Vidhyarthi Darpan, Author at Vidhyarthi Darpan

GATE 2021 Mechanical Engineering Section-II 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.

Mechanical Engineering

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

Q.1	Consider an 𝒏 × 𝒏 matrix 𝑨 and a non-zero 𝒏 × 𝟏 vector 𝒑. Their product𝑨𝒑 = 𝑎^𝟐𝒑, where 𝑎 ∈ 𝕽 and 𝑎 ∉ {−𝟏, 𝟎, 𝟏}. Based on the given information, the eigen value of 𝑨^𝟐 is:
(A)	𝑎
(B)	_𝑎𝟐
(C)	√𝑎
(D)	_𝑎𝟒

Q.2	If the Laplace transform of a function 𝒇(𝒕) is given by, then 𝒇(𝟎) is
(A)	0
(B)	𝟏/𝟐
(C)	1
(D)	𝟑/𝟐

Q.3	The mean and variance, respectively, of a binomial distribution for **n independent trials with the probability of success as** p, are
(A)	√𝑛𝑝 , 𝑛𝑝(1 − 2𝑝)
(B)	√𝑛𝑝 , √𝑛𝑝(1 − 𝑝)
(C)	𝑛𝑝 , 𝑛𝑝
(D)	𝑛𝑝 , 𝑛𝑝(1 − 𝑝)

Q.4	The Cast Iron which possesses all the carbon in the combined form as cementite is known as
(A)	Grey Cast Iron
(B)	Spheroidal Cast Iron
(C)	Malleable Cast Iron
(D)	White Cast Iron

Q.5	The size distribution of the powder particles used in Powder Metallurgy process can be determined by
(A)	Laser scattering
(B)	Laser reflection
(C)	Laser absorption
(D)	Laser penetration

Q.6	In a CNC machine tool, the function of an interpolator is to generate
(A)	signal for the lubrication pump during machining
(B)	error signal for tool radius compensation during machining
(C)	NC code from the part drawing during post processing
(D)	reference signal prescribing the shape of the part to be machined

Q.7	The machining process that involves ablation is
(A)	Abrasive Jet Machining
(B)	Chemical Machining
(C)	Electrochemical Machining
(D)	Laser Beam Machining

Q.8	A PERT network has 9 activities on its critical path. The standard deviation of each activity on the critical path is 3. The standard deviation of the critical path is
(A)	3
(B)	9
(C)	27
(D)	81

Q.9	The allowance provided in between a hole and a shaft is calculated from the difference between
(A)	lower limit of the shaft and the upper limit of the hole
(B)	upper limit of the shaft and the upper limit of the hole
(C)	upper limit of the shaft and the lower limit of the hole
(D)	lower limit of the shaft and the lower limit of the hole

Q.10	In forced convective heat transfer, Stanton number (St), Nusselt number (Nu), Reynolds number (Re) and Prandtl number (Pr) are related as
(A)	St =Nu/Re Pr
(B)	St =Nu Pr/Re
(C)	St = Nu Pr Re
(D)	St =Nu Re/Pr

Q.11	For a two-dimensional, incompressible flow having velocity components 𝒖 and 𝒗 in the 𝒙 and 𝒚 directions, respectively, the expressioncan be simplified to
(A)
(B)
(C)
(D)

Q.12	Which of the following is responsible for eddy viscosity (or turbulent viscosity) in a turbulent boundary layer on a flat plate?
(A)	Nikuradse stresses
(B)	Reynolds stresses
(C)	Boussinesq stresses
(D)	Prandtl stresses

Q.13	A two dimensional flow has velocities in 𝒙 and 𝒚 directions given by 𝒖 =𝟐𝒙𝒚𝒕 and 𝒗 = −𝒚^𝟐𝒕, where 𝒕 denotes time. The equation for streamline passing through 𝒙 = 𝟏, 𝒚 = 𝟏 is
(A)	𝑥²𝑦 =1
(B)	𝑥𝑦² =1
(C)	𝑥²𝑦² =1
(D)	𝑥/𝑦² =1

Q.17	The von Mises stress at a point in a body subjected to forces is proportional to the square root of the
(A)	total strain energy per unit volume
(B)	plastic strain energy per unit volume
(C)	dilatational strain energy per unit volume
(D)	distortional strain energy per unit volume

Q.18	Value of ∫^𝟓^.^𝟐_𝟒 𝐥𝐧 𝒙 𝒅𝒙 using Simpson’s one-third rule with interval size 0.3 is
(A)	1.83
(B)	1.60
(C)	1.51
(D)	1.06

Q.19	Value of (𝟏 + 𝒊)^𝟖, where 𝒊 = √−𝟏 , is equal to
(A)	4
(B)	16
(C)	4i
(D)	16i

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

Q.20

Consider adiabatic flow of air through a duct. At a given point in the duct, velocity of air is 300 m/s, temperature is 330 K and pressure is 180 kPa. Assume that the air behaves as a perfect gas with constant 𝒄_𝒑 =1.005 kJ/kg.K. The stagnation temperature at this point is _________K (round off to two decimal places).

Q.21

Consider an ideal vapour compression refrigeration cycle working on R-134a refrigerant. The COP of the cycle is 10 and the refrigeration capacity is 150 kJ/kg. The heat rejected by the refrigerant in the condenser is______kJ/kg (round off to the nearest integer).

Q.22

A rigid tank of volume 50 m³contains a pure substance as a saturated liquid vapour mixture at 400 kPa. Of the total mass of the mixture, 20% mass is liquid and 80% mass is vapour. Properties at 400 kPa are: 𝑻_𝒔𝒂𝒕 =143.61 ^°C, 𝒗_𝒇 = 0.001084 m³/kg, 𝒗_𝒈 = 0.46242 m³/kg. The total mass of liquid vapour mixture in the tank is kg (round off to the nearest integer).

Q.23

An object is moving with a Mach number of 0.6 in an ideal gas environment, which is at a temperature of 350 K. The gas constant is 320 J/kg.K and ratio of specific heats is 1.3. The speed of object is ______m/s (round off to the nearest integer).

Q.24

A column with one end fixed and one end free has a critical buckling load of 100 N. For the same column, if the free end is replaced with a pinned end then the critical buckling load will be _____N (round off to the nearest integer).

Q.25

A steel cubic block of side 200 mm is subjected to hydrostatic pressure of 250 N/mm². The elastic modulus is 2 × 10⁵ N/mm² and Poisson ratio is 0.3 for steel. The side of the block is reduced by ________mm (round off to two decimal places).

Q. 26 – Q.34 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).

Q.26	The value of 𝒓 𝐬𝐢𝐧 𝜽 𝒅𝒓 𝒅𝜽 is
(A)	0
(B)	1/6
(C)	4/3
(D)	𝜋

Q.27	Let the superscript T represent the transpose operation. Consider the function where 𝒙 and 𝒓 are 𝒏 × 𝟏 vectors and 𝑸 is a symmetric 𝒏 × 𝒏 matrix. The stationary point of 𝒇(𝒙) is
(A)	𝑄^𝑇𝑟
(B)	_𝑄−1_𝑟
(C)	𝑟/𝑟^𝑇𝑟
(D)	𝑟

Q.28	Consider the following differential equation(𝟏 + 𝒚) = 𝒚.The solution of the equation that satisfies the condition 𝒚(𝟏) = 𝟏 is
(A)	2𝑦𝑒^𝑦 = 𝑒^𝑥 + 𝑒
(B)	𝑦²𝑒^𝑦 = 𝑒^𝑥
(C)	𝑦𝑒^𝑦 = 𝑒^𝑥
(D)	(1 + 𝑦)𝑒^𝑦 = 2𝑒^𝑥

Q.29	A factory produces 𝒎 (𝒊 = 𝟏, 𝟐, … , 𝒎) products, each of which requires processing on 𝒏 (𝒋 = 𝟏, 𝟐, … , 𝒏) workstations. Let 𝒂_𝒊𝒋 be the amount of processing time that one unit of the 𝒊^𝒕𝒉 product requires on the 𝒋^𝒕𝒉 workstation. Let the revenue from selling one unit of the 𝒊^𝒕𝒉product be 𝒓_𝒊 and 𝒉_𝒊 be the holding cost per unit per time period for the 𝒊^𝒕𝒉 product. The planning horizon consists of 𝑻 (𝒕 = 𝟏, 𝟐, … , 𝑻) time periods. The minimum demand that must be satisfied in time period 𝒕 is 𝒅_𝒊𝒕, and the capacity of the 𝒋^𝒕𝒉 workstation in time period 𝒕 is 𝒄_𝒋𝒕. Consider the aggregate planning formulation below, with decision variables 𝑺_𝒊𝒕 (amount of product 𝒊 sold in time period 𝒕), 𝑿_𝒊𝒕 (amount of product 𝒊 manufactured in time period 𝒕) and 𝑰_𝒊𝒕 (amount of product 𝒊 held in inventory at the end of time period 𝒕. 𝑺_𝒊𝒕 ≥ 𝒅_𝒊𝒕 ∀ 𝒊, 𝒕< 𝒄𝒂𝒑𝒂𝒄𝒊𝒕𝒚 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕 >< 𝒊𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝒃𝒂𝒍𝒂𝒏𝒄𝒆 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕 >𝑿_𝒊𝒕, 𝑺_𝒊𝒕, 𝑰_𝒊𝒕 ≥ 𝟎; 𝑰_𝒊𝟎 = 𝟎 The capacity constraints and inventory balance constraints for this formulation are
(A)	𝑚∑ 𝑎_𝑖𝑗𝑋_𝑖𝑡 ≤ 𝑐_𝑗𝑡 ∀ 𝑗, 𝑡 and 𝐼_𝑖𝑡 = 𝐼_𝑖_,_𝑡₋₁ + 𝑋_𝑖𝑡 − 𝑆_𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(B)	𝑚∑ 𝑎_𝑖𝑗𝑋_𝑖𝑡 ≤ 𝑐_𝑗𝑡 ∀ 𝑖, 𝑡 and 𝐼_𝑖𝑡 = 𝐼_𝑖_,_𝑡₋₁ + 𝑋_𝑖𝑡 − 𝑑_𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(C)	𝑚∑ 𝑎_𝑖𝑗𝑋_𝑖𝑡 ≤ 𝑑_𝑖𝑡 ∀ 𝑖, 𝑡 and 𝐼_𝑖𝑡 = 𝐼_𝑖_,_𝑡₋₁ + 𝑋_𝑖𝑡 − 𝑆_𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(D)	𝑚∑ 𝑎_𝑖𝑗𝑋_𝑖𝑡 ≤ 𝑑_𝑖𝑡 ∀ 𝑖, 𝑡 and 𝐼_𝑖𝑡 = 𝐼_𝑖_,_𝑡₋₁ + 𝑆_𝑖𝑡 − 𝑋_𝑖𝑡 ∀ 𝑖, 𝑡𝑖

Q.30	Ambient pressure, temperature, and relative humidity at a location are 101 kPa, 300 K, and 60%, respectively. The saturation pressure of water at 300 K is 3.6 kPa. The specific humidity of ambient air is ______g/kg of dry air.
(A)	21.4
(B)	35.1
(C)	21.9
(D)	13.6

Q.33	A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to reduce the force transmitted to the foundation. The isolator can be considered as a combination of stiffness (K) and damper (damping factor,) in parallel. The designer has the following four isolators:1) K = 640 kN/m, = 0.702) K = 640 kN/m, = 0.073) K = 22.5 kN/m, = 0.704) K = 22.5 kN/m, = 0.07Arrange the isolators in the ascending order of the force transmitted to the foundation.
(A)	1-3-4-2
(B)	1-3-2-4
(C)	4-3-1-2
(D)	3-1-2-4

Q.35 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.35

Find the positive real root of 𝒙^𝟑 − 𝒙 − 𝟑 = 𝟎 using Newton-Raphson method. If the starting guess (𝒙_𝟎) is 2, the numerical value of the root after two iterations (𝒙_𝟐) is______ (round off to two decimal places).

Q.36

Daily production capacity of a bearing manufacturing company is 30000 bearings. The daily demand of the bearing is 15000. The holding cost per year of keeping a bearing in the inventory is ₹ 20. The setup cost for the production of a batch is ₹ 1800. Assuming 300 working days in a year, the economic batch quantity in number of bearings is ___________(in integer).

Q.37

A cast product of a particular material has dimensions 75 mm x 125 mm x 20 mm. The total solidification time for the cast product is found to be 2.0 minutes as calculated using Chvorinov’s rule having the index, n = 2. If under the identical casting conditions, the cast product shape is changed to a cylinder having diameter = 50 mm and height = 50 mm, the total solidification time will be ______minutes (round off to two decimal places).

Q.38

A spot welding operation performed on two pieces of steel yielded a nugget with a diameter of 5 mm and a thickness of 1 mm. The welding time was 0.1 s. The melting energy for the steel is 20 J/mm³. Assuming the heat conversion efficiency as 10%, the power required for performing the spot welding operation is _______kW (round off to two decimal places).

Q.39

A surface grinding operation has been performed on a Cast Iron plate having dimensions 300 mm (length)  10 mm (width)  50 mm (height). The grinding was performed using an alumina wheel having a wheel diameter of 150 mm and wheel width of 12 mm. The grinding velocity used is 40 m/s, table speed is 5 m/min, depth of cut per pass is 50 µm and the number of grinding passes is 20. The average tangential and average normal force for each pass is found to be 40 N and 60 N respectively. The value of the specific grinding energy under the aforesaid grinding conditions is ___________J/mm³(round off to one decimal place).

Q.40

In a pure orthogonal turning by a zero rake angle single point carbide cutting tool, the shear force has been computed to be 400 N. The cutting velocity, Vc = 100 m/min, depth of cut, t = 2.0 mm, feed, 𝒔_𝟎 = 0.1 mm/revolution and chip velocity, Vf = 20 m/min, the shear strength, 𝑟_𝒔 of the material will be ___________MPa (round off to two decimal places).

Q.41

The thickness, width and length of a metal slab are 50 mm, 250 mm and 3600 mm, respectively. A rolling operation on this slab reduces the thickness by 10% and increases the width by 3%. The length of the rolled slab is________mm (round off to one decimal place).

Q.42

A 76.2 mm gauge block is used under one end of a 254 mm sine bar with roll diameter of 25.4 mm. The height of gauge blocks required at the other end of the sine bar to measure an angle of 30º is ______________mm (round off to two decimal places).

Q.44

A shell and tube heat exchanger is used as a steam condenser. Coolant water enters the tube at 300 K at a rate of 100 kg/s. The overall heat transfer coefficient is 1500 W/m².K, and total heat transfer area is 400 m². Steam condenses at a saturation temperature of 350 K. Assume that the specific heat of coolant water is 4000 J/kg.K. The temperature of the coolant water coming out of the condenser is ______K (round off to the nearest integer).

Q.45

Ambient air flows over a heated slab having flat, top surface at 𝒚 = 𝟎. The local temperature (in Kelvin) profile within the thermal boundary layer is given by (𝒚) = 𝟑𝟎𝟎 + 𝟐𝟎𝟎 𝐞𝐱𝐩(−𝟓𝒚), where 𝒚 is the distance measured from the slab surface in meter. If the thermal conductivity of air is 1.0 W/m.K and that of the slab is 100 W/m.K, then the magnitude of temperature gradient |𝒅𝑻/𝒅𝒚| within the slab at 𝒚 = 𝟎 is _______ K/m (round off to the nearest integer).

