Environmental Impact Unit (EIU)”, before and after the project implementation is _________ .[*round off to 2 decimal places*]

Q.24	A system consists of four components connected functionally in a parallel configuration. The reliability of the individual components is 0.40, 0.60, 0.50 and 0.40. The system reliability, is ______ *. [round off to 3 decimal places***]

Q.25	A vehicle is moving at a speed of 12 m/s on a level road. It applies emergency brakes and starts to skid without rolling in a straight path. The deceleration of the vehicle is constant after braking and it comes to rest at a distance of 15 m. Assuming, g = 10 m/s², the coefficient of kinetic friction between the tyres and road is __. [*round off to 2 decimal places*]

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

Q.26	In a bord and pillar panel six shuttle cars, each of 10 tonne capacity, are deployed to transport coal produced by two continuous miners to a belt conveyor. Each shuttle car on an average carries 80% of its rated capacity and makes 7 round trips in an hour. The belt conveyor has a capacity such that the effective material cross section area is of 0.09 m²and runs at a speed 1.1 m/s. The broken coal has a bulk density of 1.2 tonne/m³. The ratio between the production and the capacity of the belt conveyor, in percent is
(A)	65.46
(B)	71.42
(C)	78.56
(D)	82.46

Q.30	In a longwall panel, face is supported with shields of yield capacity 460 tonne per shield. The distance from the canopy tip to coal face is 0.15 m when the support is fully advanced. The depth of web is 0.60 m. The shields are set skin to skin at the face. Length of the canopy of the shield is 3.25 m and width 1.5 m. Setting capacity is 80% of the yield capacity. The setting resistance at the maximum and minimum span of the coal face, in tonne/m², respectively are
(A)	61.33 and 72.15.
(B)	63.72 and 75.48.
(C)	76.66 and 90.19.
(D)	91.99 and 108.22.

Q.31	A 10 ml sample of wastewater is diluted with water having no BOD, to fill a 300 ml BOD bottle. The initial DO of the diluted waste water is 9.0 mg/l. If the BOD5 of the waste water sample is 60 mg/l, the final DO of the diluted waste water in mg/l, is
(A)	5.0
(B)	6.0
(C)	7.0
(D)	8.0

Q.35	The value of 𝐥𝐢𝐦 (𝒙 √𝒙^𝟐 + 𝒃^𝟐 − √𝒙^𝟒 + 𝒃^𝟒 ) is
(A)	0.
(B)	b²/2
(C)	∞.
(D)	𝑏².

Q.36	In order to check whether iron ore is supplied to the specification of 62% Fe, a steel company has conducted a hypothesis test with the null hypothesis as H0: µFe = 62% and alternative hypothesis Ha: µFe < 62%. A random sample of 5 observations reveal the following grade values of the lot, 58%, 56%, 60%, 64%, 62%. The t-test statistic for the hypothesis is
(A)	– 3.000
(B)	1.414
(C)	– 1.414
(D)	3.000

Q.40	On an old plan of scale 1:1000, leasehold area of a mine is now measured as 802 cm²using a planimeter. The plan is found to have shrunk, such that the original line of 10 cm is now measured as 9.8 cm on the plan. True lease hold mine area, in m², is *_______. [round off to the nearest integer***]

Q.41	CO is released from a point source on a level ground at a rate of 25 g/s. The average wind speed is 5 m/s. The dispersion coefficients are 150 m and 200 m in horizontal and vertical directions, respectively, at a receiver station located on the ground along the downwind direction. Assuming the plume follows Gaussian dispersion model, the concentration of CO, in µg/m³, at the station is _________ .[*round off to 2 decimal places*]

Q.42	Assume that COVID-19 growth rate of number of infections per day (𝒄) in a certain population is represented by the following differential equation.𝟏𝟎𝟎dc/𝒅𝒕− 𝟕 𝒄 = 𝟎Where, 𝒕 stands for time in days. Time taken for the number of infections per day to double, in days, is *_______. [round off to the nearest integer***]

Q.43	Ore is hoisted from 620 m depth using a single skip of 7 tonne pay load. The skip winding system has constant acceleration/deceleration of 1 m/s²and a constant speed of 10 m/s. The skip loading time and unloading time are 120 s and 60 s, respectively. Considering the overall utilization of the skip as 70 %, the maximum daily capacity of the winding system, in tonne, is ___________ *. [round off to the nearest integer***]

Q.44	In an analysis of fragmented blast muck, the mean fragment size is found to be 60 cm with uniformity index of 1.25. Considering Rosin-Ramler equation, the cumulative mass fraction, in percent, to pass the grizzly screen size of 100 cm is _______. [*round off to 2 decimal places*]

