GATE 2021 Physics Previous Year Paper

GATE 2021 Physics Previous Year Paper
General  Aptitude (GA)

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

Q.1 Arun and Aparna are here.Arun and Aparna is here.Arun’s families is here.Arun’s family is here.Which of the above sentences are grammatically CORRECT?
(A) (i) and (ii)
(B) (i) and (iv)
(C) (ii) and (iv)
(D) (iii) and (iv)
Q.2


The mirror image of the above text about the x-axis is
Q.3 Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
(A) 1/36
(B) 1/12
(C) 1/8
(D) 1/4
Q.4 ⊕ and ⊙ are two operators on numbers p and q such that𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p ⊕ q = p× qThen, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=
(A) 40
(B) -26
(C) -33
(D) -40
Q.5 Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A) 6
(B) 9
(C) 18
(D) 24

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

Q.6 On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.The distance of point Q to point O, in the same units, should be____________            
(A) 3
(B) 4
(C) 5
(D) 6
Q.7 The author said, “Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk on to the stage and start speaking. In my opinion, it is no less important for public speakers to rehearse their talks.”Based on the above passage, which one of the following is TRUE?
(A) The author is of the opinion that rehearsing is important for musicians, actors and public speakers.
(B) The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors.
(C) The author is of the opinion that rehearsing is more important only for musicians than public speakers.
(D) The author is of the opinion that rehearsal is more important for actors than musicians.
Q.8 Some football players play cricket.All cricket players play hockey.Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
(A) No football player plays hockey.
(B) Some football players play hockey.
(C) All football players play hockey.
(D) All hockey players play football.
Q.9
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.The probability that any point picked randomly within the square falls in the shaded area is___________
(A) 4-π/2
(B) 1/2
(C) π/2-1
(D) 𝜋/4
Q.10 In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum area of the triangle PQR possible, in cm2, is
(A) 18
(B) 24
(C) 48√3
(D) 144√3
Physics (PH)
  1. – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1 Choose the graph that best describes the variation of dielectric constant (𝝐𝒓) with temperature (𝑻) in a ferroelectric material.(𝑻𝑪 is the Curie temperature)
Q.2 A matter wave is represented by the wave function   𝚿(𝒙, 𝒚, 𝒛, 𝒕) = 𝑨𝒆𝒊 (𝟒𝒙+𝟑𝒚+𝟓𝒛−𝟏𝟎𝝅𝒕)where 𝑨 is a constant. The unit vector representing the direction of propagation of this matter wave is
(A)
(B)
(C)
(D)
Q.3 As shown in the figure, X-ray diffraction pattern is obtained from a diatomic chain of atoms 𝑷 and 𝑸. The diffraction condition is given by 𝒂 𝐜𝐨𝐬 𝜽 = 𝒏𝝀, where 𝒏 is the order of the diffraction peak. Here, 𝒂 is the lattice constant and 𝝀 is the wavelength of the X-rays. Assume that atomic form factors and resolution of the instrument do not depend on 𝜽. Then, the intensity of the diffraction peaks is 
(A) lower for even values of 𝑛, when compared to odd values of 𝑛
(B) lower for odd values of 𝑛, when compared to even values of 𝑛
(C) zero for odd values of 𝑛
(D) zero for even values of 𝑛
Q.4

As shown in the figure, two metal-semiconductor junctions are formed between an n-type semiconductor 𝑺 and metal 𝑴. The work functions of 𝑺 and 𝑴 are 𝝋𝑺 𝐚𝐧𝐝 𝝋𝑴, respectively with 𝝋𝑴 > 𝝋𝑺.

The 𝑰 − 𝑽 characteristics (on linear scale) of the junctions is best represented by
(A)
(B)
(C)
(D)
Q.5 Consider a tiny current loop driven by a sinusoidal alternating current. If the surface integral of its time-averaged Poynting vector is constant, then the magnitude of the time-averaged magnetic field intensity, at any arbitrary position, ⃗𝒓→, is proportional to
(A) 1/r3
(B) 1/r2
(C) 1/𝑟
(D) r
Q.6

Consider a solenoid of length 𝑳 and radius 𝑹, where 𝑹 ≪ 𝑳. A steady-current flows through the solenoid. The magnetic field is uniform inside the solenoid and zero outside.

  

Among the given options, choose the one that best represents the variation in the magnitude of the vector potential, (𝟎, 𝑨𝝋, 𝟎) at 𝒛 = 𝑳/𝟐, as a function of the radial distance (𝒓) in cylindrical coordinates.Useful information: The curl of a vector ⃗𝑭→, in cylindrical coordinates is 
(A)
(B)
(C)
(D)
Q.7 Assume that 13N (𝒁 = 𝟕) undergoes first forbidden β+decay from its ground state with spin-parity 𝑱i𝝅, to a final state 𝑱𝒇𝝅. The possible values for 𝑱i𝝅 and𝑱f𝝅, respectively, are
(A)
(B)
(C)
(D)
 
Q.8 In an experiment, it is seen that an electric-dipole (𝑬𝟏) transition can connect an initial nuclear state of spin-parity 𝑱i𝝅 = 𝟐+ to a final state 𝑱𝒇𝝅. All possible values of 𝑱f𝝅 are
(A) 1+, 2+
(B) 1+, 2+, 3+
(C) 1−, 2−
(D) 1−, 2−, 3−
 
Q.9 Choose the correct statement from the following.
(A) Silicon is a direct band gap semiconductor.
(B) Conductivity of metals decreases with increase in temperature.
(C) Conductivity of semiconductors decreases with increase in temperature.
(D) Gallium Arsenide is an indirect band gap semiconductor.