Q.49

A vertical shaft Francis turbine rotates at 300 rpm. The available head at the inlet to the turbine is 200 m. The tip speed of the rotor is 40 m/s. Water leaves the runner of the turbine without whirl. Velocity at the exit of the draft tube is3.5 m/s. The head losses in different components of the turbine are: (i) stator and guide vanes: 5.0 m, (ii) rotor: 10 m, and (iii) draft tube: 2 m. Flow rate through the turbine is 20 m³/s. Take 𝒈 = 9.8 m/s². The hydraulic efficiency of the turbine is % (round off to one decimal place).

Q.53

A cantilever beam with a uniform flexural rigidity (EI = 200 x 10⁶N.m²) is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding to the full length of the beam is 10000 N.m². The magnitude of the slope of the beam at its free end is _________micro radian (round off to the nearest integer).

Q.54

The torque provided by an engine is given by T(θ) = 12000 + 2500 sin(2θ) N.m, where θ is the angle turned by the crank from inner dead center. The mean speed of the engine is 200 rpm and it drives a machine that provides a constant resisting torque. If variation of the speed from the mean speed is not to exceed±0.5%, the minimum mass moment of inertia of the flywheel should be___________kg.m²(round off to the 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	B	1	D	11	D	21	165 to 165	31	B	41	3880.0 to 3886.0	51	0.16 to 0.19
2	A	2	C	12	B	22	134 to 136	32	A	42	200.00 to 206.00	52	1.3 to 1.4
3	A	3	D	13	B	23	228 to 230	33	C	43	6.00 to 10.00	53	48 to 52
4	B	4	D	14	B	24	800 to 840	34	B	44	337 to 341	54	560 to 580
5	d	5	A	15	D	25	0.08 to 0.12	35	1.66 to 1.68	45	10 to 10	55	152000 to 165000
6	C	6	D	16	B	26	B	36	40200 to 40300	46	2257.0 to 2259.0
7	B	7	D	17	D	27	B	37	2.60 to 3.00	47	25.0 to 25.4
8	C	8	B	18	A	28	C	38	39.00 to 40.00	48	130.00 to 134.00
9	B	9	C	19	B	29	A	39	38.0 to 39.0	49	90.0 to 92.0
10	D	10	A	20	373.00 to 377.00	30	D	40	388.00 to 400.00	50	2.1 to 2.3

GATE 2021 Mechanical Engineering Section-I 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	Consider the following sentences:After his surgery, Raja hardly could walk.After his surgery, Raja could barely walk.After his surgery, Raja barely could walk.After his surgery, Raja could hardly walk.Which of the above sentences are grammatically CORRECT?
(A)	(i) and (ii)
(B)	(i) and (iii)
(C)	(iii) and (iv)
(D)	(ii) and (iv)

Q.2	Ms. X came out of a building through its front door to find her shadow due to the morning sun falling to her right side with the building to her back.From this, it can be inferred that building is facing____________.
(A)	North
(B)	East
(C)	West
(D)	South

Q.5	“The increased consumption of leafy vegetables in the recent months is a clear indication that the people in the state have begun to lead a healthy lifestyle”Which of the following can be logically inferred from the information presented in the above statement?
(A)	The people in the state did not consume leafy vegetables earlier.
(B)	Consumption of leafy vegetables may not be the only indicator of healthy lifestyle.
(C)	Leading a healthy lifestyle is related to a diet with leafy vegetables.
(D)	The people in the state have increased awareness of health hazards causing by consumption of junk foods.

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

Q.6	Oxpeckers and rhinos manifest a symbiotic relationship in the wild. The oxpeckers warn the rhinos about approaching poachers, thus possibly saving the lives of the rhinos. Oxpeckers also feed on the parasitic ticks found on rhinos.In the symbiotic relationship described above, the primary benefits for oxpeckers and rhinos respectively are,
(A)	Oxpeckers get a food source, rhinos have no benefit.
(B)	Oxpeckers save their habitat from poachers while the rhinos have no benefit.
(C)	Oxpeckers get a food source, rhinos may be saved from the poachers.
(D)	Oxpeckers save the lives of poachers, rhinos save their own lives.

Q.7

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

Q.8	The number of hens, ducks and goats in farm P are 65, 91 and 169, respectively. The total number of hens, ducks and goats in a nearby farm Q is 416. The ratio of hens:ducks:goats in farm Q is 5:14:13. All the hens, ducks and goats are sent from farm Q to farm P.The new ratio of hens:ducks:goats in farm P is_____
(A)	5:7:13
(B)	5:14:13
(C)	10:21:26
(D)	21:10:26

Q. 10	Five persons P, Q, R, S and T are sitting in a row not necessarily in the same order. Q and R are separated by one person, and S should not be seated adjacent to Q.The number of distinct seating arrangements possible is:
(A)	4
(B)	8
(C)	10
(D)	16

Mechanical Engineering

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

Q.1	If 𝒚(𝒙) satisfies the differential equation(𝐬𝐢𝐧 𝒙)+ 𝒚 𝐜𝐨𝐬 𝒙 = 𝟏,subject to the condition 𝒚(𝝅/𝟐) = 𝝅/𝟐, then 𝒚(𝝅/𝟔) is
(A)	0
(B)	𝜋 / 6
(C)	𝜋 /3
(D)	𝜋 / 2

Q.2
(A)	1/4
(B)	1/3
(C)	1/2
(D)	1

Q.3	The Dirac-delta function (𝜹(𝒕 − 𝒕_𝟎)) for 𝒕, 𝒕_𝟎 ∈ ℝ, has the following property The Laplace transform of the Dirac-delta function 𝜹(𝒕 − 𝒂) for a > 0; 𝐿(𝜹(𝒕 − 𝒂)) = 𝑭(𝒔) is
(A)	0
(B)	∞
(C)	_𝑒𝑠𝑎
(D)	_𝑒−𝑠𝑎

Q.4	The ordinary differential equation=−𝝅𝒚 subject to an initial condition 𝒚(𝟎) = 𝟏 is solved numerically using the following scheme: where 𝒉 is the time step, 𝒕_𝒏 = 𝒏𝒉, and 𝒏 = 𝟎, 𝟏, 𝟐, …. This numerical scheme is stable for all values of 𝒉 in the interval .
(A)	0 < ℎ <
(B)	0 < ℎ < 1
(C)	0 < ℎ <
(D)	for all ℎ > 0

Q.5	Consider a binomial random variable 𝑿. If 𝑿_𝟏, 𝑿_𝟐, … , 𝑿_𝒏 are independent and identically distributed samples from the distribution of 𝑿 with sum 𝒀 =∑^𝒏 _𝒊₌_𝟏𝑿_𝒊, then the distribution of 𝒀 as 𝒏 → ∞ can be approximated as
(A)	Exponential
(B)	Bernoulli
(C)	Binomial
(D)	Normal

Q.8	The correct sequence of machining operations to be performed to finish a large diameter through hole is
(A)	drilling, boring, reaming
(B)	boring, drilling, reaming
(C)	drilling, reaming, boring
(D)	boring, reaming, drilling

Q.9	In modern CNC machine tools, the backlash has been eliminated by
(A)	preloaded ballscrews
(B)	rack and pinion
(C)	ratchet and pinion
(D)	slider crank mechanism

Q.11	In which of the following pairs of cycles, both cycles have at least one isothermal process?
(A)	Diesel cycle and Otto cycle
(B)	Carnot cycle and Stirling cycle
(C)	Brayton cycle and Rankine cycle
(D)	Bell-Coleman cycle and Vapour compression refrigeration cycle

Q.12	Superheated steam at 1500 kPa, has a specific volume of 2.75 m³/kmol and compressibility factor (𝒁) of 0.95. The temperature of steam is ____________⁰*C (round off to the nearest integer***).
(A)	522
(B)	471
(C)	249
(D)	198

Q.13	A hot steel spherical ball is suddenly dipped into a low temperature oil bath.Which of the following dimensionless parameters are required to determine instantaneous center temperature of the ball using a Heisler chart?
(A)	Biot number and Fourier number
(B)	Reynolds number and Prandtl number
(C)	Biot number and Froude number
(D)	Nusselt number and Grashoff number

Q.14	An infinitely long pin fin, attached to an isothermal hot surface, transfers heat at a steady rate of 𝑸_𝟏 to the ambient air. If the thermal conductivity of the fin material is doubled, while keeping everything else constant, the rate of steady- state heat transfer from the fin becomes 𝑸_𝟐. The ratio 𝑸_𝟐/𝑸_𝟏 is
(A)	√2
(B)	2
(C)	1/√2
(D)	1/2

Q.15	The relative humidity of ambient air at 300 K is 50% with a partial pressure of water vapour equal to 𝒑_𝒗. The saturation pressure of water at 300 K is 𝒑_𝒔𝒂𝒕. The correct relation for the air-water mixture is
(A)	𝑝_𝑣 = 0.5 𝑝_𝑠𝑎𝑡
(B)	^𝑝𝑣 ⁼^𝑝𝑠𝑎𝑡
(C)	𝑝_𝑣 = 0.622 𝑝_𝑠𝑎𝑡
(D)	𝑝_𝑣 = 2 𝑝_𝑠𝑎𝑡

Q.16	Consider a reciprocating engine with crank radius 𝑹 and connecting rod of length 𝑳. The secondary unbalance force for this case is equivalent to primary unbalance force due to a virtual crank of________
(A)	radius_𝐿2/4Rrotating at half the engine speed
(B)	radius𝑅/4rotating at half the engine speed
(C)	radius_𝑅2/4𝐿rotating at twice the engine speed
(D)	radius𝐿/2rotating at twice the engine speed

Q.17	A cantilever beam of length, 𝑳, and flexural rigidity, 𝑬I, is subjected to an end moment, 𝑴, as shown in the figure.The deflection of the beam at 𝒙 =L/2 is
(A)	𝑀𝐿²/2𝐸𝐼
(B)	𝑀𝐿²/4𝐸𝐼
(C)	𝑀𝐿²/8𝐸𝐼
(D)	𝑀𝐿²/16𝐸𝐼

Q.18	A prismatic bar 𝑷𝑸𝑹𝑺𝑻 is subjected to axial loads as shown in the figure. The segments having maximum and minimum axial stresses, respectively, are
(A)	QR and PQ
(B)	ST and PQ
(C)	QR and RS
(D)	ST and RS

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

Q.20

Robot Ltd. wishes to maintain enough safety stock during the lead time period between starting a new production run and its completion such that the probability of satisfying the customer demand during the lead time period is 95%. The lead time period is 5 days and daily customer demand can be assumed to follow the Gaussian (normal) distribution with mean 50 units and a standard deviation of 10 units. Using 𝝓⁻(𝟎. 𝟗𝟓) = 𝟏. 𝟔𝟒, where 𝝓 represents the cumulative distribution function of the standard normal random variable, the amount of safety stock that must be maintained by Robot Ltd. to achieve this demand fulfillment probability for the lead time period is _________units (round off to two decimal places).

Q.21

A pressure measurement device fitted on the surface of a submarine, located at a depth H below the surface of an ocean, reads an absolute pressure of 4.2 MPa. The density of sea water is 1050 kg/m³, the atmospheric pressure is 101 kPa, and the acceleration due to gravity is 9.8 m/s². The depth H is __________m (round off to the nearest integer).

Q.22

Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constant velocity of U = 4 m/s. Separation between the plates is 5 mm. There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m³, and the kinematic viscosity is 𝟏. 𝟐𝟓 × 𝟏𝟎⁻^𝟒 m²/s. The average shear stress in the fluid is ______Pa (round off to the nearest integer).

Q.23

A rigid insulated tank is initially evacuated. It is connected through a valve to a supply line that carries air at a constant pressure and temperature of 250 kPa and 400 K respectively. Now the valve is opened and air is allowed to flow into the tank until the pressure inside the tank reaches to 250 kPa at which point the valve is closed. Assume that the air behaves as a perfect gas with constant properties(𝒄_𝒑 = 1.005 kJ/kg.K, 𝒄_𝒗 = 0.718 kJ/kg.K, 𝑹 = 0.287 kJ/kg.K). Final temperature of the air inside the tank is ______K (round off to one decimal place).

Q. 26 – Q. 34 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).

Q.26	Consider a vector 𝒑 in 2-dimensional space. Let its direction (counter- clockwise angle with the positive 𝒙-axis) be 𝜽. Let 𝒑 be an eigenvector of a 𝟐 × 𝟐 matrix 𝑨 with corresponding eigenvalue 𝝀, 𝝀 > 𝟎. If we denote the magnitude of a vector 𝒗 by ‖𝒗‖, identify the VALID statement regarding 𝒑′, where 𝒑^′ = 𝑨𝒑.
(A)	Direction of 𝑝^′ = 𝜆𝜃, ‖𝑝′‖ = ‖𝑝‖
(B)	Direction of 𝑝^′ = 𝜃, ‖𝑝′‖ = 𝜆‖𝑝‖
(C)	Direction of 𝑝^′ = 𝜆𝜃, ‖𝑝′‖ = 𝜆‖𝑝‖
(D)	Direction of 𝑝^′ = 𝜃, ‖𝑝^′‖ = ‖𝑝‖/𝜆

Q.27	Let 𝑪 represent the unit circle centered at origin in the complex plane, and complex variable, 𝒛 = 𝒙 + 𝒊𝒚. The value of the contour integral (where integration is taken counter clockwise) is
(A)	0
(B)	2
(C)	𝜋𝑖
(D)	2𝜋𝑖

Q.28	A set of jobs A, B, C, D, E, F, G, H arrive at time t = 0 for processing on turning and grinding machines. Each job needs to be processed in sequence– first on the turning machine and second on the grinding machine, and the grinding must occur immediately after turning. The processing times of the jobs are given below.JobAB CDEFGH^Turning(minutes)248976510^Grinding(minutes)61379524If the makespan is to be minimized, then the optimal sequence in which these jobs must be processed on the turning and grinding machines is
(A)	A-E-D-F-H-C-G-B
(B)	A-D-E-F-H-C-G-B
(C)	G-E-D-F-H-C-A-B
(D)	B-G-C-H-F-D-E-A

Q.29	The fundamental thermodynamic relation for a rubber band is given by 𝒅𝑼 = 𝑻𝒅𝑺 + 𝑟𝒅𝑳, where 𝑻 is the absolute temperature, 𝑺 is the entropy, 𝑟 is the tension in the rubber band, and 𝑳 is the length of the rubber band. Which one of the following relations is CORRECT:
(A)
(B)
(C)
(D)

Q.31	A right solid circular cone standing on its base on a horizontal surface is of height H and base radius R. The cone is made of a material with specific weight w and elastic modulus E. The vertical deflection at the mid-height of the cone due to self-weight is given by
(A)	𝑤𝐻² /8𝐸
(B)	𝑤𝐻² /6𝐸
(C)	𝑤𝑅𝐻 /8𝐸
(D)	𝑤𝑅𝐻 /6𝐸

Q.33	Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes is
(A)	0.39
(B)	0.86
(C)	0.50
(D)	0.61

Q.34	Let (𝒙) = 𝒙^𝟐 − 𝟐𝒙 + 𝟐 be a continuous function defined on 𝒙 ∈ [𝟏, 𝟑]. The point 𝒙 at which the tangent of 𝒇(𝒙) becomes parallel to the straight line joining 𝒇(𝟏) and 𝒇(𝟑) is
(A)	0
(B)	1
(C)	2
(D)	3

Q.35 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks)

Q.36

A true centrifugal casting operation needs to be performed horizontally to make copper tube sections with outer diameter of 250 mm and inner diameter of 230 mm. The value of acceleration due to gravity, g = 10 m/s². If a G-factor (ratio of centrifugal force to weight) of 60 is used for casting the tube, the rotational speed required is _____ rpm (round off to the nearest integer).