Q.45

A single-acting reciprocating ram pump, while running at 120 rpm, delivers water at a rate of 10 litres per second. Considering the ram diameter is 150 mm and stroke length is 300 mm, the volumetric efficiency of the pump, in percent is _______ .[round off to one decimal place]

Q.46

In a sand stowing arrangement, the slurry has a sand concentration of 35% by volume. The specific gravity of sand grain is 2.6. The concentration of sand by weight, in percent, in the slurry is ___________ . [round off to one decimal place]

Q.49

A random variable X is defined by

The value of (𝑿^𝟐) is _______. [round off to one decimal place]

Q.51

Rate of fuel consumption 𝒇_𝒄 (litres per hour) of a truck varies with truck speed 𝒙, (kmph) as given below𝒇_c = 𝟐𝟎 + 𝒙^𝟐/𝟓𝟎The fuel price is Rs. 70 per litre. Other costs amount to Rs. 500 per hour. If the truck travels 100 km from a coal mine to a thermal plant, the speed of the truck, in kmph, that minimizes the total cost is _________. [round off to one decimal place]

Q.54

A coal mine operating in three shifts produces 400 tonnes of coal per day with a face OMS of 1.0 from panel A, and 200 tonnes of coal with face OMS of 1.0 from panel B. The panel A and panel B are in parallel with resistance0.6 Ns^-2m^-8and 0.5 Ns^-2m^-8, respectively. If the panels are supplied with minimum permissible quantity as per CMR 2018, the requisite regulator resistance to meet the conditions in Ns^-2m^-8is _______. [round off to 2 decimal places]

Q.55

A set of three steel bars of equal cross-sectional area of 0.01 m²are loaded, as shown in the figure. The elastic modulus of steel is 200 GPa. The overall change of length of the complete set of bars, in mm, is _______ . [round off to 3 decimal places]

Answer Key

Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans
1	B	1	D	11	B	21	5.000 to 5.200	31	C	41	50.00 to 55.00	51	35.0 to 38.0
2	B	2	C	12	B	22	999 to 1005	32	A	42	9 to 11	52	112000 to 112000
3	D	3	D	13	C	23	10.40 to 10.40	33	D	43	1250 to 1350	53	6.0 to 6.0
4	D	4	A	14	A	24	0.918 to 0.938	34	D	44	71.00 to 75.00	54	1.80 to 2.00
5	C	5	B	15	A	25	0.45 to 0.50	35	B	45	92.0 to 96.0	55	0.045 to 0.055
6	C	6	B	16	D	26	C	36	C	46	57.00 to 59.80
7	A	7	C	17	A	27	A	37	35.50 to 38.30	47	9.00 to 11.00
8	B	8	D	18	C	28	B	38	1.20 to 1.35	48	2.20 to 2.30
9	C	9	A	19	B	29	B	39	22.5 to 24.0	49	6.0 to 6.0
10	D	10	D	20	14.00 to 15.00	30	A	40	83450 to 83550	50	5 to 5

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GATE 2021 Statistics Previous Year Paper

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Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

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

Mining Engineering (MN

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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

Q.9	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

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

Q.1	Let 𝑿 be a non-constant positive random variable such that (𝑿) = 𝟗. Then which one of the following statements is true?
(A)
(B)
(C )
(D)

Q.5
(A)	𝑆 has a monotone likelihood ratio in 𝑇₁ and 𝐻₀ is rejected if 𝑇₁ > Xn,𝛼2
(B)	𝑆 has a monotone likelihood ratio in 𝑇₁ and 𝐻₀ is rejected if 𝑇₁ > Xn,1-𝛼2
(C)	𝑆 has a monotone likelihood ratio in 𝑇₂ and 𝐻₀ is rejected if 𝑇₂ > Xn,𝛼2
(D)	𝑆 has a monotone likelihood ratio in 𝑇₂ and 𝐻₀ is rejected if 𝑇₂ > Xn,1- 𝛼2

Q.6
(A)	Under 𝐻₀, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 20/3
(B)	Under 𝐻₀, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 20/3
(C)	Under 𝐻₀, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 16/3
(D)	Under 𝐻₀, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 16/3

Q.7
(A)	4Φ_𝜌(0,0) − 1
(B)	4Φ_𝜌(0,0)
(C)	4Φ_𝜌(0,0) + 1
(D)	Φ_𝜌(0,0)

Q.8
(A)
(B)
(C)
(D)

Q.9
(A)	𝑍 follows 𝑊₄(7, 𝐼₄) distribution
(B)	𝑍 follows 𝑊₄(4, 𝐼₄) distribution
(C)	𝑍 follows 𝑊₇(4, 𝐼₇) distribution
(D)	𝑍 follows 𝑊7(7, 𝐼7) distribution

Q.10

Q.11	Let Then the value of 𝒆^𝑰+𝝅 equals (round off to 𝟐 decimal places).