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

Q.10

A two-dimensional square lattice has lattice constant 𝒂.𝒌 represents the wavevector in reciprocal space. The 

coordinates (𝒌𝒙, 𝒌𝒚)of reciprocal space where band gap(s) can occur, are
Q.11

As shown in the figure, an electromagnetic wave with intensity 𝑰𝑰 is incident at the interface of two media having refractive indices 𝒏𝟏 = 𝟏 and 𝒏𝟐 = √𝟑. The wave is reflected with intensity 𝑰𝑹 and transmitted with intensity 𝑰𝑻. Permeability of each medium is the same. (Reflection coefficient 𝑹 = 𝑰𝑹/𝑰𝑰 and Transmission coefficient 𝑻 = 𝑰𝑻/𝑰𝑰).

Choose the correct statement(s).
(A) 𝑅 = 0 if 𝜃𝐼 = 0o and polarization of incident light is parallel to the plane of incidence.
(B) 𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is parallel to the plane of incidence.
(C) 𝑅 = 0 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the plane of incidence.
(D) 𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the plane of incidence.
Q.12 A material is placed in a magnetic field intensity 𝑯. As a result, bound current density 𝑱𝒃 is induced and magnetization of the material is 𝑴. The magnetic flux density is 𝑩. Choose the correct option(s) valid at the surface of the material.
(A) 𝛁. 𝑴 = 0
(B) 𝛁. 𝑩 = 0
(C) 𝛁. 𝑯 = 0
(D) 𝛁. 𝑱𝑏 = 0
Q.13 For a finite system of Fermions where the density of states increases with energy, the chemical potential
(A) decreases with temperature
(B) increases with temperature
(C) does not vary with temperature
(D) corresponds to the energy where the occupation probability is 0.5
Q.14 Among the term symbols𝟒𝑺𝟏,   𝟐𝑫𝟕/𝟐,   𝟑𝑺𝟏 and 𝟐𝑫𝟓/𝟐choose the option(s) possible in the 𝑳𝑺 coupling notation.
(A) 4S1
(B) 2𝐷7/2
(C) 3𝑆1
(D) 2𝐷5/2
 
Q.15 To sustain lasing action in a three-level laser as shown in the figure, necessary condition(s) is(are) 
(A) lifetime of the energy level 1 should be greater than that of energy level 2
(B) population of the particles in level 1 should be greater than that of level 0
(C) lifetime of the energy level 2 should be greater than that of energy level 0
(D) population of the particles in level 2 should be greater than that of level 1
   
Q.16 If 𝒚𝒏(𝒙) is a solution of the differential equation𝒚′′ − 𝟐𝒙𝒚′ + 𝟐𝒏𝒚 = 𝟎where 𝒏 is an integer and the prime ( ′) denotes differentiation with respect to 𝒙, then acceptable plot(s) of 𝝍𝒏(𝒙) = 𝒆−𝒙 ⁄𝟐 𝒚𝒏(𝒙), is(are)
(A)
(B)
(C)
(D)

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

Q.17 The donor concentration in a sample of n-type silicon is increased by a factor of 100. Assuming the sample to be non-degenerate, the shift in the Fermi level (in meV) at 300 K (rounded off to the nearest integer) is  ________.(Given: 𝒌𝐁𝑻= 25 meV at 300 K)
   
Q.18 Two observers O and Oʹ observe two events P and Q. The observers have a constant relative speed of 0.5c. In the units, where the speed of light, c, is taken as unity, the observer O obtained the following coordinates:Event P: 𝒙 = 5, 𝒚 = 3, 𝒛 = 5, 𝒕 = 3 Event Q: 𝒙 = 5,𝒚 = 1, 𝒛 = 3, 𝒕 = 5The length of the space-time interval between these two events, as measured by Oʹ, is 𝑳. The value of |𝑳| (in integer) is ________________.
   
Q.19 A light source having its intensity peak at the wavelength 289.8 nm is calibrated as 10,000 K which is the temperature of an equivalent black body radiation. Considering the same calibration, the temperature of light source (in K) having its intensity peak at the wavelength 579.6 nm (rounded off to the nearest integer) is   ______.
Q.20 A hoop of mass 𝑴 and radius 𝑹 rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass 𝒎 slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is_______.                           
   
Q.21 Three non-interacting bosonic particles of mass 𝒎 each, are in a one- dimensional infinite potential well of width 𝒂. The energy of the third excitedstate of the system is 𝒙 ×. The value of 𝒙 (in integer) is _________.
   
Q.22 The spacing between two consecutive S-branch lines of the rotational Raman spectra of hydrogen gas is 𝟐𝟒𝟑. 𝟐 𝐜𝐦−𝟏. After excitation with a laser of wavelength 𝟓𝟏𝟒. 𝟓 𝐧𝐦, the Stoke’s line appeared at 𝟏𝟕𝟔𝟏𝟏. 𝟒 𝐜𝐦−𝟏 for a particular energy level. The wavenumber (rounded off to the nearest integer), in 𝐜𝐦−𝟏, at which Stoke’s line will appear for the next higher energy level is______________ .
   
Q.23 The transition line, as shown in the figure, arises between𝟐𝑫𝟑/𝟐 and 𝟐𝑷𝟏/𝟐 states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is______.              
Q.24

Consider the atomic system as shown in the figure, where the Einstein 𝑨 coefficients for spontaneous emission for the levels are 𝑨𝟐→𝟏 = 𝟐 × 𝟏𝟎𝟕 s–1 and 𝑨𝟏→𝟎 = 𝟏𝟎𝟖 s–1. If 1014 atoms/cm3 are excited from level 0 to level 2 and a steady state population in level 2 is achieved, then the steady state population at level 1 will be 𝒙 × 𝟏𝟎𝟏𝟑 cm–3. The value of 𝒙 (in integer) is_______________.

         
   
Q.25 If  𝒂⃗→  and  ⃗𝒃→  are  constant  vectors,  𝒓⃗→    and  ⃗𝒑→    are  generalized  positions  and conjugate momenta, respectively, then for the transformation 𝑸 =  𝒂⃗ ∙ 𝒓⃗ and 𝑷 =  ⃗𝒃 ∙ 𝒓⃗ to be canonical, the value of ⃗𝒂→ ∙ 𝒃→ (in integer) is  _______________.