Q.37

The resistance spot welding of two 1.55 mm thick metal sheets is performed using welding current of 10000 A for 0.25 s. The contact resistance at the interface of the metal sheets is 0.0001 Ω. The volume of weld nugget formed after welding is 70 mm³. Considering the heat required to melt unit volume of metal is 12 J/mm³, the thermal efficiency of the welding process is__________% (round off to one decimal place).

Q.38

An orthogonal cutting operation is performed using a single point cutting tool with a rake angle of 12º on a lathe. During turning, the cutting force and the friction force are 1000 N and 600 N, respectively. If the chip thickness and the uncut chip thickness during turning are 1.5 mm and 0.75 mm, respectively, then the shear force is _____________N (round off to two decimal places).

Q.39

In a grinding operation of a metal, specific energy consumption is 15 J/mm³. If a grinding wheel with a diameter of 200 mm is rotating at 3000 rpm to obtain a material removal rate of 6000 mm³/min, then the tangential force on the wheel is ____________N (round off to two decimal places).

Q.40

A 200 mm wide plate having a thickness of 20 mm is fed through a rolling mill with two rolls. The radius of each roll is 300 mm. The plate thickness is to be reduced to 18 mm in one pass using a roll speed of 50 rpm. The strength coefficient (K) of the work material flow curve is 300 MPa and the strain hardening exponent, n is 0.2. The coefficient of friction between the rolls and the plate is 0.1. If the friction is sufficient to permit the rolling operation then the roll force will be _________kN (round off to the nearest integer).

Q.41

The XY table of a NC machine tool is to move from P(1,1) to Q(51,1); all coordinates are in mm. The pitch of the NC drive leadscrew is 1 mm. If the backlash between the leadscrew and the nut is 1.8º, then the total backlash of the table on moving from P to Q is ___________mm (round off to two decimal places).

Q.42

Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the workstation is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the workstation at any given point of time is ________(round off to the nearest integer).

Q.43

An uninsulated cylindrical wire of radius 1.0 mm produces electric heating at the rate of 5.0 W/m. The temperature of the surface of the wire is 75^°C when placed in air at 25^°C. When the wire is coated with PVC of thickness 1.0 mm, the temperature of the surface of the wire reduces to 55 ^°C. Assume that the heat generation rate from the wire and the convective heat transfer coefficient are same for both uninsulated wire and the coated wire. The thermal conductivity of PVC is W/m.K (round off to two decimal places).

Q.44

A solid sphere of radius 10 mm is placed at the centroid of a hollow cubical enclosure of side length 30 mm. The outer surface of the sphere is denoted by 1 and the inner surface of the cube is denoted by 2. The view factor 𝑭_𝟐𝟐 for radiation heat transfer is __________(rounded off to two decimal places).

Q.45

Consider a steam power plant operating on an ideal reheat Rankine cycle. The work input to the pump is 20 kJ/kg. The work output from the high pressure turbine is 750 kJ/kg. The work output from the low pressure turbine is 1500 kJ/kg. The thermal efficiency of the cycle is 50 %. The enthalpy of saturated liquid and saturated vapour at condenser pressure are 200 kJ/kg and 2600 kJ/kg, respectively. The quality of steam at the exit of the low pressure turbine is __________% (round off to the nearest integer).

Q.46

In the vicinity of the triple point, the equation of liquid-vapour boundary in the 𝑷 − 𝑻 phase diagram for ammonia is 𝐥𝐧 𝑷 = 𝟐𝟒. 𝟑𝟖 − 𝟑𝟎𝟔𝟑/𝑻, where P is pressure (in Pa) and 𝑻 is temperature (in K). Similarly, the solid-vapour boundary is given by 𝐥𝐧 𝑷 = 𝟐𝟕. 𝟗𝟐 − 𝟑𝟕𝟓𝟒/𝑻. The temperature at the triple point is __________K(round off to one decimal place).

Q.48

A single jet Pelton wheel operates at 300 rpm. The mean diameter of the wheel is 2 m. Operating head and dimensions of jet are such that water comes out of the jet with a velocity of 40 m/s and flow rate of 5 m³/s. The jet is deflected by the bucket at an angle of 165°. Neglecting all losses, the power developed by the Pelton wheel is _____MW (round off to two 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	D	1	C	11	B	21	397 to 399	31	A	41	0.20 to 0.30	51	249.0 to 251.0
2	D	2	C	12	C	22	79 to 81	32	B	42	4 to 4	52	1.9 to 2.1
3	D	3	D	13	A	23	555 to 565	33	A	43	0.10 to 0.12	53	199.0 to 201.0
4	D	4	A	14	A	24	0 to 0	34	C	44	0.76 to 0.78	54	1.9 to 2.1
5	C	5	D	15	A	25	0.085 to 0.090	35	0.5 to 0.5	45	92 to 96	55	6.4 to 6.9
6	C	6	B	16	C	26	B	36	660 to 664	46	195.1 to 195.3
7	A	7	B	17	C	27	C	37	30 to 36	47	11.2 to 11.4
8	C	8	A	18	D	28	A	38	625 to 750	48	2.50 to 2.80
9	C	9	A	19	C	29	B	39	45.00 to 50.00	49	4.90 to 5.10
10	D	10	B	20	331 to 333	30	B	40	775 to 784	50	124.0 to 126.0

GATE 2021 Mathematics 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 ratio of boys to girls in a class is 7 to 3.Among the options below, an acceptable value for the total number of students in the class is:
(A)	21
(B)	37
(C)	50
(D)	73

Q.3	Consider the following sentences:Everybody in the class is prepared for the exam.Babu invited Danish to his home because he enjoys playing chess.Which of the following is the CORRECT observation about the above two sentences?
(A)	(i) is grammatically correct and (ii) is unambiguous
(B)	(i) is grammatically incorrect and (ii) is unambiguous
(C)	(i) is grammatically correct and (ii) is ambiguous
(D)	(i) is grammatically incorrect and (ii) is ambiguous

Q.5	_______is to *surgery* as *writer* is to____ Which one of the following options maintains a similar logical relation in the above sentence?
(A)	Plan, outline
(B)	Hospital, library
(C)	Doctor, book
(D)	Medicine, grammar

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

Q.6	We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm x 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is_______
(A)	𝜋 / 2
(B)	3/𝜋
(C)	9 / 𝜋
(D)	3𝜋

Q.8	There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.The probability that at least two chocolates are identical is
(A)	0.3024
(B)	0.4235
(C)	0.6976
(D)	0.8125

Q.9	Given below are two statements 1 and 2, and two conclusions I and II. Statement 1: All bacteria are microorganisms. Statement 2:All pathogens are microorganisms. Conclusion I: Some pathogens are bacteria. Conclusion II: All pathogens are not bacteria. 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)	Neither conclusion I nor II is correct.

Q.10	Some people suggest anti-obesity measures (AOM) such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality.Which one of the following statements summarizes the passage?
(A)	The proposed AOM addresses the core problems that cause obesity.
(B)	If obesity reduces, poverty will naturally reduce, since obesity causes poverty.
(C)	AOM are addressing the core problems and are likely to succeed.
(D)	AOM are addressing the problem superficially.

Mathematics (MA)

– Q.14 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	Let 𝑨 be a 𝟑 × 𝟒 matrix and 𝑩 be a 𝟒 × 𝟑 matrix with real entries such that 𝑨𝑩 is non-singular. Consider the following statements: P: Nullity of 𝑨 is 𝟎.Q: 𝑩𝑨 is a non-singular matrix.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.2	Let (𝒛) = 𝒖(𝒙, 𝒚) + 𝒊 𝒗(𝒙, 𝒚) for 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ, where 𝒙 and 𝒚 are real numbers, be a non-constant analytic function on the complex plane ℂ. Let 𝒖_𝒙,𝒗_𝒙 and 𝒖_𝒚, 𝒗_𝒚 denote the first order partial derivatives of 𝒖(𝒙, 𝒚) = 𝑹𝒆(𝒇(𝒛))and 𝒗(𝒙, 𝒚) = 𝑰𝒎(𝒇(𝒛)) with respect to real variables 𝒙 and 𝒚, respectively. Consider the following two functions defined on ℂ: 𝒈_𝟏(𝒛) = 𝒖_𝒙(𝒙, 𝒚) − 𝒊 𝒖_𝒚 (𝒙, 𝒚) 𝐟𝐨𝐫 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ, 𝒈_𝟐(𝒛) = 𝒗_𝒙(𝒙, 𝒚) + 𝒊 𝒗_𝒚(𝒙, 𝒚) 𝐟𝐨𝐫 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ. Then
(A)	both 𝑔₁(𝑧) and 𝑔₂(𝑧) are analytic in ℂ
(B)	𝑔₁(𝑧) is analytic in ℂ and 𝑔₂(𝑧) is NOT analytic in ℂ
(C)	𝑔₁(𝑧) is NOT analytic in ℂ and 𝑔₂(𝑧) is analytic in ℂ
(D)	neither 𝑔₁(𝑧) nor 𝑔₂(𝑧) is analytic in ℂ

Q.3	Let 𝑻(𝒛) = , 𝒂𝒅 − 𝒃𝒄 ≠ 𝟎, be the Möbius transformation which maps the points 𝒛_𝟏 𝟎, 𝒛_𝟐 = −𝒊, 𝒛_𝟑 = ∞ in the 𝒛-plane onto the points 𝒘_𝟏 = 𝟏𝟎,𝒘_𝟐 = 𝟓 − 𝟓𝒊, 𝒘_𝟑 = 𝟓 + 𝟓𝒊 in the 𝒘-plane, respectively. Then the image of the set 𝑺 = {𝒛 ∈ ℂ ∶ 𝑹𝒆(𝒛) < 𝟎} under the map 𝒘 = 𝑻(𝒛) is
(A)	{𝑤 ∈ ℂ ∶ \|𝑤\| < 5}
(B)	{𝑤 ∈ ℂ ∶ \|𝑤\| > 5}
(C)	{𝑤 ∈ ℂ ∶ \|𝑤 − 5\| < 5}
(D)	{𝑤 ∈ ℂ ∶ \|𝑤 − 5\| > 5}

Q.4	Let 𝑹 be the row reduced echelon form of a 𝟒 × 𝟒 real matrix 𝑨 and let the third column of 𝑹 be . Consider the following statements:P: If is a solution of 𝑨𝐱 = 𝟎, then = 𝟎.Q: For all 𝐛 ∈ ℝ^𝟒, 𝒓𝒂𝒏[𝑨\| 𝐛] = 𝒓𝒂𝒏𝒌[𝑹\| 𝐛]. Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.5	The eigenvalues of the boundary value problem 𝒙 ∈ (𝟎, 𝝅), 𝝀 > 𝟎, 𝒚(𝟎) = 𝟎, 𝒚(𝝅) − (𝝅) = 𝟎, are given by
(A)	𝜆 = (𝑛𝜋)², 𝑛 = 1,2,3, …
(B)	𝜆 = 𝑛², 𝑛 = 1,2,3, …
(C)	𝜆 = 𝑘_𝑛², where 𝑘_𝑛 , 𝑛 = 1,2,3, … are the roots of 𝑘 − tan(𝑘𝜋) = 0
(D)	𝜆 = 𝑘_𝑛², where 𝑘_𝑛 , 𝑛 = 1,2,3, … are the roots of 𝑘 +tan(𝑘𝜋) = 0

C:\Users\PC6~1\AppData\Local\Temp\ksohtml12188\wps82.jpg

Q.9	Let {𝒆_𝒏 ∶ 𝒏 = 𝟏, 𝟐, 𝟑, … } be an orthonormal basis of a complex Hilbert space𝑯. Consider the following statements:P: There exists a bounded linear functional 𝒇: 𝑯 → ℂ such that 𝒇(𝒆_𝒏 ) = for 𝒏 = 𝟏, 𝟐, 𝟑, … .Q: There exists a bounded linear functional 𝒈: 𝑯 → ℂ such that 𝒈(𝒆_𝒏 ) =for 𝒏 = 𝟏, 𝟐, 𝟑, … .Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.11	Consider the following statements: Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.12	Let 𝒇: ℝ^𝟑 → ℝ be a twice continuously differentiable scalar field such that 𝒅𝒊𝒗(𝛁𝒇) = 𝟔. Let 𝑺 be the surface 𝒙^𝟐 + ^𝟐 + 𝒛^𝟐 = 𝟏 and 𝒏̂ be unit outward normal to 𝑺. Then the value of ∬_𝑺 (𝛁𝒇 ⋅ 𝒏̂) 𝒅𝑺 is
(A)	2 𝜋
(B)	4 𝜋
(C)	6 𝜋
(D)	8 𝜋

Q.13	Consider the following statements:P: Every compact metrizable topological space is separable. Q: Every Hausdorff topology on a finite set is metrizable.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.14	Consider the following topologies on the set ℝ of all real numbers:Τ_𝟏 = {𝑼 ⊂ ℝ ∶ 𝟎 ∉ 𝑼 𝐨𝐫 𝑼 = ℝ},Τ_𝟐 = {𝑼 ⊂ ℝ ∶ 𝟎 ∈ 𝑼 𝐨𝐫 𝑼 = ∅} ,Τ_𝟑 = Τ_𝟏 ∩ Τ_𝟐.Then the closure of the set {𝟏} in (ℝ, Τ_𝟑) is
(A)	{1}
(B)	{0,1}
(C)	ℝ
(D)	ℝ\{0}

Q.15 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.16

Let ᴦ denote the boundary of the square region 𝑹 with vertices (𝟎, 𝟎), (𝟐, 𝟎), (𝟐, 𝟐) and (𝟎, 𝟐) oriented in the counter- clockwise direction. Then∮_ᴦ(𝟏 − 𝒚^𝟐) 𝒅𝒙 + 𝒙 𝒅𝒚 = .

Q.17

The number of 𝟓-Sylow subgroups in the symmetric group 𝑺_𝟓 of degree 𝟓 is__________.

Q.18

Let 𝑰 be the ideal generated by 𝒙^𝟐 + 𝒙 + 𝟏 in the polynomial ring 𝑹 = ℤ[𝒙], where ℤ_𝟑 denotes the ring of integers modulo 𝟑. Then the number of units in the quotient ring 𝑹/𝑰 is .

Q.19

Let 𝑻: ℝ^𝟑 → ℝ^𝟑 be a linear transformation such that

Then the rank of 𝑻 is________ .

Q.20

Let 𝒚(𝒙) be the solution of the following initial value problem

Then (𝟒) = .

Q.21

Let𝒇(𝒙) = 𝒙^𝟒 + 𝟐 𝒙^𝟑 − 𝟏𝟏 𝒙^𝟐 − 𝟏𝟐 𝒙 + 𝟑𝟔 𝐟𝐨𝐫 𝒙 ∈ ℝ.The order of convergence of the Newton-Raphson method

with 𝒙_𝟎 = 𝟐. 𝟏, for finding the root 𝑎 = 𝟐 of the equation 𝒇(𝒙) = 𝟎 is______.

Q.23

Consider the Linear Programming Problem 𝑷:

Maximize 𝟐𝒙_𝟏 + 𝟑𝒙_𝟐 subject to

𝟐𝒙_𝟏 + 𝒙_𝟐 ≤ 𝟔,

−𝒙_𝟏 + 𝒙_𝟐 ≤ 𝟏,

𝒙_𝟏 + 𝒙_𝟐 ≤ 𝟑,

𝒙𝟏 ≥ 𝟎 and 𝒙𝟐 ≥ 𝟎.

Then the optimal value of the dual of 𝑷 is equal to .