Q.12

Q.13

Q.14	Let 𝑨 and 𝑩 be two events such that (𝑩) = ¾ and 𝑷(𝑨 𝖴 𝑩^𝒄) = ½. If 𝑨 and 𝑩 are independent, then 𝑷(𝑨) equals_________(round off to𝟐 decimal places).

Q.16

Q.17	If the marginal probability density function of the 𝒌^𝒕𝒉 order statistic of a random sample of size 8 from a uniform distribution on [𝟎, 𝟐] is then 𝒌 equals

Q.18

Q.20	Let {𝑿_𝒏}_𝒏≥𝟎 be a time-homogeneous discrete time Markov chain with state space {𝟎, 𝟏} and transition probability matrix

Q.21	Let {𝟎, 𝟐} be a realization of a random sample of size 𝟐 from a binomial distribution with parameters 𝟐 𝐚𝐧𝐝 𝒑, where 𝒑 ∈ (𝟎, 𝟏). To test 𝑯₁ : 𝒑 = ½ against 𝑯₀ : 𝒑 ≠ ½, the observed value of the likelihood ratio test statistic equals_________(round off to 𝟐 decimal places).

Q.22

Q.39
(A)	𝑇/𝑛²
(B)	T/n
(C)	(𝑛 + 1)𝑇/2𝑛
(D)	(𝑛 + 1)²𝑇/4𝑛²

Q.23

Q.24	Let 𝑿_𝟏, 𝑿_𝟐 and 𝑿_𝟑 be a random sample from a bivariate normal distribution with unknown mean vector 𝝁 and unknown variance- covariance matrix 𝚺, which is a positive definite matrix. The 𝒑-value corresponding to the likelihood ratio test for testing 𝑯_𝟎: 𝝁 = 𝟎 against 𝑯₁: 𝝁 = 𝟎 based on the renzation 12, 4-2, -50 sample equals_________(round off to 𝟐 decimal places).

Q.26
(A)	𝑓 is not continuous at 0
(B)	𝑓 is not differentiable at 0
(C)	𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 0
(D)	𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 1

Q.29	Let 𝑴 be the collection of all × 𝟑 real symmetric positive definite matrices. Consider the set where 𝟎 denotes the 𝟑 × 𝟑 zero matrix. Then the number of elements in𝑺 equals
(A)	0
(B)	1
(C)	8
(D)	∞

Q.34	Let (𝑿, 𝒀) have the joint probability density function Then which one of the following statements is NOT true?
(A)
(B)	𝑃(𝑋 + 𝑌 > 4) = ¾
(C)	𝐸(𝑋 + 𝑌) = 4 log_𝑒 2
(D)	𝐸(𝑌 \| 𝑋 = 2) = 4

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

Q.11	Match the measuring feature with the corresponding measuring instrument in the table below.
(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.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^-1at the fixed end is
(A)	1.10
(B)	0.02
(C)	0.09
(D)	0.05

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
(A)	4935
(B)	2000
(C)	5167
(D)	1238

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 ν = 0.33 are press fitted upon one another. If radial interference δ = 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

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. 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.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. The maximum profit the company can make is
(A)	45
(B)	48
(C)	42
(D)	54

Q.9	In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.The probability that any point picked randomly within the square falls in the shaded area is___________
(A)	4-π/2
(B)	1/2
(C)	π/2-1
(D)	𝜋/4

Q.3	As shown in the figure, X-ray diffraction pattern is obtained from a diatomic chain of atoms 𝑷 and 𝑸. The diffraction condition is given by 𝒂 𝐜𝐨𝐬 𝜽 = 𝒏𝝀, where 𝒏 is the order of the diffraction peak. Here, 𝒂 is the lattice constant and 𝝀 is the wavelength of the X-rays. Assume that atomic form factors and resolution of the instrument do not depend on 𝜽. Then, the intensity of the diffraction peaks is
(A)	lower for even values of 𝑛, when compared to odd values of 𝑛
(B)	lower for odd values of 𝑛, when compared to even values of 𝑛
(C)	zero for odd values of 𝑛
(D)	zero for even values of 𝑛

Q.6	Consider a solenoid of length 𝑳 and radius 𝑹, where 𝑹 ≪ 𝑳. A steady-current flows through the solenoid. The magnetic field is uniform inside the solenoid and zero outside. Among the given options, choose the one that best represents the variation in the magnitude of the vector potential, (𝟎, 𝑨_𝝋, 𝟎) at 𝒛 = 𝑳/𝟐, as a function of the radial distance (𝒓*) in cylindrical coordinates.Useful information: The curl of a vector* ^⃗𝑭^→, in cylindrical coordinates is**
(A)
(B)
(C)
(D)