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

Q.26

The above combination of logic gates represents the operation
(A) OR
(B) NAND
(C) AND
(D) NOR
Q.27 In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobility of the hole to electron is
(A) 4:9
(B) 4:11
(C) 9:4
(D) 11:4
Q.28 Consider a spin 𝑺 = ℏ⁄𝟐 particle in the state |𝝓⟩ = . The probability that a measurement finds the state with 𝑺𝒙 = +ℏ⁄𝟐 is
(A) 5/18
(B) 11/18
(C) 15/18
(D) 17/18
Q.29

An electromagnetic wave having electric field 𝑬 = 𝟖 𝐜𝐨(𝒌𝒛 − ω𝒕) 𝒚̂ 𝐕 𝐜𝐦−𝟏 is incident at 𝟗𝟎°(normal incidence) on a square slab from vacuum (with refractive index 𝒏𝟎 = 𝟏. 𝟎) as shown in the figure. The slab is composed of two different materials with refractive indices 𝒏𝟏and 𝒏𝟐. Assume that the permeability of each medium is the same. After passing through the slab for the first time, the electric field amplitude, in 𝐕 𝐜𝐦−𝟏, of the electromagnetic

wave, which emerges from the slab in region 2, is closest to

       
(A) 11/1.6
(B) 11/3.2
(C) 11/13.8
(D) 11/25.6
Q.30

Consider a point charge +𝑸 of mass 𝒎 suspended by a massless, inextensible string of length 𝒍 in free space (permittivity 𝗌𝟎) as shown in the figure. It is placed at a height 𝒅 (𝒅 > 𝒍) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane.

If 𝜽 represents the angular position and 𝒑𝜽 its corresponding canonical momentum, then the correct Hamiltonian of the system is
(A) (𝑝𝜃2/2𝑚𝑙2 )    _ [𝑄2/16𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃) 
(B) (P𝜃2/2𝑚𝑙2 ) – [𝑄2+ /8𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)]-𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(C) (𝑝𝜃2/2𝑚𝑙2) -[𝑄2/8𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
(D) (𝑝𝜃2/2𝑚𝑙2) -[𝑄2/16𝜋𝜀0(𝑑 − 𝑙 cos 𝜃)] + 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
Q.31

Consider two concentric conducting spherical shells as shown in the figure. The inner shell has a radius a and carries a charge +𝑸. The outer shell has a radius b and carries a charge −𝑸. The empty space between them is half- filled by a hemispherical shell of a dielectric having permittivity ε𝟏. The remaining space between the shells is filled with air having the permittivity ε𝟎.

  
The electric field at a radial distance 𝒓 from the center and between the shells (𝒂 < 𝒓 < 𝒃) is
(A)   
(B)
(C)
 
(D)
 
Q.32


For the given sets of energy levels of nuclei X and Y whose mass numbers are odd and even, respectively, choose the best suited interpretation.
(A) Set I: Rotational band of X Set II: Vibrational band of Y
(B) Set I: Rotational band of Y Set II: Vibrational band of X
(C) Set I: Vibrational band of X Set II: Rotational band of Y
(D) Set I: Vibrational band of Y Set II: Rotational band of X
Q.33 Consider a system of three distinguishable particles, each having spin 𝑺=𝟏/𝟐 such that 𝑺𝒛 = ±𝟏/𝟐 with corresponding magnetic moments 𝝁𝒛 = ±𝝁. When the system is placed in an external magnetic field 𝑯 pointing along the𝒛-axis, the total energy of the system is 𝝁𝑯. Let 𝒙 be the state where the first spin has 𝑺𝒛 = 𝟏/𝟐. The probability of having the state 𝒙 and the mean magnetic moment (in the +𝒛 direction) of the system in state 𝒙 are
(A)
(B)
(C)
(D)
Q.34

Consider a particle in a one-dimensional infinite potential well with its walls at 𝒙 = 𝟎 and 𝒙 = 𝑳. The system is perturbed as shown in the figure.

The first order correction to the energy eigenvalue is
(A) 𝑉0 /4
(B) 𝑉0 /3
(C) 𝑉0 /2
(D) 𝑉0 /5
Q.35 Consider a state described by 𝝍(𝒙, 𝒕) = 𝝍𝟐(𝒙, 𝒕) + 𝝍𝟒(𝒙, 𝒕), where 𝝍𝟐(𝒙, 𝒕) and 𝝍𝟒(𝒙, 𝒕) are respectively the second and fourth normalized harmonic oscillator wave functions and 𝑚 is the angular frequency of the harmonic oscillator. The wave function 𝝍(𝒙, 𝒕 = 𝟎) will be orthogonal to 𝝍(𝒙, 𝒕) at time 𝒕 equal to
(A) 𝜋/2𝜔
(B) 𝜋/𝜔
(C) 𝜋/4𝜔
(D) 𝜋/6𝜔
Q.36



(A) ħ𝜔/2+ 𝛽−1 ln[1 − exp(𝛽ħ𝜔)]
(B) ħ𝜔/2+ 𝛽−1 ln[1 − exp(−𝛽ħ𝜔)]
(C) ħ𝜔/2+ 𝛽−1 ln[1 + exp(−𝛽ħ𝜔)]
(D) 𝛽−1 ln[1 − exp(−𝛽ħ𝜔)]
Q.37 A system of two atoms can be in three quantum states having energies 0, 𝜖 and 2𝜖. The system is in equilibrium at temperature 𝑇 = (𝑘𝐵𝛽)−1. Match the following Statistics with the Partition function. 
(A) CD:Z1, CI:Z2, FD:Z3, BE:Z4
(B) CD:Z2, CI:Z3, FD:Z4, BE:Z1
(C) CD:Z3, CI:Z4, FD:Z1, BE:Z2
(D) CD:Z4, CI:Z1, FD:Z2, BE:Z3
Q.38 The free energy of a ferromagnet is given by 𝑭 = 𝑭𝟎 + 𝒂𝟎(𝑻 − 𝑻𝑪)𝑴𝟐 +𝒃𝑴𝟒, where 𝑭𝟎, 𝒂𝟎, and 𝒃 are positive constants, 𝑴 is the magnetization, 𝑻 is the temperature, and 𝑻𝑪 is the Curie temperature. The relation between𝑴𝟐 and 𝑻 is best depicted by
(A)
(B)
(C)
(D)
Q.39 Consider a spherical galaxy of total mass 𝑴 and radius 𝑹, having a uniform matter distribution. In this idealized situation, the orbital speed 𝒗 of a star of mass 𝒎 (𝒎 ≪ 𝑴) as a function of the distance 𝒓 from the galactic centre is best described by(𝑮 is the universal gravitational constant)
(A)
(B)
(C)
(D)
Q.40 Consider the potential 𝑼(𝒓) defined aswhere 𝑎 and 𝑼𝟎 are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with 𝑼(𝒓) for a (wave-vector) momentum transfer 𝒒 and 𝑎 → 𝟎, is proportional to(Useful integral:
(A) q−2
(B) q−1
(C) 𝑞
(D) q2
Q.41