Q.24

Consider the Linear Programming Problem 𝑷:

Minimize 𝟐𝒙_𝟏 − 𝟓𝒙_𝟐subject to

𝟐𝒙_𝟏 + 𝟑𝒙_𝟐 + 𝒔_𝟏 = 𝟏𝟐,

−𝒙_𝟏 + 𝒙_𝟐 + 𝒔_𝟐 = 𝟏,

−𝒙_𝟏 + 𝟐𝒙_𝟐 + 𝒔_𝟑 = 𝟑,

𝒙_𝟏 ≥ 𝟎, 𝒙_𝟐 ≥ 𝟎, 𝒔_𝟏 ≥ 𝟎, 𝒔_𝟐 ≥ 𝟎, and 𝒔_𝟑 ≥ 𝟎.

If is a basic feasible solution of 𝑷, then 𝒙_𝟏 + 𝒔_𝟏 + 𝒔_𝟐 + 𝒔_𝟑 = .

Q.25

Let 𝑯 be a complex Hilbert space. Let 𝒖, 𝒗 ∈ 𝑯 be such that 〈𝒖, 𝒗〉 = 𝟐. Then

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

Q.26	Let ℤ denote the ring of integers. Consider the subring𝑹 = {𝒂 + 𝒃 √−𝟏𝟕 ∶𝒂, 𝒃 ∈ ℤ} of the field ℂ of complex numbers. Consider the following statements: P: 𝟐 + √−𝟏𝟕 is an irreducible element. Q: 𝟐 + √−𝟏𝟕 is a prime element.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.27	Consider the second-order partial differential equation (PDE) Consider the following statements: P: The PDE is parabolic on the ellipse + 𝒚^𝟐 = 𝟏. Q: The PDE is hyperbolic inside the ellipse + 𝒚^𝟐 = 𝟏.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.28	If 𝒖(𝒙, 𝒚) is the solution of the Cauchy problem 𝒖(𝒙, 𝟎) = −𝒙^𝟐, 𝒙 > 𝟎,then 𝒖(𝟐, 𝟏) is equal to
(A)	1 − 2 𝑒⁻²
(B)	1 + 4 𝑒⁻²
(C)	1 − 4 𝑒⁻²
(D)	1 + 2 𝑒⁻²

Q.30	The critical point of the differential equation +β^𝟐𝒚 = 𝟎, 𝑎 > β > 𝟎,is a
(A)	node and is asymptotically stable
(B)	spiral point and is asymptotically stable
(C)	node and is unstable
(D)	saddle point and is unstable

Q.31	The initial value problem 𝒕 > 𝟎, 𝒚(𝟎) = 𝟏,where 𝒇(𝒕, 𝒚) = 𝟏𝟎 𝒚, is solved by the following Euler method𝒚_𝒏+𝟏 = 𝒚_𝒏 + 𝒉 𝒇(𝒕_𝒏, 𝒚_𝒏), 𝒏 ≥ 𝟎,with step-size h. Then 𝒚_𝒏 → 𝟎 as 𝒏 → ∞, provided
(A)	0 < ℎ < 0.2
(B)	0.3 < ℎ < 0.4
(C)	0.4 < ℎ < 0.5
(D)	0.5 < ℎ < 0.55

Q.32	Consider the Linear Programming Problem 𝑷: Maximize 𝒄_𝟏𝒙_𝟏 + 𝒄_𝟐𝒙_𝟐subject to 𝒂_𝟏𝟏𝒙_𝟏 + 𝒂_𝟏𝟐𝒙_𝟐 ≤ 𝒃_𝟏, 𝒂_𝟐𝟏𝒙_𝟏 + 𝒂_𝟐𝟐𝒙_𝟐 ≤ 𝒃_𝟐, 𝒂_𝟑𝟏𝒙_𝟏 + 𝒂_𝟑𝟐𝒙_𝟐 ≤ 𝒃_𝟑, 𝒙_𝟏≥ 𝟎 and 𝒙_𝟐 ≥ 𝟎, where 𝒂𝒊𝒋, 𝒃𝒊 and 𝒄𝒋 are real numbers (𝒊 = 𝟏, 𝟐, 𝟑; 𝒋 =𝟏, 𝟐).Let be a feasible solution of 𝑷 such that 𝒑𝒄_𝟏 + 𝒒𝒄_𝟐= 𝟔 and let all feasible solutionsof 𝑷 satisfy −𝟓 ≤ 𝒄_𝟏𝒙+𝒄_𝟐𝒙_𝟐 ≤ 𝟏𝟐. Then, which one of the following statements is NOT true?
(A)	𝑃 has an optimal solution
(B)	The feasible region of 𝑃 is a bounded set
(C)	If is a feasible solution of the dual of 𝑃, then 𝑏₁𝑦₁ + 𝑏₂𝑦₂ + 𝑏₃𝑦₃ ≥ 6
(D)	The dual of 𝑃 has at least one feasible solution

Q.33	Let 𝑳^𝟐[−𝟏, 𝟏] be the Hilbert space of real valued square integrable functions on [−𝟏, 𝟏] equipped with the norm ‖𝒇‖ = (∫_−𝟏^𝟏\|𝒇(𝒙)\|^𝟐𝒅𝒙)^1/2.Consider the subspace 𝑴 = {𝒇 ∈ 𝑳²[−𝟏, 𝟏] ∶ ∫_−𝟏^𝟏𝒇(𝒙)𝒅𝒙 = 𝟎}.For (𝒙) = 𝒙^𝟐, define 𝒅 = 𝐢𝐧𝐟 {‖𝒇 − 𝒈‖ ∶ 𝒈 ∈ 𝑴 }. Then
(A)
(B)
(C)
(D)

Q.34	Let 𝑪[𝟎, 𝟏] be the Banach space of real valued continuous functions on [𝟎, 𝟏] equipped with the supremum norm. Define 𝑻: 𝑪[𝟎, 𝟏] → 𝑪[𝟎, 𝟏] by Let (𝑻) denote the range space of 𝑻. Consider the following statements:P: 𝑻 is a bounded linear operator.Q: 𝑻^−𝟏: (𝑻) → 𝑪[𝟎, 𝟏] exists and is bounded. Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.35	Let 𝑃^𝟏 = {𝒙 = (𝒙(𝟏), 𝒙(𝟐), … , 𝒙(𝒏), … ) \| ∑_𝒏=𝟏^∞ \|𝒙(𝒏)\| < ∞} be the sequence space equipped with the norm ‖𝒙‖ = ∑_𝒏=𝟏^∞\|𝒙(𝒏)\|. Consider the subspace and the linear transformation 𝑻: 𝑿 → 𝑃^𝟏 given by(𝑻𝒙)(𝒏) = 𝒏 (𝒏) for 𝒏 = 𝟏, 𝟐, 𝟑, … . Then
(A)	𝑇 is closed but NOT bounded
(B)	𝑇 is bounded
(C)	𝑇 is neither closed nor bounded
(D)	𝑇⁻¹ exists and is an open map

Q.36	Let 𝒇_𝒏: [𝟎, 𝟏𝟎] → ℝ be given by 𝒇_n(𝒙) = 𝒏 𝒙^𝟑 𝒆^−𝒏𝒙 for 𝒏 = 𝟏, 𝟐, 𝟑, … . Consider the following statements:P: (𝒇_𝒏) is equicontinuous on [𝟎, 𝟏𝟎].Q: ∑_𝒏=𝟏^∞𝒇_𝒏 does NOT converge uniformly on [𝟎, 𝟏𝟎].Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.37	Let 𝒇: ℝ^𝟐 → ℝ be given by Consider the following statements:P: 𝒇 is continuous at (𝟎, 𝟎) but 𝒇 is NOT differentiable at (𝟎, 𝟎).Q: The directional derivative 𝑫_𝒖(𝟎, 𝟎) of 𝒇 at (𝟎, 𝟎) exists in the direction of every unit vector 𝒖 ∈ ℝ^𝟐.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

Q.38	Let 𝑽 be the solid region in ℝ^𝟑 bounded by the paraboloid 𝒚 = (𝒙^𝟐 + 𝒛^𝟐) and the plane 𝒚 = 𝟒. Then the value of ∭_𝑽𝟏𝟓 √𝒙^𝟐 + 𝒛^𝟐 𝒅𝑽 is
(A)	128 𝜋
(B)	64 𝜋
(C)	28 𝜋
(D)	256 𝜋

Q.39	Let 𝒇: ℝ^𝟐 → ℝ be given by (𝒙, 𝒚) = 𝟒𝒙𝒚 − 𝟐 𝒙^𝟐 − 𝒚^𝟒. Then 𝒇 has
(A)	a point of local maximum and a saddle point
(B)	a point of local minimum and a saddle point
(C)	a point of local maximum and a point of local minimum
(D)	two saddle points

Q.40	The equation 𝒙𝒚 − 𝒛 𝐥𝐨𝐠 𝒚 + 𝒆^𝒙𝒛 = 𝟏 can be solved in a neighborhood of the point (𝟎, 𝟏, 𝟏) as 𝒚 = (𝒙, 𝒛) for some continuously differentiable function 𝒇. Then
(A)	∇𝑓(0, 1) = (2, 0)
(B)	∇𝑓(0, 1) = (0, 2)
(C)	∇𝑓(0, 1) = (0, 1)
(D)	∇𝑓(0, 1) = (1, 0)

Q.41	Consider the following topologies on the set ℝ of all real numbers.Τ_𝟏 is the upper limit topology having all sets (𝒂, 𝒃] as basis.Τ_𝟐 = {𝑼 ⊂ ℝ ∶ ℝ\𝑼 𝐢𝐬 𝐟𝐢𝐧𝐢𝐭𝐞} 𝖴 {∅}.Τ_𝟑 is the standard topology having all sets (𝒂, 𝒃) as basis. Then
(A)	Τ₂ ⊂ Τ₃ ⊂ Τ₁
(B)	Τ₁ ⊂ Τ₂ ⊂ Τ₃
(C)	Τ₃ ⊂ Τ₂ ⊂ Τ₁
(D)	Τ₂ ⊂ Τ₁ ⊂ Τ₃

Q.42	Let ℝ denote the set of all real numbers. Consider the following topological spaces.𝑿_𝟏 = (ℝ, Τ_𝟏), where Τ_𝟏 is the upper limit topology having all sets (𝒂, 𝒃] as basis.𝑿_𝟐 = (ℝ, Τ_𝟐), where Τ_𝟐 = {𝑼 ⊂ ℝ ∶ ℝ\𝑼 𝐢𝐬 𝐟𝐢𝐧𝐢𝐭𝐞} 𝖴 {∅}. Then
(A)	both 𝑋₁ and 𝑋₂ are connected
(B)	𝑋₁ is connected and 𝑋₂ is NOT connected
(C)	𝑋₁ is NOT connected and 𝑋₂ is connected
(D)	neither 𝑋₁ nor 𝑋₂ is connected

Q.43	Let 〈∙, ∙〉: ℝ^𝒏 × ℝ^𝒏 → ℝ be an inner product on the vector space ℝ^𝒏 over ℝ. Consider the following statements:P: \|〈𝒖, 𝒗〉\| ≤ (〈𝒖, 𝒖〉 + 〈𝒗, 𝒗〉) for all 𝒖, 𝒗 ∈ ℝ^𝒏.Q: If 〈𝒖, 𝒗〉 = 〈𝟐𝒖, −𝒗〉 for all 𝒗 ∈ ℝ^𝒏, then 𝒖 = 𝟎.Then
(A)	both P and Q are TRUE
(B)	P is TRUE and Q is FALSE
(C)	P is FALSE and Q is TRUE
(D)	both P and Q are FALSE

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

Q.44

Let 𝑮 be a group of order 𝟓^𝟒 with center having 𝟓^𝟐 elements. Then the number of conjugacy classes in 𝑮 is____.

Q.45

Let 𝑭 be a finite field and 𝑭^× be the group of all nonzero elements of 𝑭 under multiplication. If 𝑭^× has a subgroup of order 𝟏𝟕, then the smallest possible order of the field 𝑭 is______.

Q.46

Let 𝑹 = {𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ ∶ 𝟎 < 𝒙 < 𝟏 a𝐧𝐝 − 𝟏𝟏 𝝅 < 𝒚 < 𝟏𝟏 𝝅} and ᴦ be the positively oriented boundary of 𝑹. Then the value of the integral

is ___________.

Q.47

Let 𝑫 = {𝒛 ∈ ℂ ∶ |𝒛| < 𝟐𝝅} and 𝒇: 𝑫 → ℂ be the function defined by

If (𝒛) = ∑^∞_𝒏=𝟎𝒂_𝒏 𝒛^𝒏 for 𝒛 ∈ 𝑫, then 𝟔𝒂_𝟐=____________ .

Q.48

The number of zeroes (counting multiplicity) of 𝑷(𝒛) = 𝟑𝒛^𝟓 + 𝟐𝒊 𝒛^𝟐 + 𝟕𝒊 𝒛 +𝟏 in the annular region {𝒛 ∈ ℂ ∶ 𝟏 < |𝒛| < 𝟕} is____________.

Q.49

Let 𝑨 be a square matrix such that 𝐝𝐞(𝒙𝑰 − 𝑨) = 𝒙^𝟒(𝒙 − 𝟏)^𝟐(𝒙 − 𝟐)^𝟑, where 𝐝𝐞𝐭(𝑴) denotes the determinant of a square matrix 𝑴.If 𝐫𝐚𝐧(𝑨^𝟐) < 𝐫𝐚𝐧𝐤(𝑨^𝟑) = 𝐫𝐚𝐧𝐤(𝑨^𝟒), then the geometric multiplicity of the eigenvalue 𝟎 of 𝑨 is .

Q.50

If 𝒚 = ∑^∞_𝒌=𝟎𝒂_𝒌𝒙^𝒌, (𝒂_𝟎 ≠ 𝟎) is the power series solution of the differential equation− 𝟐𝟒 𝒙^𝟐𝒚 = 𝟎, then =_________.

Q.52

Let 𝑽 = {𝒑 ∶ (𝒙) = 𝒂_𝟎 + 𝒂_𝟏𝒙 + 𝒂_𝟐𝒙^𝟐, 𝒂_𝟎, 𝒂_𝟏, 𝒂_𝟐 ∈ ℝ } be the vector space of all polynomials of degree at most 𝟐 over the real field ℝ. Let 𝑻: 𝑽 →𝑽 be the linear operator given by 𝑻(𝒑) = (𝒑(𝟎) − 𝒑(𝟏)) + (𝒑(𝟎) + 𝒑(𝟏)) 𝒙 + 𝒑(𝟎) 𝒙^𝟐.Then the sum of the eigenvalues of 𝑻 is _____ .

Q.53

The quadrature formula𝟐∫ 𝒙 𝒇(𝒙) 𝒅𝒙 ≈ 𝑎 𝒇(𝟎) + 𝖰 𝒇(𝟏) + 𝒇(𝟐)𝟎is exact for all polynomials of degree ≤ 𝟐. Then 𝟐β−🇾= .

Q.54

For each 𝒙 ∈ (𝟎, 𝟏], consider the decimal representation 𝒙 = ∙ 𝒅_𝟏𝒅_𝟐𝒅_𝟑 ⋯ 𝒅_𝒏 ⋯. Define 𝒇: [𝟎, 𝟏] → ℝ by 𝒇(𝒙) = 𝟎 if 𝒙 is rational and 𝒇(𝒙) = 𝟏𝟖 𝒏 if 𝒙 is irrational, where 𝒏 is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of 𝒙. Then the Lebesgue integral ∫^𝟏_𝟎 (𝒙) 𝒅𝒙 = _______.