Q.11	As shown in the figure, an electromagnetic wave with intensity 𝑰_𝑰 is incident at the interface of two media having refractive indices 𝒏_𝟏 = 𝟏 and 𝒏_𝟐 = √𝟑. The wave is reflected with intensity 𝑰_𝑹 and transmitted with intensity 𝑰_𝑻. Permeability of each medium is the same. (Reflection coefficient 𝑹 = 𝑰_𝑹/𝑰_𝑰 and Transmission coefficient 𝑻 = 𝑰_𝑻/𝑰_𝑰). Choose the correct statement(s).
(A)	𝑅 = 0 if 𝜃_𝐼 = 0^o and polarization of incident light is parallel to the plane of incidence.
(B)	𝑇 = 1 if 𝜃_𝐼 = 60^o and polarization of incident light is parallel to the plane of incidence.
(C)	𝑅 = 0 if 𝜃_𝐼 = 60^o and polarization of incident light is perpendicular to the plane of incidence.
(D)	𝑇 = 1 if 𝜃_𝐼 = 60^o and polarization of incident light is perpendicular to the plane of incidence.

Q.16	If 𝒚_𝒏(𝒙) is a solution of the differential equation𝒚^′′ − 𝟐𝒙𝒚^′ + 𝟐𝒏𝒚 = 𝟎where 𝒏 is an integer and the prime ( ′) denotes differentiation with respect to 𝒙, then acceptable plot(s) of 𝝍_𝒏(𝒙) = 𝒆^−𝒙 ^⁄𝟐 𝒚_𝒏(𝒙), is(are)
(A)
(B)
(C)
(D)

Q.20	A hoop of mass 𝑴 and radius 𝑹 rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass 𝒎 slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is_______.

Q.21	Three non-interacting bosonic particles of mass 𝒎 each, are in a one- dimensional infinite potential well of width 𝒂. The energy of the third excitedstate of the system is 𝒙 ×. The value of 𝒙 (in integer) is _________.

Q.22	The spacing between two consecutive S-branch lines of the rotational Raman spectra of hydrogen gas is 𝟐𝟒𝟑. 𝟐 𝐜𝐦^−𝟏. After excitation with a laser of wavelength 𝟓𝟏𝟒. 𝟓 𝐧𝐦, the Stoke’s line appeared at 𝟏𝟕𝟔𝟏𝟏. 𝟒 𝐜𝐦^−𝟏 for a particular energy level. The wavenumber (rounded off to the nearest integer), in 𝐜𝐦^−𝟏, at which Stoke’s line will appear for the next higher energy level is______________ .

Q.23	The transition line, as shown in the figure, arises between^𝟐𝑫_𝟑/𝟐 and ^𝟐𝑷_𝟏/𝟐 states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is______.

Q.26	The above combination of logic gates represents the operation
(A)	OR
(B)	NAND
(C)	AND
(D)	NOR

Q.29	An electromagnetic wave having electric field 𝑬 = 𝟖 𝐜𝐨(𝒌𝒛 − ω𝒕) 𝒚̂ 𝐕 𝐜𝐦^−𝟏 is incident at 𝟗𝟎^°(normal incidence) on a square slab from vacuum (with refractive index 𝒏_𝟎 = 𝟏. 𝟎) as shown in the figure. The slab is composed of two different materials with refractive indices 𝒏_𝟏and 𝒏_𝟐. Assume that the permeability of each medium is the same. After passing through the slab for the first time, the electric field amplitude, in 𝐕 𝐜𝐦^−𝟏, of the electromagnetic wave, which emerges from the slab in region 2, is closest to
(A)	11/1.6
(B)	11/3.2
(C)	11/13.8
(D)	11/25.6

Q.30	Consider a point charge +𝑸 of mass 𝒎 suspended by a massless, inextensible string of length 𝒍 in free space (permittivity 𝗌_𝟎) as shown in the figure. It is placed at a height 𝒅 (𝒅 > 𝒍) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane. If 𝜽 represents the angular position and 𝒑_𝜽 its corresponding canonical momentum, then the correct Hamiltonian of the system is
(A)	(𝑝_𝜃²/2𝑚𝑙²)^_ [𝑄²/16𝜋𝜀₀(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(B)	(P_𝜃²/2𝑚𝑙²) – [𝑄²+ /8𝜋𝜀₀(𝑑 − 𝑙 cos 𝜃)]-𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(C)	(𝑝_𝜃²/2𝑚𝑙²) -[𝑄²/8𝜋𝜀₀(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(D)	(𝑝_𝜃²/2𝑚𝑙²) -[𝑄²/16𝜋𝜀₀(𝑑 − 𝑙 cos 𝜃)] + 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)