As shown in the figure, inverse magnetic susceptibility (𝟏/𝝌) is plotted as a function of temperature (𝑻) for three different materials in paramagnetic states.

(Curie temperature of ferromagnetic material =𝑻𝐂  Néel temperature of antiferromagnetic material= 𝑻𝐍)Choose the correct statement from the following
(A) Material 1 is antiferromagnetic (𝑇 < 𝑇N), 2 is paramagnetic, and 3 is ferromagnetic (𝑇 < 𝑇C).
(B) Material 1 is paramagnetic, 2 is antiferromagnetic (𝑇 < 𝑇N), and 3 is ferromagnetic (𝑇 < 𝑇C).
(C) Material 1 ferromagnetic (𝑇 < 𝑇C), 2 is antiferromagnetic (𝑇 < 𝑇N), and 3 is paramagnetic.
(D) Material 1 is ferromagnetic (𝑇 < 𝑇C), 2 is paramagnetic, and 3 is antiferromagnetic (𝑇 < 𝑇N).

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

Q.42 A function (𝒕) is defined only for 𝒕 ≥ 𝟎. The Laplace transform of 𝒇(𝒕) is
𝐿(𝒇; 𝒔) = ∫∞𝟎 𝒆−𝒔𝒕 𝒇(𝒕) 𝒅𝒕
whereas the Fourier transform of 𝒇(𝒕) is
𝒇̃(𝑚) = ∫∞𝟎   𝒇(𝒕) 𝒆−𝒊𝑚𝒕 𝒅𝒕 .
The correct statement(s) is(are)
(A) The variable 𝑠 is always real.
(B) The variable 𝑠 can be complex.
(C) (𝑓; 𝑠) and 𝑓̃(𝜔) can never be made connected.
(D) (𝑓; 𝑠) and 𝑓̃(𝜔) can be made connected.
 
Q.43 𝑷 and 𝑸 are two Hermitian matrices and there exists a matrix 𝑹, which diagonalizes both of them, such that 𝑹𝑷𝑹−𝟏 = 𝑺𝟏 and 𝑹𝑸𝑹−𝟏 = 𝑺𝟐 , where 𝑺𝟏 and 𝑺𝟐 are diagonal matrices. The correct statement(s) is(are)
(A) All the elements of both matrices 𝑆1 and 𝑆2 are real.
(B) The matrix 𝑃𝑄 can have complex eigenvalues.
(C) The matrix 𝑄𝑃 can have complex eigenvalues.
(D) The matrices 𝑃 and 𝑄 commute.
Q.44

A uniform block of mass 𝑴 slides on a smooth horizontal bar. Another mass 𝒎 is connected to it by an inextensible string of length l of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as x and θ, as shown in the figure.

If 𝒑𝒙 and 𝒑𝜽 are the generalized momenta corresponding to x and θ, respectively, then the correct option(s) is(are)
(A) 𝑝𝑥 = (𝑚 + 𝑀)𝑥̇ + 𝑚𝑙 cos 𝜃 𝜃̇
(B) 𝑝𝜃 = 𝑚𝑙2𝜃̇ − 𝑚𝑙 cos 𝜃 𝑥̇
(C) 𝑝𝑥 is conserved
(D) 𝑝𝜃 is conserved
Q.45 The Gell-Mann – Okuba mass formula defines the mass of baryons as 𝑴 = 𝑴   + 𝒂𝒀 + 𝒃 [𝑰(𝑰 + 𝟏) − 1/4 𝒀𝟐], where 𝑴 , 𝒂 and 𝒃 are constants, 𝑰 represents the isospin and 𝒀 represents the hypercharge. If the mass of 𝚺 hyperons is same as that of 𝚲 hyperons, then the correct option(s) is(are)
(A) 𝑀 ∝ 𝐼(𝐼 + 1)
(B) 𝑀 ∝ 𝑌
(C) 𝑀 does not depend on 𝐼
(D) 𝑀 does not depend on 𝑌
Q.46 The time derivative of a differentiable function 𝒈(𝒒𝒊, 𝒕) is added to a Lagrangian 𝑳(𝒒𝒊, 𝒒̇𝒊, 𝒕) such that, where 𝒒𝒊, 𝒒̇ 𝒊, 𝒕 are the generalized coordinates, generalized velocities and time, respectively. Let 𝒑𝒊 be the generalized momentum and 𝑯 the Hamiltonian associated with (𝒒𝒊, 𝒒̇ 𝒊, 𝒕). If 𝒑′ and 𝑯′ are those associated with 𝑳′, then the correct option(s) is(are)
(A) Both 𝐿 and 𝐿′ satisfy the Euler-Lagrange’s equations of motion
(B)
(C) If 𝑝𝑖 is conserved, then 𝑝′ is necessarily conserved
(D)

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

 
Q.47 A linear charged particle accelerator is driven by an alternating voltage source operating at 10 MHz. Assume that it is used to accelerate electrons. After a few drift-tubes, the electrons attain a velocity 𝟐. 𝟗 × 𝟏𝟎𝟖 m s–1. The minimum length of each drift-tube, in m, to accelerate the electrons further (rounded off to one decimal place) is  _______________.
   