Q.55

Let𝒙̃=be an optimal solution of the following Linear Programming Problem 𝑷:

Maximize 𝟒𝒙_𝟏 + 𝒙_𝟐 − 𝟑𝒙_𝟑subject to

𝟐𝒙_𝟏 + 𝟒𝒙_𝟐 + 𝒂𝒙_𝟑 ≤ 𝟏𝟎,

𝒙𝟏 − 𝒙𝟐 + 𝒃𝒙𝟑 ≤ 𝟑,

𝟐𝒙_𝟏 + 𝟑𝒙_𝟐 + 𝟓𝒙_𝟑 ≤ 𝟏𝟏,

𝒙_𝟏 ≥ 𝟎, 𝒙_𝟐 ≥ 𝟎 and 𝒙_𝟑 ≥ 𝟎,

where 𝒂, 𝒃 are real numbers. If 𝒚̃ = is an optimal solution of the dual of 𝑷, then 𝒑 + 𝒒 + 𝒓 =_________(Round off to two 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	D	11	A	21	1 to 1	31	A	41	A	51	MTA
2	A	2	B	12	D	22	1 to 1	32	B	42	C	52	1 to 1
3	C	3	C	13	A	23	8 to 8	33	A	43	A	53	2 to 2
4	A	4	D	14	C	24	5 to 5	34	B	44	145 to 145	54	2 to 2
5	C	5	C	15	4 to 4	25	2 to 2	35	A	45	103 to 103	55	3.14 to 3.18
6	C	6	A	16	12 to 12	26	B	36	B	46	11 to 11
7	C	7	A	17	6 to 6	27	A	37	A	47	3 to 3
8	C	8	C	18	6 to 6	28	C	38	A	48	4 to 4
9	C OR D	9	B	19	2 to 2	29	A	39	A	49	MTA
10	D	10	D	20	32 to 32	30	A	40	A	50	2 to 2

* MTA means Marks to All

GATE 2021 Instrumentation 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	Getting to the top is than staying on top.
(A)	more easy
(B)	much easy
(C)	easiest
(D)	easier

Q.3	In a company, 35% of the employees drink coffee, 40% of the employees drink tea and 10% of the employees drink both tea and coffee. What % of employees drink neither tea nor coffee?
(A)	15
(B)	25
(C)	35
(D)	40

Q.4	⊕ and ⊙ are two operators on numbers p and q such that If 𝒙 ⊕ 𝒚 = 𝟐 ⊙ 𝟐, then x =
(A)	y2
(B)	y
(C)	3y2
(D)	2 y

Q.5	Four persons P, Q, R and S are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and R cannot sit adjacent to each other. S should be seated to the right of Q. The number of distinct seating arrangements possible is:
(A)	2
(B)	4
(C)	6
(D)	8

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


Q.6	Statement: Either P marries Q or X marries YAmong the options below, the logical NEGATION of the above statement is:
(A)	P does not marry Q and X marries Y.
(B)	Neither P marries Q nor X marries Y.
(C)	X does not marry Y and P marries Q.
(D)	P marries Q and X marries Y.

Q.7	Consider two rectangular sheets, Sheet M and Sheet N of dimensions 6 cm x 4 cm each. Folding operation 1: The sheet is folded into half by joining the short edges of the current shape. Folding operation 2: The sheet is folded into half by joining the long edges of the current shape. Folding operation 1 is carried out on Sheet M three times. Folding operation 2 is carried out on Sheet N three times.The ratio of perimeters of the final folded shape of Sheet N to the final folded shape of Sheet M is .
(A)	13 : 7
(B)	3 : 2
(C)	7 : 5
(D)	5 : 13

Q.10	Humans have the ability to construct worlds entirely in their minds, which don’t exist in the physical world. So far as we know, no other species possesses this ability. This skill is so important that we have different words to refer to its different flavors, such as imagination, invention and innovation.Based on the above passage, which one of the following is TRUE?
(A)	No species possess the ability to construct worlds in their minds.
(B)	The terms imagination, invention and innovation refer to unrelated skills.
(C)	We do not know of any species other than humans who possess the ability to construct mental worlds.
(D)	Imagination, invention and innovation are unrelated to the ability to construct mental worlds.

Instrumentation Engineering (IN)

Q.1 – Q.8 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	Consider the row vectors 𝒗 = (𝟏, 𝟎) and 𝒘 = (𝟐, 𝟎). The rank of the matrix𝑴 = 𝟐𝒗^𝑻𝒗 + 𝟑𝒘^𝑻𝒘, where the superscript T denotes the transpose, is
(A)	1
(B)	2
(C)	3
(D)	4

Q.2	Consider the sequence 𝒙𝒏 = 𝟎. 𝟓𝒙𝒏−𝟏 + 𝟏, 𝒏 = 𝟏, , … … with 𝒙𝟎 = 𝟎. Then 𝐥𝐢𝐦 𝒙_𝒏 is 𝒏→∞
(A)	0
(B)	1
(C)	2
(D)	∞

Q.3	An infinitely long line, with uniform positive charge density, lies along the z- axis. In cylindrical coordinates (𝒓, ∅, 𝐳), at any point ^⃗𝑷^→not on the z-axis, the direction of the electric field is
(A)	𝑟̂
(B)	_∅̂
(C)	𝑧̂
(D)

Q.5	In an ac main, the rms voltage Vac, rms current Iac and power Wac are measured as: Vac = 100 V ± 1%, Iac = 1 A ± 1% and Wac = 50 W ± 2% (errors are with respect to readings). The percentage error in calculating the power factor using these readings is
(A)	1%
(B)	2%
(C)	3%
(D)	4%

Q.7	Input-output characteristic of a temperature sensor is exponential for a
(A)	Thermistor
(B)	Thermocouple
(C)	Resistive Temperature Device (RTD)
(D)	Mercury thermometer

Q.8	The signal 𝐬𝐢𝐧(√𝟐𝝅𝒕) is
(A)	periodic with period T = √2𝜋
(B)	not periodic
(C)	periodic with period T = 2𝜋
(D)	periodic with period T = 4𝜋²

Q.9 – Q.11 Multiple Select Question (MSQ), carry ONE mark each (no negative marks).

Q.9	The step response of a circuit is seen to have an oscillatory behaviour at the output with oscillations dying down after some time. The correct inference(s) regarding the transfer function from input to output is/are
(A)	that it is of at least second order.
(B)	that it has at least one pole-pair that is underdamped.
(C)	that it does not have a real pole.
(D)	that it is a first order system.

Q.10	For a 4-bit Flash type Analog to Digital Convertor (ADC) with full scale input voltage range “V”, which of the following statement(s) is/are true?
(A)	The ADC requires 15 comparators.
(B)	The ADC requires one 4 to 2 priority encoder and 4 comparators.
(C)	A change in the input voltage by will always flip MSB of the output.
(D)	A change in the input voltage by will always flip the LSB of the output.

Q.11	A 16-bit microprocessor has twenty address lines (A0 to A19) and 16 data lines. The higher eight significant lines of the data bus of the processor are tied to the 8-data lines of a 16 Kbyte memory that can store one byte in each of its 16K address locations. The memory chip should map onto contiguous memory locations and occupy only 16 Kbyte of memory space. Which of the following statement(s) is/are correct with respect to the above design?
(A)	If the 16 Kbyte of memory chip is mapped with a starting address of 80000H, then the ending address will be 83FFFH.
(B)	The active high chip-select needed to map the 16 Kbyte memory with a starting address at F0000H is given by the logic expression (A₁₉ · A₁₈ · A₁₇ · A₁₆).
(C)	The 16 Kbyte memory cannot be mapped with contiguous address locations with a starting address as 0F000H using only A₁₉ to A₁₄ for generating chip select.
(D)	The above chip cannot be interfaced as the width of the data bus of the processor and the memory chip differs.

Q.12 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).


Q.12	A single-phase transformer has a magnetizing inductance of 250 mH and a core loss resistance of 300 Ω, referred to primary side. When excited with a 230 V, 50 Hz sinusoidal supply at the primary, the power factor of the input current drawn, with secondary on open circuit, is (rounded off to two decimal places).

Q.13	Taking N as positive for clockwise encirclement, otherwise negative, the number of encirclements N of (−𝟏 , 𝟎) in the Nyquist plot of 𝑮(𝒔) = is. _______________.

Q.14	The diode used in the circuit has a fixed voltage drop of 0.6 V when forward biased. A signal vs is given to the ideal OpAmp as shown. When vs *is at its positive peak, the output (vOA) of the OpAmp in volts is* .

Q.21	An amplitude modulation (AM) scheme uses tone modulation, with modulation index of 0.6. The power efficiency of the AM scheme is ________% (rounded off to one decimal place).

Q.22	When the movable arm of a Michelson interferometer in vacuum (𝒏 = 𝟏) is moved by 𝟑𝟐𝟓 𝛍𝐦, the number of fringe crossings is 1000. The wavelength of the laser used in nanometers is ________ .

Q.23	Consider the function (𝒙) = −𝒙^𝟐 + 𝟏𝟎𝒙 + 𝟏𝟎𝟎. The minimum value of the function in the interval [𝟓, 𝟏𝟎] is________ .

Q.24	Let (𝒛) =defined in the complex plane. The integral ∮𝒄 𝒇(𝒛)𝒅𝒛 over the contour of a circle c with center at the origin and unit radius is________.

Q.25	The determinant of the matrix M shown below is ___________.

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

Q.26	𝒇(𝒛) = (𝒛 − 𝟏)^−𝟏 − 𝟏 + (𝒛 − 𝟏) − (𝒛 − 𝟏)^𝟐 + ⋯ is the series expansion of
(A)	−1/𝑧(𝑧 − 1) for \|𝑧 − 1\| < 1
(B)	1/ 𝑧(𝑧 − 1) for \|𝑧 − 1\| < 1
(C)	1/(z-1)²for \|𝑧 − 1\| < 1
(D)	-1 / (z-1)for \|𝑧 − 1\| < 1

Q.27	A single-phase transformer has maximum efficiency of 98 %. The core losses are 80 W and the equivalent winding resistance as seen from the primary side is 0.5 Ω. The rated current on the primary side is 25 A. The percentage of the rated input current at which the maximum efficiency occurs is
(A)	35.7%
(B)	50.6%
(C)	80.5%
(D)	100%

Q.28	A slip-ring induction motor is expected to be started by adding extra resistance in the rotor circuit. The benefit that is derived by adding extra resistance in the rotor circuit in comparison to the rotor being shorted is
(A)	The starting torque would be higher.
(B)	The power factor at start will be lower.
(C)	The starting current is higher.
(D)	The losses at starting would be lower.

Q.29	Consider a unity feedback configuration with a plant and a PID controller as shown in the figure. 𝑮(𝒔) =and 𝑪(𝒔) = with K being scalar. The closed loop is
(A)	only stable for 𝐾 > 0
(B)	only stable for 𝐾 between -1 and +1
(C)	only stable for 𝐾 < 0
(D)	stable for all values of 𝐾

Q.31	A Boolean function F of three variables X, Y, and Z is given asF (X, Y, Z) = (Xʹ + Y + Z)˖(X + Yʹ + Zʹ )˖(Xʹ + Y + Zʹ )˖( Xʹ Yʹ Zʹ + Xʹ Y Zʹ + X YZʹ )Which one of the following is true?
(A)	F (X, Y, Z) = (X + Y + Zʹ)˖( Xʹ + Yʹ +Zʹ )
(B)	F (X, Y, Z) = (Xʹ + Y)˖(X + Y ʹ+ Zʹ )
(C)	F (X, Y, Z) = Xʹ Zʹ + Y Zʹ
(D)	F (X, Y, Z) = Xʹ Yʹ Z + X Y Z

Q.32	A 10½ digit Counter-timer is set in the ‘frequency mode’ of operation (with T_s = 1s). For a specific input, the reading obtained is 1000. Without disconnecting this input, the Counter-timer is changed to operate in the ‘Period mode’ and the range selected is microseconds (µs, with fs = 1 MHz). The counter will then display
(A)	0
(B)	10
(C)	100
(D)	1000

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


Q.37	Consider that X and Y are independent continuous valued random variables with uniform PDF given by X ~ U(2, 3) and Y ~ U(1, 4). Then P(Y ≤ X) is equal to (rounded off to two decimal places).

Q.38	Given A=. The value of the determinant │A4 -5A3 +6A2 +2 I│=_________.

Q.39	The figure below shows an electrically conductive bar of square cross-section resting on a plane surface. The bar of mass of 1 kg has a depth of 0.5 m along the y direction. The coefficient of friction between the bar and the surface is 0.1. Assume the acceleration due to gravity to be 10 m/s². The system faces a uniform flux density B = −𝟏 𝒛̂ T. At time t = 0, a current of 10 A is switched onto the bar and is maintained. When the bar has moved by 1 m, its speed in metre per second is _______ (rounded off to one decimal place).
Q.40	A toroid made of CRGO has an inner diameter of 10 cm and an outer diameter of 14 cm. The thickness of the toroid is 2 cm. 200 turns of copper wire is wound on the core. µo = 4π×10^-7 H/m and µR of CRGO is 3000. When a current of 5 mA flows through the winding, the flux density in the core in millitesla is .

Q.41	An air cored coil having a winding resistance of 10 Ω is connected in series with a variable capacitor Cx. The series circuit is excited by a 10 V sinusoidal voltage source of angular frequency 1000 rad/s. As the value of the capacitor is varied, a maximum voltage of 30 V was observed across it. Neglecting skin- effect, the value of the inductance of the coil in millihenry is_________ .

Q.42	A household fan consumes 60 W and draws a current of 0.3125 A (rms) when connected to a 230 V (rms) ac, 50 Hz single phase mains. The reactive power drawn by the fan in VAr is ___(rounded off to the nearest integer).

Q.43	Given (𝒕) = 𝒆^−𝟑𝒕𝒖(𝒕) ∗ 𝒖(𝒕 + 𝟑)*, where denotes convolution operation. The value of (𝒕) 𝐚𝐬 𝒕 → ∞ is (rounded off to two decimal places).**

Q.45	A sinusoid (√𝟐 𝐬𝐢𝐧 𝒕) 𝝁(𝒕), where 𝝁(𝒕) is the step input, is applied to a system with transfer-function 𝑮(𝒔) = The amplitude of the steady state output is________.

Q.46	Consider a system with transfer-function. A unit step function (𝒕) is applied to the system,which results in an output y(𝒕).If (𝒕) = 𝒚(𝒕) − 𝝁(𝒕), then 𝐥𝐢𝐦 𝒆(𝒕) is . ^𝒕→∞

Q.47	The circuit shown below uses an ideal OpAmp. Output VO in volt is___________(rounded off to one decimal place).

Q.50	A 3½ digit, rectifier type digital meter is set to read in its 2000 V range. A symmetrical square wave of frequency 50 Hz and amplitude ±100 V is measured using the meter. The meter will read .

Q.51	A bar primary current transformer of rating 1000/1 A, 5VA, UPF has 995 secondary turns. It exhibits zero ratio error and phase error of 30 minutes at 1000 A with rated burden. The watt loss component of the primary excitation current in ampere is_______(rounded off to one decimal place).

Q.52	In the bridge circuit shown, the voltmeter V showed zero when the value of the resistors are: R1 = 100 Ω, R2 = 110 Ω, and R3 = 90 Ω. If (R1/R2) = (RA/RB), the value of R4 in ohm is ________.