Q.48 The Coulomb energy  component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is  _____________.
(Given:e2/𝟒𝝅E𝟎= 𝟏. 𝟒𝟒 MeV fm)
   
Q.49 For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices 𝝈𝟏, 𝝈𝟐 for particles 1 and 2, respectively. The value of (𝝈𝟏 ⋅ 𝝈𝟐) (in integer) is  ________.
Q.50

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

The value of 𝑰𝒏 (in integer) is  _________.
Q.51 The normalized radial wave function of the second excited state of hydrogen atom is where 𝒂 is the Bohr radius and 𝒓 is the distance from the center of the atom.The distance at which the electron is most likely to be found is 𝒚 × 𝒂. The value of 𝒚 (in integer) is  ____________.
   
Q.52 Consider an atomic gas with number density 𝒏 = 𝟏𝟎𝟐𝟎 𝐦−𝟑, in the ground state at 300 K. The valence electronic configuration of atoms is 𝒇𝟕. The paramagnetic susceptibility of the gas 𝝌 = 𝒎 × 𝟏𝟎−𝟏𝟏. The value of 𝒎 (rounded off to two decimal places) is   ________.(Given: 𝐌𝐚𝐠𝐧𝐞𝐭𝐢𝐜 𝐩𝐞𝐫𝐦𝐞𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐟𝐫𝐞𝐞 𝐬𝐩𝐚𝐜𝐞 𝝁𝟎 = 𝟒𝝅 × 𝟏𝟎−𝟕 𝐇 𝐦−𝟏𝐁𝐨𝐡𝐫 𝐦𝐚𝐠𝐧𝐞𝐭𝐨𝐧 𝝁𝐁 = 𝟗. 𝟐𝟕𝟒 × 𝟏𝟎−𝟐𝟒𝐀 𝐦𝟐𝐁𝐨𝐥𝐭𝐳𝐦𝐚𝐧𝐧 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝒌𝐁 = 𝟏. 𝟑𝟖𝟎𝟕 × 𝟏𝟎−𝟐𝟑 𝐉 𝐊−𝟏 )
Q.53

Consider a cross-section of an electromagnet having an air-gap of 𝟓 cm as shown in the figure. It consists of a magnetic material (𝝁 = 𝟐𝟎𝟎𝟎𝟎𝝁𝟎) and is driven by a coil having 𝑵𝑰 = 𝟏𝟎𝟒A, where 𝑵 is the number of turns and 𝑰 is the current in Ampere.

Ignoring the fringe fields, the magnitude of the magnetic field ⃗𝑩⃗→ (in Tesla, rounded off to two decimal places) in the air-gap between the magnetic poles is ____________ .
   
Q.54 The spin ⃗𝑺→ and orbital angular momentum  ⃗𝑳→ of an atom precess about 𝑱→, the total angular momentum. 𝑱→ precesses about an axis fixed by a magnetic field⃗𝑩𝟏→ = 𝟐𝑩𝟎𝒛̂  , where 𝑩𝟎  is a constant. Now the magnetic field is changed to 𝑩⃗𝟐→ = 𝑩𝟎(𝒙̂ + √𝟐𝒚̂ + 𝒛̂).  Given  the  orbital  angular  momentum  quantum number 𝒍 = 𝟐 and spin quantum number 𝒔 = 𝟏⁄𝟐, 𝜽 is the angle between𝑩⃗𝟏 and j→ for the largest possible values of total angular quantum number 𝒋 and its z-component 𝒋𝒛. The value of 𝜽 (in degree, rounded off to the nearest integer) is  ____.
   
Q.55 The spin-orbit effect splits the 𝟐𝑷→𝟐𝑺 transition (wavelength, 𝝀 =𝟔𝟓𝟐𝟏 Å) in Lithium into two lines with separation of ∆𝝀 = 𝟎. 𝟏𝟒 Å. The corresponding positive value of energy difference between the above two lines, in eV, is 𝒎 × 𝟏𝟎−𝟓. The value of 𝒎 (rounded off to the nearest integer) is   _________ .(Given: Planck’s constant, 𝐡 = 𝟒. 𝟏𝟐𝟓 × 𝟏𝟎−𝟏𝟓𝐞𝐕 𝐬Speed of light, 𝐜 = 𝟑 × 𝟏𝟎𝟖 𝐦 𝐬−𝟏 )

Answer Key 

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

GATE 2021 Petroleum Engineering Previous Year Paper

GATE 2021 Petroleum Engineering Previous Year Paper

General Aptitude (GA)

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

Q.1 The current population of a city is 11,02,500. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago?
(A) 9,92,500
(B) 9,95,006
(C) 10,00,000
(D) 12,51,506
Q.2
(A) 3
(B) 7
(C) 9
(D) 11
Q.3

 

The least number of squares that must be added so that the line P-Q becomes the line of symmetry is____________
(A) 4
(B) 3
(C) 6
(D) 7
Q.4 Nostalgia is to anticipation as    ____ is to_________Which one of the following options maintains a similar logical relation in the above sentence?
(A) Present, past
(B) Future, past
(C) Past, future
(D) Future, present
Q.5 Consider the following sentences:I woke up from sleep.I woked up from sleep.I was woken up from sleep.I was wokened up from sleep.Which of the above sentences are grammatically CORRECT?
(A) (i) and (ii)
(B) (i) and (iii)
(C) (ii) and (iii)
(D) (i) and (iv)
  1. 6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).
Q.6 Given below are two statements and two conclusions. Statement 1: All purple are green.Statement 2: All black are green. Conclusion I: Some black are purple. Conclusion II: No black is purple.Based on the above statements and conclusions, which one of the following options is logically CORRECT?
(A) Only conclusion I is correct.
(B) Only conclusion II is correct.
(C) Either conclusion I or II is correct.
(D) Both conclusion I and II are correct.
Q.7 Computers are ubiquitous. They are used to improve efficiency in almost all fields from agriculture to space exploration. Artificial intelligence (AI) is currently a hot topic. AI enables computers to learn, given enough training data. For humans, sitting in front of a computer for long hours can lead to health issues.Which of the following can be deduced from the above passage?Nowadays, computers are present in almost all places.Computers cannot be used for solving problems in engineering.For humans, there are both positive and negative effects of using computers.Artificial intelligence can be done without data.
(A) (ii) and (iii)
(B) (ii) and (iv)
(C) (i), (iii) and (iv)
(D) (i) and (iii)
Q.8 Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in cubic units, is________________  
(A) 𝜋/3
(B) 2𝜋/3
(C) 3𝜋2
(D) 3π

Q.9


 

The number of minutes spent by two students, X and Y, exercising every day in a given week are shown in the bar chart above.The number of days in the given week in which one of the students spent a minimum of 10% more than the other student, on a given day, is
(A) 4
(B) 5
(C) 6
(D) 7
Q.10

Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above.The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is
(A) 2 : 3
(B) 3 : 4
(C) 4 : 5
(D) 5 : 6
Petroleum Engineering (PE)

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

Q.1 MOPU (in the context of offshore drilling and production systems) stands for
(A) Mobile Offshore Process Unit
(B) Mobile Offshore Piping Unit
(C) Mobile Offshore Production Unit
(D) Mobile Oil Production Unit
Q.2 Which ONE of the following statements is INCORRECT?
(A) Conductor is the outer casing of a well
(B) Riser is used for transporting fluid
(C) Conductor and riser have the same functions
(D) Conductor is used for shielding the well flow lines from external forces
Q.3 Which ONE of the following offshore installations uses Dynamic Positioning System (DPS) for station keeping?
(A) Jacket
(B) Jack-up
(C) Semi-submersible
(D) Tension Leg Platform
Q.4 Which ONE of the following is NOT a primary safety system for offshore installations?
(A) Emergency Shut Down
(B) Isolation
(C) Fire Protection
(D) Blowdown
Q.5 The primary function of the thruster in the Dynamic Positioning System (DPS) of an offshore installation is
(A) To apply thrust in the direction opposite to the resultant environmental force
(B) To apply thrust in the same direction as the resultant environmental force
(C) To apply thrust in the direction opposite to the motion
(D) To apply thrust in the same direction as the motion
 
Q.6 Select the CORRECT firefighting system for electrical switchgear room in an offshore facility.
(A) Wet chemical
(B) Halon system
(C) Foam
(D) Water sprinklers
Q.7 Which ONE of the following options can be used to quantify secondary porosity?
(A) Sonic and Gamma Ray Logs
(B) Sonic and Neutron Logs
(C) Sonic and Caliper Logs
(D) Density and Neutron Logs
 
Q.8 Among the options given below, what is the typical temperature range for significant oil generation in a source rock associated with conventional crude-oil reservoirs?
(A) 10 oC – 40 oC
(B) 60 oC – 175 oC
(C) 225 oC – 325 oC
(D) 350 oC – 425 oC
Q.9 In the original Darcy’s law as proposed by Henry Darcy, which of the following drives the fluid flow through a fully saturated sand column?
(A) Pressure-gradient or hydraulic-gradient
(B) Viscous force per unit volume
(C) Capillary force per unit volume
(D) Inertial force per unit volume
Q.10 At which ONE of the following scales is Darcy’s law for fluid flow through a porous medium applicable?
(A) Nano-scale
(B) Molecular-scale
(C) Microscopic-scale
(D) Macroscopic-scale
 
Q.11

Pressure – Temperature phase diagram of CO2 is shown below. Identify the correct phases from the given options.

       
(A) I = Solid Phase, II = Liquid Phase, III = Gas Phase, IV = Supercritical Phase
(B) I = Gas Phase, II = Supercritical Phase, III = Solid Phase, IV = Liquid Phase
(C) I = Supercritical Phase, II = Liquid Phase, III = Solid Phase, IV = Gas Phase
(D) I = Gas Phase, II = Solid Phase, III = Liquid Phase, IV = Supercritical Phase
Q.12 A measure of the potential of crude oil to form surfactants for Enhanced Oil Recovery (EOR) is given by the Total Acid Number (TAN). TAN is the mass of  _________(in milligrams) that is required to neutralize one gram of crude oil.
(A) Ca(OH)2
(B) NaCl
(C) KOH
(D) NaOH
 
Q.13 In Water-Alternating-Gas (WAG) injection, the purpose of the injection is to_____I  _____ the “relative permeability” of gas and to  _____ II   _____ the “mobility” of the gas.
(A) I = reduce, II = enhance
(B) I = reduce, II = reduce
(C) I = enhance, II = reduce
(D) I = enhance, II = enhance

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

Q.14 Solids that may possibly form in the offshore pipelines during the production of oil and gas from deep-water reservoirs are
(A) Wax
(B) Char
(C) Hydrates
(D) Asphaltenes
 
Q.15 Oil and gas pipelines, which are at an elevated pressure (about 3 MPa) and sub-ambient temperature (below 298 K), may get blocked by the formation of solid hydrates. One of the strategies adopted to inhibit the formation of hydrates is the injection of Thermodynamic Hydrate Inhibitors (THIs) into the reservoir fluid.Identify all suitable chemicals that are commonly used as THIs.
(A) Sodium Chloride
(B) Methanol
(C) Polyvinylpyrrolidone
(D) Sodium Dodecyl Sulphate
Q.16

When CO2 and liquid water are brought in contact with each other, they may form solid hydrates. The three-phase hydrate boundary is shown in the Pressure -Temperature plot given below.Identify the correct statements.G = Gas