Answer Key

Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans	Q.No.	Ans
1	D	1	A	11	A; C	21	15.0 to 15.5	31	C	41	30 to 30	51	4.9 to 5.1
2	B	2	C	12	0.24 to 0.26	22	650 to 650	32	D	42	39 to 40	52	99 to 99
3	C	3	A	13	-1 to -1	23	100 to 100	33	B	43	0.3 to 0.35	53	2.45 to 2.55
4	B	4	C	14	1 to 1	24	0 to 0	34	D	44	10 to 10	54	120 to 120
5	C	5	D	15	10 to 10	25	4 to 4	35	B	45	0.95 to 1.05	55	1.30 to 1.35
6	B	6	C	16	0.2 to 0.2	26	B	36	B	46	1 to 1
7	A	7	A	17	4.9 to 5.1	27	B	37	0.45 to 0.55	47	1.0 to 1.1
8	A	8	B	18	1000.25 to 1000.25	28	A	38	4 to 4	48	1 to 1
9	B	9	A; B	19	1333 to 1360	29	A	39	2.7 to 2.9	49	0 to 0
10	C	10	A; D	20	10 to 10	30	C	40	10 to 10	50	111 to 111

GATE 2021 Geophysics 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 people _________were at the demonstration were from all sections of society.
(A)	whose
(B)	which
(C)	who
(D)	whom

Q.3	For a regular polygon having 10 sides, the interior angle between the sides of the polygon, in degrees, is:
(A)	396
(B)	324
(C)	216
(D)	144

Q.4	Which one of the following numbers is exactly divisible by (11¹³+1)?
(A)	11²⁶+1
(B)	11³³+1
(C)	11³⁹-1
(D)	11⁵²-1

Q.5	Oasis is to sand as island is to________Which one of the following options maintains a similar logical relation in the above sentence?
(A)	Stone
(B)	Land
(C)	Water
(D)	Mountain

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

Q.6	The importance of sleep is often overlooked by students when they are preparing for exams. Research has consistently shown that sleep deprivation greatly reduces the ability to recall the material learnt. Hence, cutting down on sleep to study longer hours can be counterproductive.Which one of the following statements is the CORRECT inference from the above passage?
(A)	Sleeping well alone is enough to prepare for an exam. Studying has lesser benefit.
(B)	Students are efficient and are not wrong in thinking that sleep is a waste of time.
(C)	If a student is extremely well prepared for an exam, he needs little or no sleep.
(D)	To do well in an exam, adequate sleep must be part of the preparation.

Q.8	Let X be a continuous random variable denoting the temperature measured. The range of temperature is [0, 100] degree Celsius and let the probability density function of X be f(x) = 0.01 for 0 ≤ X ≤ 100.The mean of X is_________
(A)	2.5
(B)	5.0
(C)	25.0
(D)	50.0

Q.10	Seven cars P, Q, R, S, T, U and V are parked in a row not necessarily in that order. The cars T and U should be parked next to each other. The cars S and V also should be parked next to each other, whereas P and Q cannot be parked next to each other. Q and S must be parked next to each other. R is parked to the immediate right of V. T is parked to the left of U.Based on the above statements, the only INCORRECT option given below is:
(A)	There are two cars parked in between Q and V.
(B)	Q and R are not parked together.
(C)	V is the only car parked in between S and R.
(D)	Car P is parked at the extreme end.

Geophysics (GG)

Q.1- Q.15 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	Which of the given planets has the highest average density?
(A)	Mercury
(B)	Venus
(C)	Earth
(D)	Mars

Q.2	In a multi-electrode resistivity tomography (ERT) survey, using equally spaced electrodes, which of the given configurations will provide the maximum number of data points?
(A)	Wenner array
(B)	Axial Dipole-dipole array
(C)	Axial Pole-dipole array
(D)	Schlumberger array

Q.3	In Electromagnetic methods of prospecting, which one of the given options is CORRECT about frequency and type of current source for the Primary field used?
(A)	High frequency A.C.
(B)	Low frequency A.C.
(C)	Both high frequency A.C. and D.C.
(D)	Low frequency D.C.

Q.4	‘Group’ is a unit of:
(A)	Lithostratigraphy
(B)	Sequence stratigraphy
(C)	Biostratigraphy
(D)	Chronostratigraphy

Q.5	Furongian is an Epoch of:
(A)	Cambrian
(B)	Ordovician
(C)	Triassic
(D)	Cretaceous

Q.6	The stage of textural maturity of a clay-rich sandstone containing poorly- sorted and angular framework grains is:
(A)	Mature
(B)	Supermature
(C)	Immature
(D)	Submature

Q.7	Which one of the following structures indicates Synsedimentary deformation?
(A)	Festoon bedding
(B)	Flaser bedding
(C)	Tabular bedding
(D)	Convolute bedding

Q.8	Low value in SP log as observed in dispersed shales is mainly due to the impeded movement of:
(A)	Na⁺ ion
(B)	Cl ion
(C)	K⁺ ion
(D)	OH^– ion

Q.9	In Radiometric survey, the g-ray spectrometer count rate depends on:
(A)	Cracks present in the target rock volume
(B)	Solid angle of the target rock about the spectrometer
(C)	Temperature in the target rock
(D)	Pressure in the target rock

Q.10	The dimension of radiant emittance of a blackbody as per Stefan- Boltzmann law is:
(A)	_M0_L1_T-1
(B)	_M1_L-1_T-2
(C)	_M1_L2_T-2
(D)	_M1_L0_T-3

Q.11	A surface geological process that can create a landform called Cirque is:
(A)	aeolian deposition
(B)	fluvial deposition
(C)	glacial erosion
(D)	deposition of volcanic ash

Q.12	If α and β are P- and S-wave velocities, respectively, then α²– (4/3)β²is equal to:(k is the bulk modulus, μ is shear modulus and ρ is density)
(A)	k/ρ
(B)	μ/ρ
(C)	k + μ/ρ
(D)	k – μ/ρ

Q.13	Which one of the following phases is P-wave that converts to S-wave during passage through the solid inner core?
(A)	PKIKP
(B)	PKJKP
(C)	PKiKP
(D)	PKPPcP

Q.14	In reduction of gravity data, the latitude correction is maximum at:
(A)	35° latitude
(B)	45° latitude
(C)	55° latitude
(D)	65° latitude

Q.15	The most coaliferous unit of the Gondwana Supergroup is:
(A)	Talchir Formation
(B)	Barakar Formation
(C)	Karharbari Formation
(D)	Panchet Formation

Q.16 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.16

A vertical borehole encounters a shale bed of uniform thickness occurring at a depth of 5 m and dipping 60^o. The borehole pierces through this shale bed for a length of 10 m to reach a sandstone layer below. The true thickness of the shale bed is_______m. [in integer]

Q.17

The mass and volume of a fully dried soil sample are 2200 gm and 1100 cm³, respectively. If the specific gravity of the soil particles is 2.5 and water density is 1 gm/cm³, the void ratio of the soil is______ . [round off to 2 decimal places]

Q.18

A constant-head permeability test was performed on a vertical sand column of height 40 cm and cross-sectional area of 25 cm². During the test, when the loss of head was 50 cm, the volume of water collected in 2 minutes was 300 cm³. Applying Darcy’s law, the calculated coefficient of permeability of the sand column is_______ cm/sec. [round off to 2 decimal places]

Q.19

The radius (r) of the oblate spheroid at 45° latitude with ellipticity of polar flattening of 1/298.25 and equatorial radius of 6378140 m is_____km. [round off to 2 decimal places]

Q.20

Light passes through two media with refractive indices of 1.75 and 1.55, respectively. The thickness of both the media is 30 m. The resultant path difference of the yellow light component (λ = 589 nm) is________ μm. (Take π = 3.141) [round off to one decimal places]

Q.21

The water table in an unconfined aquifer at a place near the coast is 1 m above the Mean Sea Level. Given the densities of fresh and saline water as 1.001 and 1.025 g/cc, respectively, the fresh-saline water interface at the same location should be at a depth of ____________ m from the water table. [round off to one decimal place]

Q.22

The volume percentage of galena and quartz in an ore body of Pb are 90 and 10, respectively. The densities of galena and quartz are 7.6 and 2.65 g/cc, respectively. The grade of the ore body in terms of weight percent of Pb is______. (Atomic weights of Pb = 206 and S = 32) [round off to 2 decimal places]

Q.23

Normal moveout (NMO) for reflected phase of seismic data is 2 milliseconds. Consider the diffraction source at the edge of the same reflector, where the shot point is directly above diffraction source. In this case, the NMO due to diffraction is milliseconds [in integer].

Q.24

In a 2D seismic survey, first receiver location is at (1000 m, 4000 m), second receiver location is at (2000 m, 4000 m) and the source location is at (2000 m, 1000 m). Consider P-wave velocity as 5000 m/sec. The difference in first arrival time of P-wave phase for the two receivers is_______ seconds. [round off to 2 decimal places].

Q.25

The potential difference measured between potential electrodes using Wenner array is 500 mV when a current of 2 A is passed through the subsurface between current electrodes. If the computed apparent resistivity is 100Ωm then the distance between the current electrodes will be________m. [round off to 2 decimal places] (Use π = 3.141)

Q.26 – Q.42 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).

Q.26	In a horizontally stratified cuboid rock sample (stratified in vertical z direction with various layers of different resistivity), bulk resistivity is measured in three perpendicular directions. If ρ₁, ρ₂, and ρ₃ are the bulk resistivities measured perpendicular to xy, xz and yz planes, respectively, then
(A)	ρ₁ < ρ₂ = ρ₃
(B)	ρ₁ > ρ₂= ρ₃
(C)	ρ₁ = ρ₂ ≠ ρ₃
(D)	ρ₁ ≠ ρ₂ ≠ ρ₃

Q.27	Which one is the CORRECT sequence of electromagnetic methods in terms of depth of investigation? P – AFMAG method Q – VLF methodR – GPR methodS – Magnetotelluric method
(A)	P > Q > S > R
(B)	P > S > R > Q
(C)	S > P > Q > R
(D)	S > Q > R > P

Q.28	Which Norm gives the maximum weight to the data points having maximum deviation/outlier from the smoothly fitted curve during linearized inversion?
(A)	L1-Norm
(B)	L2-Norm
(C)	Lp-Norm
(D)	L∞-Norm

Q.29	Which one of the following statements is CORRECT for the Quenching agent used in the tube of Geiger-Muller counter?
(A)	It enhances the emission of secondary electrons from the cathode.
(B)	It reduces the emission of secondary electrons from the cathode.
(C)	It enhances the emission of secondary electrons from the anode.
(D)	It reduces the emission of secondary electrons from the anode.

Q.30	Which one of the following statements is CORRECT regarding the property of Laplacian operator for vector/scalar fields?
(A)	Laplacian of a vector field is zero if the Laplacian of each of its components are zero.
(B)	Laplacian of a vector field is zero if the Laplacian of any one of its component is zero.
(C)	If the Laplacian of a scalar field is zero then the scalar field is not harmonic.
(D)	If the Laplacian of a scalar field is finite (non-zero) then the scalar field is harmonic.

Q.31	The most desirable interaction of ү-ray with matter for ү-ray spectroscopy is
(A)	Photoelectric effect only
(B)	Both Photoelectric effect and Compton scattering
(C)	Both Compton Scattering and Pair production
(D)	Photoelectric effect, Compton scattering and Pair production

Q.32	Which one of the following is the CORRECT sequence for a 2D seismic reflection data processing prior to Time-depth conversion?
(A)	Migration → Deconvolution → Filtering → Equalization → Coherency
(B)	Deconvolution → Migration → Filtering → Coherency → Equalization
(C)	Filtering → Deconvolution → Migration → Equalization → Coherency
(D)	Deconvolution → Filtering → Equalization → Migration → Coherency

Q.33	Choose the CORRECT procedure to avoid the area of cracked, altered formation in Sonic log.
(A)	Measure interval transit times using long-spacing sonic tools.
(B)	Use more number of sets of sources.
(C)	Measure interval transit times using short-spacing sonic tools.
(D)	Use more number of sets of detectors.

Q.34	The factor that DOES NOT influence measurement of Nuclear Magnetic Resonance log is
(A)	mineral composition of the rock.
(B)	bound water (irreducible water).
(C)	free water.
(D)	pore fluid pressure.

Q.35	Consider a time-invariant geophysical filter with the given input as:𝒙(𝒕) = 𝒆⁻^𝑎𝒕 𝒘𝒉𝒆𝒏 𝒕 ≥ 𝟎 ; 𝒙(𝒕) = 𝟎 𝒘𝒉𝒆𝒏 𝒕 < 𝟎 and output𝒚(𝒕) = 𝒆⁻^𝛽𝒕 𝒘𝒉𝒆𝒏 𝒕 ≥ 𝟎 ; 𝒚(𝒕) = 𝟎 𝒘𝒉𝒆𝒏 𝒕 < 𝟎. The transfer function for the given input and output of time-invariant filter will be:
(A)	𝛼 + 𝑖𝜔𝛽 – 𝑖𝜔
(B)	𝛼 + 𝑖𝜔𝛽 + 𝑖𝜔
(C)	𝛼 – 𝑖𝜔𝛽 + 𝑖𝜔
(D)	𝛼 – 𝑖𝜔𝛽 – 𝑖𝜔

Q.37	If a mountain range is 100% isostatically compensated (Airy’s type), what would be the expected nature of the Bouguer anomaly and free air anomaly?
(A)	Bouguer anomaly is very large and negative; free air anomaly is small and positive.
(B)	Bouguer anomaly is very large and negative; free air anomaly is large and positive.
(C)	Bouguer anomaly is exactly zero; free air anomaly is very large and positive.
(D)	Bouguer anomaly is very large and negative; free air anomaly is large and negative.

Q.38	Which of the following is INCORRECT for recorded nuclear explosion event?
(A)	The first P-wave from an explosion to arrive at any seismic station, irrespective of Azimuth, should be compressional.
(B)	Nuclear explosions are not as good as earthquakes at generating surface waves or S-waves.
(C)	In general, earthquakes have Mb values same those for nuclear explosions with same Ms values.
(D)	Nuclear explosions have all been shallower than 2 km depth.

Q.39	Focal depth can be determined from measurement of the difference in the travel time between:
(A)	pP and P
(B)	PP and P
(C)	PcP and P
(D)	PPP and P

Q.40	Of the following options, at which discontinuity both P-wave and S-wave have maximum velocity drop?
(A)	Conard
(B)	Mohorovicic
(C)	Gutenberg
(D)	Lehman

Q.41	In data enhancement techniques, what is the advantage of magnetic anomaly being ‘Reduced to the pole’?
(A)	Enhances the signal to noise ratio.
(B)	Estimates the depth to the basement.
(C)	Takes care of variation of the magnetic anomaly with latitude.
(D)	Helps in pseudo-gravity transformation.