Phase, H = Hydrate Phase, L = Liquid Phase

    
(A) Hydrates are stable in region I
(B) Hydrates are stable in region II
(C) Hydrates are stable in region III
(D) Hydrates are stable in region IV
Q.17 Heavy oil recovered from reservoirs can be represented by CXH1.5X. Suitable processes to reduce the density of heavy oil are
(A) Carbon Rejection
(B) Pyrolysis
(C) Hydrogenation
(D) Filtration
 
Q.18 Identify the CORRECT statements for a 𝒏 × 𝒏 matrix.
(A) Under elementary row operations, the rank of the matrix remains invariant
(B) Under elementary row operations, the eigenvalues of the matrix remain the same
(C) If the elements in a row can be written as a linear combination of two or more rows, then the matrix is singular.
(D) The rank of the matrix is equal to 𝑛 if the determinant of the matrix is zero.
Q.19 During a drilling operation, kick occurs if
(A) the shear ram in the Blow Out Preventer (BOP) does not work.
(B) the formation pressure is equal to the drilling fluid pressure.
(C) the volume of the mud used to fill the hole is less than that of the pipe being pulled out.
(D) the formation pressure is more than the drilling fluid pressure.

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

 
Q.20 The value of is_________(correct up to one decimal place).
   
Q.21 Given two complex numbers, 𝒛𝟏 = 𝟒 + 𝟑𝒊 𝐚𝐧𝐝 𝒛𝟐 = 𝟐 − 𝟓𝒊, the real part of (𝒛𝟏𝒛̅̅𝟐̅) is       .
   
Q.22 The number of ‘three-digit numbers’ that can be formed using the digits from 1 to 9 without the repetition of each digit is                                  .
   
Q.23 The estimate for the root of the function 𝒇(𝒙) = 𝒆𝟐𝒙 + 𝟐𝒙 after one iteration with an initial guess of 𝒙𝟎 = 𝟎, using the Newton-Raphson method is__________(correct up to two decimal places).
   
Q.24 A saturated oil reservoir has an average reservoir pressure of 3000 psia, tested for flowing bottom-hole pressure (BHP) of 2000 psia and production rate of 500 STB/day. The maximum reservoir deliverability based on Vogel’s equation for two-phase flow is_____________STB/day.
   
Q.25 If the specific heat ratio of natural gas is 1.28, the critical pressure ratio (ratio of outlet pressure to upstream pressure) through a choke is_____(round off to two decimal places).

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

Q.26 Match the suitable artificial lift methods (GROUP I) to meet the requirements (GROUP II) given in the table.   
(A) P – I, Q – II, R – IV, S – III
(B) P – II, Q – I, R – IV, S – III
(C) P – I, Q – II, R – III, S – IV
(D) P – II, Q – I, R – III, S – IV
Q.27 Match the Enhanced Oil Recovery (EOR) methods (GROUP I) with the corresponding laboratory tests (GROUP II) 
(A) P – III, Q – II, R – I, S – IV
(B) P – II, Q – IV, R – I, S – III
(C) P – III, Q – IV, R – II, S – I
(D) P – III, Q – II, R – IV, S – I
Q.28 Identify the following well test methods corresponding to the transient pressure profiles in the figures given below. (BHP: Bottom-hole pressure, BOPD: Barrels of oil per day)Flow-after-flow testInterference testFall-off testModified isochronal test
           
                         (I)                                                              (II)



               
                          (III)                                                                      (iv)
(A) P – IV, Q – II, R – I, S – III
(B) P – II, Q – III, R – IV, S – I
(C) P – II, Q – III, R – I, S – IV
(D) P – IV, Q – I, R – III, S – II
Q.29

Match the following wire-line logging methods (GROUP I) based on the physical principles of measurements (GROUP II).

 
(A) P – II, Q – I, R – IV, S – III
(B) P – III, Q – I, R – IV, S – II
(C) P – III, Q – II, R – IV, S – I
(D) P – II, Q – I, R – III, S – IV
Q.30

Match the following rock types (GROUP I) with their respective chemical compositions (GROUP II) from the given options.