Q.43 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.43

If a gravity determination is made at an elevation of 150 m above mean sea level, the Bouguer correction required for a density contrast of 250 kg m^-3with the surroundings is _____________ mgal. [round off to 2 decimal places]

Q.44

An infinite horizontal cylinder of radius 40 km is buried at a depth of 100 km and yields the same maximum gravity anomaly as that of an infinite horizontal cylinder of radius 1 km, buried at a depth of 1 km having density contrast with the surroundings of 200 kgm^-3. The density contrast of the deeper cylinder with respect to the surrounding is kg/m³. [round off to one decimal place]

Q.45

An earthquake causes an average of 25 m strike slip displacement over a 50 km long, 25 km deep portion of a transform fault. Assuming that the rock rigidity is 3 x 10¹⁰Nm^-2, the moment magnitude (Mw) of the earthquake is ___________.[round off to 2 decimal places]

Q.46

Lithological unit X is sandwiched between Y1 above and Y2 below it. Now consider a log across lithology X, where Gamma ray (GR) reading is given by 100API; Y1 lithology, where minimum GR reading is 10API; and Y2 lithology of shale, where GR reading is 200API. Then the shale-free fractional volume in the X lithology will be____ . [round off to 2 decimal places]

Q.47

In a 2D seismic survey, 25 receivers are placed in a group and 25 sources are placed in another group, where random noise is present. The signal to noise ratio for this arrangement will be________. [in integer]

Q.48

In a VSP survey, the tube wave passage through borehole causes cross- section area change from 0.79 m²to 1.13 m². The transmission coefficient will be _______. [round off to 2 decimal places]

Q.49

In a cratonic region, radioactive heat generation decreases exponentially with depth. Assuming characteristic depth as 10 km and surface heat generation as 3 Wm^-3and neglecting mantle heat flow, the heat production per unit volume for a 30 km thick layer will be μWm^-3. [round off to 2 decimal places]

Q.50

In an Induced Polarization survey, 50 milliseconds chargeability was measured for steady state voltage (full saturation reached) of 200 V between potential electrodes. When the current was switched off, the voltage across potential electrodes drops instantaneously (time t = 0 s) to a level Va and thereafter decays linearly with time and becomes zero in 10 seconds. The magnitude of instantaneous voltage Va (at time t = 0 s) will be mV. [in integer]

Q.51

Apparent resistivity sounding data for Schlumberger array is theoretically generated by the teacher for the following 4-layer model as: ρ1 = 100 Ωm,ρ₂ = 20Ωm, ρ₃ = 500Ωm, ρ₄ = 10Ωm and layer thicknesses h₁ = 50 m,h₂= 20 m and h₃ = 50 m. If the student interprets this theoretical sounding data for ρ₃ as 750Ωm, then according to the Principle of Equivalence, the thickness h3 would be m. [round off to 2 decimal places]

Q.52

A 3D conducting body is located at a depth of 50 m in a homogeneous medium of resistivity 500 Ωm. A frequency of f1 Hz is appropriate to detect this conducting body in a plane wave EM survey. When the same conducting body is located in a host medium of resistivity 100 m at the same depth then a frequency of f2 Hz is found to be appropriate. Then the value of will be_________. [round off to 2 decimal places]

Q.53

The apparent resistivity and phase computed for MT measurement at 10^-3Hz frequency is 500 Ωm and 30⁰, respectively. Ratio of Imaginary to Real component of the Impedance tensor is . [round off to 2 decimal places]

Q.54

The diagonal elements of a covariance matrix computed for a linearized inverse problem having model parameters m1, m2, m3, m4, m5 are 49, 15, 3, 200, 40, respectively. The standard deviation (uncertainty) in the estimation of model parameters m4 is____. [round off to 2 decimal places]

Q.55

Electric current density incident at an angle 40 from vertical at the horizontal interface between two layers with resistivity 1 = 100 m and2 = 500 m (from layer 1 to layer 2). The current density will enter into the second layer at an angle degrees from vertical. [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	C	21	42.6 to 42.7	31	A	41	C	51	33.00 to 34.00
2	C	2	C	12	A	22	83.30 to 83.34	32	D	42	C	52	0.19 to 0.21
3	D	3	B	13	B	23	4 to 4	33	A	43	1.55 to 1.65 OR -1.65 to -1.55	53	0.57 to 0.59
4	D	4	A	14	B	24	0.03 to 0.03	34	A	44	12.5 to 12.5	54	14.00 to 15.00
5	C	5	A	15	B	25	190.00 to 192.00	35	A	45	3.24 to 3.26 OR 7.90 to 7.92	55	9.45 to 9.55
6	D	6	C	16	5 to 5	26	B	36	C	46	0.52 to 0.53
7	C	7	D	17	0.25 to 0.25	27	C	37	A	47	25 to 25
8	D	8	B	18	0.08 to 0.08	28	D	38	C	48	0.82 to 0.82
9	D	9	B	19	6367.44 to 6367.46	29	B	39	A	49	0.95 to 0.95
10	A	10	D	20	64.0 to 64.0	30	A	40	C	50	2000 to 2000

GATE 2021 Geology Previous Year Paper

GATE 2021 Geology 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 people _______were at the demonstration from all sections of society.
(A)	whose
(B)	which
(C)	who
(D)	whom

Q.3	For a regular polygon having 10 sides, the interior angle between the sides of the polygon, in degrees, is:
(A)	396
(B)	324
(C)	216
(D)	144

Q.4	Which one of the following numbers is exactly divisible by (11¹³+1)?
(A)	11²⁶ +1
(B)	11³³ +1
(C)	11³⁹ −1
(D)	11⁵² −1

Q.5	Oasis is to sand as island is to _____________Which one of the following options maintains a similar logical relation in the above sentence?
(A)	Stone
(B)	Land
(C)	Water
(D)	Mountain

Q6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).

Q.6	The importance of sleep is often overlooked by students when they are preparing for exams. Research has consistently shown that sleep deprivation greatly reduces the ability to recall the material learnt. Hence, cutting down on sleep to study longer hours can be counterproductive.Which one of the following statements is the CORRECT inference from the above passage?
(A)	Sleeping well alone is enough to prepare for an exam. Studying has lesser benefit.
(B)	Students are efficient and are not wrong in thinking that sleep is a waste of time.
(C)	If a student is extremely well prepared for an exam, he needs little or no sleep.
(D)	To do well in an exam, adequate sleep must be part of the preparation.

Q.8	Let X be a continuous random variable denoting the temperature measured. The range of temperature is [0, 100] degree Celsius and let the probability density function of X be f(x) = 0.01 for 0 ≤ X ≤ 100.The mean of X is
(A)	2.5
(B)	5.0
(C)	25.0
(D)	50.0

Q.10	Seven cars P, Q, R, S, T, U and V are parked in a row not necessarily in that order. The cars T and U should be parked next to each other. The cars S and V also should be parked next to each other, whereas P and Q cannot be parked next to each other. Q and S must be parked next to each other. R is parked to the immediate right of V. T is parked to the left of U.Based on the above statements, the only INCORRECT option given below is:
(A)	There are two cars parked in between Q and V.
(B)	Q and R are not parked together.
(C)	V is the only car parked in between S and R.
(D)	Car P is parked at the extreme end.

Geology (GG)

– Q.15 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).

Q.1	Which of the given planets has the highest average density?
(A)	Mercury
(B)	Venus
(C)	Earth
(D)	Mars

Q.2	In a multi-electrode resistivity tomography (ERT) survey, using equally spaced electrodes, which of the given configurations will provide the maximum number of data points?
(A)	Wenner array
(B)	Axial Dipole-dipole array
(C)	Axial Pole-dipole array
(D)	Schlumberger array

Q.3	In Electromagnetic methods of prospecting, which one of the given options is CORRECT about frequency and type of current source for the Primary field used?
(A)	High frequency A.C.
(B)	Low frequency A.C.
(C)	Both high frequency A.C. and D.C.
(D)	Low frequency D.C.

Q.4	‘Group’ is a unit of:
(A)	Lithostratigraphy
(B)	Sequence stratigraphy
(C)	Biostratigraphy
(D)	Chronostratigraphy

Q.5	Furongian is an Epoch of:
(A)	Cambrian
(B)	Ordovician
(C)	Triassic
(D)	Cretaceous

Q.6	The stage of textural maturity of a clay-rich sandstone containing poorly- sorted and angular framework grains is:
(A)	Mature
(B)	Supermature
(C)	Immature
(D)	Submature

Q.7	Which one of the following structures indicates Synsedimentary deformation?
(A)	Festoon bedding
(B)	Flaser bedding
(C)	Tabular bedding
(D)	Convolute bedding

Q.8	Low value in SP log as observed in dispersed shales is mainly due to the impeded movement of:
(A)	Na⁺ ion
(B)	Cl ion
(C)	K⁺ ion
(D)	OH^– ion

Q.9	In Radiometric survey, the g-ray spectrometer count rate depends on:
(A)	Cracks present in the target rock volume
(B)	Solid angle of the target rock about the spectrometer
(C)	Temperature in the target rock
(D)	Pressure in the target rock

Q.10	The dimension of radiant emittance of a blackbody as per Stefan- Boltzmann law is:
(A)	_M0_L1_T-1
(B)	_M1_L-1_T-2
(C)	_M1_L2_T-2
(D)	_M1_L0_T-3

Q.11	A surface geological process that can create a landform called Cirque is:
(A)	aeolian deposition
(B)	fluvial deposition
(C)	glacial erosion
(D)	deposition of volcanic ash

Q.12	If α and β are P- and S-wave velocities, respectively, then α²– (4/3)β²is equal to:(κ is the bulk modulus, μ is shear modulus and ρ is density)
(A)	κ/ρ
(B)	μ/ρ
(C)	κ + μ/ρ
(D)	κ – μ/ρ

Q.13	Which one of the following phases is P-wave that converts to S-wave during passage through the solid inner core?
(A)	PKIKP
(B)	PKJKP
(C)	PKiKP
(D)	PKPPcP

Q.14	In reduction of gravity data, the latitude correction is maximum at:
(A)	35° latitude
(B)	45° latitude
(C)	55° latitude
(D)	65° latitude

Q.15	The most coaliferous unit of the Gondwana Supergroup is:
(A)	Talchir Formation
(B)	Barakar Formation
(C)	Karharbari Formation
(D)	Panchet Formation

Q.16 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.16

Q.17

Q.18

Q.19

The radius (r) of the oblate spheroid at 45° latitude with ellipticity of polar flattening of 1/298.25 and equatorial radius of 6378140 m is________ km. [round off to 2 decimal places]

Q.20

Light passes through two media with refractive indices of 1.75 and 1.55, respectively. The thickness of both the media is 30 μm. The resultant path difference of the yellow light component (λ = 589 nm) is _______μm. (Take π = 3.141) [round off to one decimal places]

Q.21

The water table in an unconfined aquifer at a place near the coast is 1 m above the Mean Sea Level. Given the densities of fresh and saline water as1.001 and 1.025 g/cc, respectively, the fresh-saline water interface at the same location should be at a depth of _______m from the water table. [round off to one decimal place]

Q.22

The volume percentage of galena and quartz in an ore body of Pb are 90 and 10, respectively. The densities of galena and quartz are 7.6 and 2.65 g/cc, respectively. The grade of the ore body in terms of weight percent of Pb is _________ . (Atomic weights of Pb = 206 and S = 32) [round off to 2 decimal places]

Q.23

Q.24

Q.25

The potential difference measured between potential electrodes using Wenner array is 500 mV when a current of 2 A is passed through the subsurface between current electrodes. If the computed apparent resistivity is 100 Ωm then the distance between the current electrodes will be_________m. [round off to 2 decimal places] (Use π = 3.141)

Q.26 – Q.42 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).

Q.26	Which one of the following statements is CORRECT?
(A)	Taphonomy refers to the study of fossilization pathways from death of an organism to its recovery as a fossil.
(B)	Biostratinomy refers to the study of fossilization pathways from burial of an organism under sediments to its recovery as a fossil.
(C)	Biostratinomy is an integral component of biostratigraphy and refers to the characterization of strata based on fossil content.
(D)	Taphonomy refers to the study of fossilization pathways from death of an organism to its burial under the sediments.

Q.27	Based on the three statements given below, choose the CORRECT option:Statement I: Gunderdehi Formation is a stratigraphic unit of the Chhattisgarh Supergroup.Statement II: Raniganj Formation is a coal-bearing Triassic unit of the Gondwana Supergroup.Statement III: Pitepani Volcanics is a stratigraphic unit of the Dongargarh Supergroup.
(A)	All the statements are correct
(B)	Statement I is correct, but statements II and III are incorrect
(C)	Statements I and III are correct, but statement II is incorrect
(D)	Statements II and III are correct but statement I is incorrect

Q.28	Which one of the following equid genera was a one-toed grazer?
(A)	Merychippus
(B)	Parahippus
(C)	Pliohippus
(D)	Mesohippus

Q.30	Tillite with faceted boulders and green shale with dropstones characterize the lithology of:
(A)	Lameta Formation
(B)	Bagra Formation
(C)	Talchir Formation
(D)	Panchet Formation

Q.35	With regard to occurrence of groundwater in an area, which of the given statements is CORRECT?
(A)	Vadose water occurs in the zone of saturation.
(B)	The zone of aeration lies below the zone of saturation.
(C)	The water table marks the uppermost surface of the vadose zone.
(D)	The depth of the perched water table is less than that of the water table.

Q.36	There are indications of presence of a massive tabular multimetal sulfide ore body at a shallow depth from the surface. Which of the following would be the most efficient geophysical method to confirm the presence of the ore body?
(A)	resistivity sounding
(B)	ground geomagnetic survey
(C)	self-potential method of geophysical prospecting
(D)	ground gravity survey

Q.37	The following reaction takes place in the amphibolite grade of metamorphism of pelitic rocks: kyanite + chlorite ⇔ staurolite + quartz +H₂O Which of the following is a CORRECT statement on this reaction?
(A)	The reaction can be represented as a sharp univariant boundary.
(B)	Initially chlorite and staurolite are Fe-rich and will gradually become Mg-rich with increasing temperature.
(C)	With increasing temperature chlorite becomes Mg-rich and staurolite becomes Fe –rich.
(D)	The reaction is independent of fugacity of H₂O.

Q.39	The symmetry elements of a point group are: 3 crystallographic axes of 2- fold symmetry and 3 mirror planes perpendicular to the crystallographic axes. The Hermann – Mauguin notation of the point group is:
(A)	2m2m2m
(B)	2mm
(C)	2/m2/m2/m
(D)	2/m

Q.40	An aqueous polyphase (L + V + solid) inclusion contains a halite daughter crystal at room temperature and pressure. Which of the given statements is CORRECT in relation to this inclusion?
(A)	The salinity of the bulk aqueous fluid can be determined from the temperature of melting of ice.
(B)	The salinity of the bulk aqueous fluid can be determined from the temperature of dissolution of halite.
(C)	The density in all cases can be determined from the temperature of liquid-vapor homogenization.
(D)	The density in all cases can be determined from the temperature of dissolution of the halite daughter crystal.

Q.42	A mantle source rock melts at a time t0 giving rise to melt (M) and residue (R). Which of the following statements is CORRECT about evolution of the (¹⁴³Nd/¹⁴⁴Nd) and (⁸⁷Sr/⁸⁶Sr) isotope ratio in M (that crystallized to form a rock) and R?
(A)	The growth of Nd isotope ratio versus time is faster in R than M and the Sr isotope ratio grows slower in R than M.
(B)	The growth of Nd isotope ratio versus time is slower in R than M and the Sr isotope ratio grows faster in R than M.
(C)	Both the Nd and Sr isotope ratios grow at identical rates in R and M.
(D)	The growth of Nd and Sr isotope ratio in M and R would depend on the initial concentrations of Sm and Rb in the mantle source rock.

Q.43 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.43

The mole percentages of SiO2, Al2O3 and K2O in a granitic rock are 84.21,7.89 and 7.89, respectively. The molar proportion (in %) of K-feldspar in the rock is_______ . [ round off to one decimal place]

Q.44

In a zone of active normal faulting, the maximum and minimum in situ principal stresses (compressive in nature) are 30 MPa (s1) and 10 MPa (s3), respectively. The fault plane striking N-S has a dip amount of 60^otowards E. Considering Anderson theory of faulting and using the given information, the calculated normal stress on the fault plane is________MPa. [in integer]

Q.45

A circular tunnel is being excavated in a blocky rock mass by drilling and blasting. An excavation disturbed zone (EDZ) around the tunnel extends 0.70 m into the rock from the excavation surface. Considering the unit weight of the rock as 25 kN/m³, the support pressure required at the crown of the tunnel to stabilize the loose blocks of the EDZ is_________kPa. [round off to one decimal place]

Q.46

Under uniaxial compression, a cylindrical quartzite specimen (length = 122 mm and diameter = 60 mm) showed linear elastic behaviour. The uniaxial compressive strength and the modulus ratio of the rock are 150 MPa and 500, respectively. The axial strain at 75 MPa during the loading was_____________milli-strain. [in integer]

Q.48

A stratified confined aquifer consists of three parallel homogeneous and isotropic horizontal layers with thickness of 10 m, 5 m and 5 m. The layers have the same width. The hydraulic conductivities of the strata are 15 m/day, 20 m/day and 30 m/day, respectively. The water flow follows Darcy’s law and is parallel to the strata. Considering the same hydraulic gradient for all the layers, the effective hydraulic conductivity of the aquifer is m/day. [in integer]

Q.49

A drainage basin of fourth order covers an area of 35 km². Within the basin, the total lengths of the 1^storder, 2^ndorder and 3^rdorder drainages are 11.5 km, 8.5 km and 4.2 km, respectively. If the drainage density of the basin is 0.8 km^-1, the total length of the 4^thorder drainage is ______km. [round off to one decimal place]

Q.51

The heat flux at the Earth’s surface is 60 mWm^-2. If the thermal conductivity at the surface is 2.5 Wm^-1°C^-1, the geothermal gradient is __________°C/km. [in integer]

Q.52

A rock formed at time t0 = 0 with number of ¹⁴C atoms = 10⁵. The number of ¹⁴C atoms (in log10) after a time of 8×10³ years is___________. [round off to 3 decimal places] (Use a decay constant of 1.25 × 10^-4yr^-1)

Q.53

In the given reaction, 2Fe²⁺ + 3H2O → Fe2O3 + 6H⁺ + 2e^–consider ideal condition, take concentration of Fe²⁺as 10^-5molal, E⁰= 0.98 V and pH=6. The value of (2.303 × R × T)/F = 0.059 (where F is the Faraday constant). The value of Eh on the Fe²⁺ / hematite boundary at25 °C is _________V. [round off to 2 decimal places]

Q.54

The first and second dissociation constants of H2CO3 are 6.761 × 10^-7 and 4.68 × 10^-11, respectively. If the concentration of H2CO3 is 1 molal and pH = 6, the ∑CO2 in the solution (assuming ideal condition) is _____ molal. [round off to 3 decimal places]

Q.55

A satellite orbits the earth at an altitude of 700 km on the equatorial plane of the earth and it revolves in the same direction as the direction of rotation of the earth. Considering the radius of a spherical earth as 6300 km and the acceleration due to gravity as 10 m/s², the tangential velocity of the satellite in the orbit is____ km/s. [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	C	21	42.6 to 42.7	31	A	41	C	51	24 to 24
2	C	2	C	12	A	22	83.30 to 83.34	32	C	42	A	52	4.565 to 4.567
3	D	3	B	13	B	23	4 to 4	33	B	43	29.9 to 30.1	53	0.21 to 0.22
4	D	4	A	14	B	24	0.03 to 0.03	34	A	44	15 to 15	54	1.675 to 1.677
5	C	5	A	15	B	25	190.00 to 192.00	35	D	45	17.5 to 17.5	55	7.52 to 7.53
6	D	6	C	16	5 to 5	26	A	36	C	46	1 to 1
7	C	7	D	17	0.25 to 0.25	27	C	37	B	47	55 to 55
8	D	8	B	18	0.08 to 0.08	28	C	38	A	48	20 to 20
9	D	9	B	19	6367.44 to 6367.46	29	A	39	C	49	3.8 to 3.8
10	A	10	D	20	64.0 to 64.0	30	C	40	B	50	1.67 to 1.69

Author: Vidhyarthi Darpan

GATE 2021 Mechanical Engineering Section-II Previous Year Paper

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Q.44 The input signal shown below

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Q.14	A plane truss *PQRS* (𝑷𝑸 = 𝑹𝑺, 𝐚𝐧𝐝 ∠𝑷𝑸𝑹 = 𝟗𝟎°) is shown in the figure. The forces in the members PR and RS, respectively, are _
(A)	𝐹√2 (tensile) and 𝐹 (tensile)
(B)	𝐹√2 (tensile) and 𝐹 (compressive)
(C)	𝐹 (compressive) and 𝐹√2 (compressive)
(D)	𝐹 (tensile) and 𝐹√2 (tensile)

Q.15	Consider the mechanism shown in the figure. There is rolling contact without slip between the disc and ground. Select the correct statement about instantaneous centers in the mechanism.
(A)	Only points P, Q, and S are instantaneous centers of mechanism
(B)	Only points P, Q, S and T are instantaneous centers of mechanism
(C)	Only points P, Q, R, S, and U are instantaneous centers of mechanism
(D)	All points P, Q, R, S, T and U are instantaneous centers of mechanism

Q.16	The controlling force curves P, Q and R for a spring controlled governor are shown in the figure, where 𝒓_𝟏 and 𝒓_𝟐 are any two radii of rotation. The characteristics shown by the curves are
(A)	P – Unstable; Q – Stable; R – Isochronous
(B)	P – Unstable; Q – Isochronous; R – Stable
(C)	P– Stable; Q – Isochronous; R – Unstable
(D)	P – Stable; Q – Unstable; R – Isochronous

Q.31	A plane frame *PQR* (fixed at 𝑷 and free at 𝑹) is shown in the figure. Both members (PQ and QR) have length, 𝑳, and flexural rigidity, 𝑬𝑰. Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end, 𝑹, is
(A)	5𝐹𝐿³/3𝐸𝐼
(B)	4𝐹𝐿³/3𝐸𝐼
(C)	2𝐹𝐿³/3𝐸𝐼
(D)	𝐹𝐿³ 3𝐸𝐼

Q.32	A power transmission mechanism consists of a belt drive and a gear train as shown in the figure. Diameters of pulleys of belt drive and number of teeth (T) on the gears 2 to 7 are indicated in the figure. The speed and direction of rotation of gear 7, respectively, are
(A)	255.68 rpm; clockwise
(B)	255.68 rpm; anticlockwise
(C)	575.28 rpm; clockwise
(D)	575.28 rpm; anticlockwise

Q.34	Consider the system shown in the figure. A rope goes over a pulley. A mass, 𝒎, is hanging from the rope. A spring of stiffness, k, is attached at one end of the rope. Assume rope is inextensible, massless and there is no slip between pulley and rope. The pulley radius is 𝒓 and its mass moment of inertia is 𝑱. Assume that the mass is vibrating harmonically about its static equilibrium position. The natural frequency of the system is
(A)
(B)
(C)
(D)

Q.3	In the above figure, O is the center of the circle and, M and N lie on the circle.The area of the right triangle MON is 50 cm². What is the area of the circle in cm²?
(A)	2π
(B)	50π
(C)	75π
(D)	100π

Q.9	The distribution of employees at the rank of executives, across different companies C1, C2, …, C6 is presented in the chart given above. The ratio of executives with a management degree to those without a management degree in each of these companies is provided in the table above. The total number of executives across all companies is 10,000.The total number of management degree holders among the executives in companies C2 and C5 together is ______________.
A)	225
(B)	600
(C)	1900
(D)	2500

Q.6	The loading and unloading response of a metal is shown in the figure. The elastic and plastic strains corresponding to 200 MPa stress, respectively, are
(A)	0.01 and 0.01
(B)	0.02 and 0.01
(C)	0.01 and 0.02
(D)	0.02 and 0.02

Q.7	In a machining operation, if a cutting tool traces the workpiece such that the directrix is perpendicular to the plane of the generatrix as shown in figure, the surface generated is
(A)	plane
(B)	cylindrical
(C)	spherical
(D)	a surface of revolution

Q.10	Consider the surface roughness profile as shown in the figure. The center line average roughness (Ra, in µm) of the measured length (L) is
(A)	0
(B)	1
(C)	2
(D)	4

Q.19	Shear stress distribution on the cross-section of the coil wire in a helical compression spring is shown in the figure. This shear stress distribution represents
(A)	direct shear stress in the coil wire cross-section
(B)	torsional shear stress in the coil wire cross-section
(C)	combined direct shear and torsional shear stress in the coil wire cross-section
(D)	combined direct shear and torsional shear stress along with the effect of stress concentration at inside edge of the coil wire cross-section

Q.30	Consider a two degree of freedom system as shown in the figure, where **PQ is a rigid uniform rod of length, 𝒃 and mass, 𝒎. Assume that the spring deflects only horizontally and force F is applied horizontally at** Q. For this system, the Lagrangian, L is
(A)
(B)
(C)
(D)

Q.32	A tappet valve mechanism in an IC engine comprises a rocker arm ABC that is hinged at B as shown in the figure. The rocker is assumed rigid and it oscillates about the hinge B. The mass moment of inertia of the rocker about B is 10^-4 kg.m². The rocker arm dimensions are a = 3.5 cm and b = 2.5 cm. A pushrod pushes the rocker at location A, when moved vertically by a cam that rotates at N rpm. The pushrod is assumed massless and has a stiffness of 15 N/mm. At the other end C, the rocker pushes a valve against a spring of stiffness 10 N/mm. The valve is assumed massless and rigid. Resonance in the rocker system occurs when the cam shaft runs at a speed of *rpm (round off to the nearest integer***).
(A)	496
(B)	4739
(C)	790
(D)	2369

Q.7	Details of prices of two items P and Q are presented in the above table. The ratio of cost of item P to cost of item Q is 3:4. Discount is calculated as the difference between the marked price and the selling price. The profit percentage is calculated as the ratio of the difference between selling price and cost, to the cost (𝐏𝐫𝐨𝐟𝐢𝐭 % = × 𝟏𝟎𝟎). The discount on item Q, as a percentage of its marked price, is________
(A)	25
(B)	12.5
(C)	10
(D)	5

Q.8	Consider the fixed-point iteration $C:\Users\PC6~1\AppData\Local\Temp\ksohtml12188\wps82.jpg$ with ,and the initial approximation x₀=3.25Then, the order of convergence of the fixed-point iteration method is
(A)	1
(B)	2
(C)	3
(D)	4

Q.8	Five line segments of equal lengths, PR, PS, QS, QT and RT are used to form a star as shown in the figure above.The value of θ, in degrees, is ________
(A)	36
(B)	45
(C)	72
(D)	108

Q.9	A function, λ, is defined by
(A)	−1
(B)	0
(C)	163
(D)	16

Q.4	The input-output relationship of an LTI system is given below. For an input x[n] shown below the peak value of the output when x[n] passes through h is .
(A)	2
(B)	4
(C)	5
(D)	6

Q.15	The transistor Q1 has a current gain β1 = 99 and the transistor Q2 has a current gain β2 = 49. The current IB2 in microampere is _______________ .

Q.16	A 300 V, 5 A, LPF wattmeter has a full scale of 300 W. The wattmeter can be used for loads supplied by 300 V ac mains with a maximum power factor of _____________(rounded off to one decimal place).

Q.17	A 10-bit ADC has a full-scale of 10.230 V, when the digital output is (11 1111 1111)₂. The quantization error of the ADC in millivolt is ___________.

Q.18	A strain gage having nominal resistance of 1000 Ω has a gage factor of 2.5. If the strain applied to the gage is 100 µm/m, its resistance in ohm will change to _________(rounded off to two decimal places).

Q.19	Given: Density of mercury is 13,600 kg/m³and acceleration due to gravity is 9.81 m/s². Atmospheric pressure is 101 kPa. In a mercury U-tube manometer, the difference between the heights of the liquid in the U-tube is 1 cm. The differential pressure being measured in pascal is ________(rounded off to the nearest integer).

Q.20	A piezoresistive pressure sensor has a sensitivity of 1 (mV/V)/kPa. The sensor is excited with a dc supply of 10 V and the output is read using a 3 ½ digit 200 mV full-scale digital multimeter. The resolution of the measurement set-up, in pascal is _________ .

Q.30	The output Vo of the ideal OpAmp used in the circuit shown below is 5 V. Then the value of resistor RL in kilo ohm (kΩ) is
(A)	2.5
(B)	5
(C)	25
(D)	50

Q.33	A J-type thermocouple has an output voltage V_θ = (13650 + 50 θ_x)µV, where θ_x is the junction temperature in Celsius (^oC). The thermocouple is used with reference junction compensation, as shown in the figure. The Instrumentation amplifier used has a gain G = 20. If θ*_Ref* is 1⁰C, for an inputθ_xof 100 ^oC, the output V_o of the instrumentation amplifier in millivolt is
(A)	98 mV
(B)	99 mV
(C)	100 mV
(D)	101 mV

Q.34	A laser pulse is sent from ground level to the bottom of a concrete water tank at normal incidence. The tank is filled with water up to 2 m below the ground level. The reflected pulse from the bottom of the tank travels back and hits the detector. The round-trip time elapsed between sending the laser pulse, the pulse hitting the bottom of the tank, reflecting back and sensed by the detector is 100 ns. The depth of the tank from ground level marked as 𝒙 in metre is ______________.(Refractive index of water 𝐧_{𝐰𝐚𝐭𝐞𝐫} = 𝟏. 𝟑 and velocity of light in air 𝐜_𝐚𝐢𝐫 = 𝟑 × 𝟏𝟎^𝟖𝐦/𝐬)
(A)	9
(B)	10
(C)	11
(D)	12

Q.35	A 4 × 1 multiplexer with two selector lines is used to realize a Boolean function F having four Boolean variables X, Y, Z and W as shown below. S0 and S1 denote the least significant bit (LSB) and most significant bit (MSB) of the selector lines of the multiplexer respectively. I₀, I₁,I₂, I₃ are the input lines of the multiplexer. The canonical sum of product representation of F is

(A)	F (X, Y, Z, W) = Σ m(0,1,3,14,15)
(B)	F (X, Y, Z, W) = Σ m(0,1,3,11,14)
(C)	F (X, Y, Z, W) = Σ m(2,5,9,11,14)
(D)	F (X, Y, Z, W) = Σ m(1,3,7,9,15)

Q.36	Given below is the diagram of a synchronous sequential circuit with one J-K flip-flop and one T flip-flop with their outputs denoted as A and B respectively, with JA = (Aʹ + Bʹ), KA = (A + B), and TB = A. Starting from the initial state (AB = 00), the sequence of states (AB) visited by the circuit is

(A)	00 → 01→ 10 → 11→ 00 …
(B)	00 → 10→ 01 → 11→ 00 …
(C)	00 → 10→ 11 → 01→ 00 …
(D)	00 → 01→ 11 → 00 …

Q.7	In the figure shown above, each inside square is formed by joining the midpoints of the sides of the next larger square. The area of the smallest square (shaded) as shown, in cm²is:
(A)	12.50
(B)	6.25
(C)	3.125
(D)	1.5625

Q.9	The number of students passing or failing in an exam for a particular subject are presented in the bar chart above. Students who pass the exam cannot appear for the exam again. Students who fail the exam in the first attempt must appear for the exam in the following year. Students always pass the exam in their second attempt.The number of students who took the exam for the first time in the year 2 and the year 3 respectively, are______.
A)	65 and 53
(B)	60 and 50
(C)	55 and 53
(D)	55 and 48

Q.36	Which of the given figures is the Hilbert transform of the Dirac delta function 𝜹(𝝃) :
(A)	Figure 1
(B)	Figure 2
(C)	Figure 3
(D)	Figure 4

Q.42	Match the source in Group – I with their half-width (X_1/2) / δ𝒈_𝐦𝐚𝐱 𝐚𝐧𝐝 depth (d) relation in Group –II
(A)	P-1, Q-2, R-4, S-3
(B)	P-1, Q-4, R-3, S-2
(C)	P-3, Q-4, R-1, S-2
(D)	P-3, Q-1, R-4, S-2

Q.29	Match the following invertebrate genera in Group I with their corresponding Class/Phylum in Group II:
(A)	P-4, Q-3, R-1, S-2
(B)	P-4, Q-1, R-2, S-3
(C)	P-4, Q-3, R-2, S-1
(D)	P-3, Q-1, R-4, S-2