 
(A) P – III, Q – II, R – IV, S – I
(B) P – II, Q – III, R – I, S – IV
(C) P – III, Q – I, R – IV, S – II
(D) P – III, Q – I, R – II, S – IV
Q.31 The following equations describe the transient fluid flow in a typical petroleum reservoir system. Here, 𝒑 is pressure, 𝒙 𝐚𝐧𝐝 𝒓 𝐚𝐫𝐞 𝐭𝐡𝐞 spatial coordinates in rectangular and cylindrical systems respectively, 𝐚𝐧𝐝 𝒕 is time. Also,𝝓 (𝐩𝐨𝐫𝐨𝐬𝐢𝐭𝐲), 𝝁 (𝐯𝐢𝐬𝐜𝐨𝐬𝐢𝐭𝐲),𝒄𝒇(𝐟𝐨𝐫𝐦𝐚𝐭𝐢𝐨𝐧 𝐜𝐨𝐦𝐩𝐫𝐞𝐬𝐬𝐢𝐛𝐢𝐥𝐢𝐭𝐲), 𝒄𝒕(𝐭𝐨𝐭𝐚𝐥 𝐜𝐨𝐦𝐩𝐫𝐞𝐬𝐬𝐢𝐛𝐢𝐥𝐢𝐭𝐲) 𝐚𝐧𝐝 𝒌(𝐩𝐞𝐫𝐦𝐞𝐚𝐛𝐢𝐥𝐢𝐭𝐲) are constant coefficients.Match the equations (GROUP I) with their corresponding descriptions (GROUP II).  
(A) P – IV, Q – I, R – II, S – III
(B) P – IV, Q – III, R – II, S – I
(C) P – III, Q – IV, R – II, S – I
(D) P – III, Q – IV, R – I, S – II
Q.32 Select the INCORRECT statement related to Enhanced Oil Recovery (EOR) techniques from the following options.
(A) Alkaline flooding recovers crude oil by reduction of interfacial tension (IFT) and reversal of wettability of rocks.
(B) In-situ combustion recovers crude oil by the application of heat, thus lowering the viscosity of the crude oil.
(C) Nitrogen flue gas flooding recovers crude oil by vaporizing the lighter components of the crude oil.
(D) Polymer flooding recovers crude oil by reducing the viscosity and increasing the mobility of water.
Q.33 Match the petroleum reservoir forming traps (GROUP I) with their general classifications (GROUP II). 
(A) P – III, Q – II, R – IV, S – I
(B) P – II, Q – III, R – IV, S – I
(C) P – III, Q – I, R – II, S – IV
(D) P – II, Q – III, R – I, S – IV
Q.34 An Ideal Pressure Buildup Test yields a single straight line for all times, where shut-in Bottom-Hole Pressure (Pws) is plotted against 𝐥𝐨𝐠𝟏𝟎. Here 𝒕𝒑 is the well production time and 𝚫𝒕 is the time elapsed since shut-in. However, in an actual Pressure Buildup Test, a non-linear curve is obtained which can be logically divided into distinct regions.Choose the INCORRECT option from the following.
(A) A late-time region, in which the radius of investigation has reached the well’s drainage boundaries.
(B) A middle-time region during which the pressure transient has moved away from the wellbore and into the bulk formation.
(C) An early-time region during which a pressure transient is moving through the formation nearest the wellbore.
(D) An early-time region during which a pressure transient is moving away from the drainage boundary.
Q.35 When two immiscible fluid phases are placed in contact with a solid surface, one phase usually is attracted to solid more strongly than the other phase.The more strongly attracted phase is called the ‘wetting phase’.The inter-molecular interaction of the non-wetting phase with the solid is_____I       than its intra-molecular interaction. Due to this, the non-wetting phase tends to occupy the       II       of the reservoir.
(A) I = stronger, II = smaller pores
(B) I = stronger, II = larger pores
(C) I = weaker, II = smaller pores
(D) I = weaker, II = larger pores
Q.36 Coal bed methane is methane gas adsorbed in coal seams. To desorb the methane from the coal seam it should be exposed to CO2 and (or) N2. Which ONE of the following is an appropriate reason to enhance the desorption process?
(A) N2 is used because it has low kinetic energy compared to CH4
(B) CO2 is used as it has high kinetic energy compared to CH4
(C) CO2 is used as it strongly binds with coal compared to CH4
(D) N2 is used as it strongly binds with coal compared to CH4
Q.37 Match the platforms (GROUP I) with appropriate support systems (GROUP II). 
(A) P – IV, Q – III, R – I, S – II
(B) P – II, Q – III, R – I, S – IV
(C) P – III, Q – I, R – IV, S – II
(D) P – I, Q – IV, R – II, S – III

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

Q.38 Select the CORRECT statements from the following. Well testing operations on a typical crude oil reservoir
(A) do not measure rock and fluid properties of the reservoir when the well is flowing or shut-in.
(B) measure variation in pressure response of the reservoir with time when the well is flowing or shut-in.
(C) measure productivity index and partial well completion.
(D) do not measure length and conductivity of hydraulic fractures.
Q.39 Crude oil from oil sands contains bitumen and asphaltene and this crude is heavy and viscous at room temperature. Assume that one such crude oil is represented by CXH1.2XOY. For easier transportation through pipelines it should be processed further. Identify the processes which help in transportation of this crude oil.
(A) Drying
(B) Vis-breaking
(C) Coking process
(D) Hydro-treating

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

Q.40 Given matrix 𝑨 =. The eigenvalue corresponding to the eigenvectoris ______________ .
   
Q.41 The maximum value of the function 𝒇(𝒙) = 𝒙𝟒 − 𝟖𝒙𝟐 + 𝟐 for −𝟐 ≤ 𝒙 ≤ 𝟐 is_______.
   
Q.42 Given the second order ordinary differential equation: 𝒚′′ + 𝟑𝒚′ − 𝟒𝒚 = 𝟎 with the initial conditions 𝒚(𝟎) = 𝟑, 𝐚𝐧𝐝 𝒚′(𝟎) = −𝟕, the value of y(1) is________(round off to two decimal places).
   
Q.43 The directional derivative of 𝒇(𝒙, 𝒚, 𝒛) = 𝒙𝟐 + 𝟑𝒚𝟐 + 𝒛𝟐, at point (2,1,0) along the unit vector in x-direction, 𝒊̂, is __.
Q.44 A productivity test conducted on a crude oil well indicates a stabilized flow rate of 150 STB/day (water-free oil production) at a bottom-hole flowing pressure of 935 psig. After shutting the well for 24 hours, the bottom-hole pressure reached a static value of 1250 psig.The Absolute Open Flow (AOF) potential of the well is__________STB/day.
Q.45 A porous medium (shown schematically in the figure) has the following properties.Length L = 600 m, Width W = 8 m, Height h = 0.5 m, Permeability k = 100 mD, Porosity ϕ = 15%.

An incompressible fluid having a viscosity of 2 cP is flowing through a porous medium at the inlet and exit pressures of 7 × 106 Pa and 6 × 106 Pa, respectively.The actual fluid velocity through the porous medium is × 10-7 m/s. (1 Darcy = 10-12 m2)
Q.46 A tubing with an inner diameter of 2.259 inch delivers oil from a well at the rate of 1000 bbl/day. The API gravity and viscosity of the oil are 40o and 1.2 cP, respectively. The tubing makes an angle of 15o with the vertical. Assuming a fanning friction factor of 0.006, the pressure-drop over a length of 1000 ft tubing is  ______ psi (round off to nearest integer).[1 bbl = 5.615 ft3]
Q.47 A cylindrical crude oil reservoir with a radius of 3000 ft is under water influx from a cylindrical aquifer with an estimated radius of 9000 ft. The reservoir has the following properties.Aquifer thickness, h = 40 ft Porosity, ϕ = 15%Formation compressibility, Cf = 4.5 × 10-6 psi-1 Water compressibility, Cw = 4.0 × 10-6 psi-1Assuming a pot reservoir model with fractional encroachment angle as unity, the water influx into the reservoir for a pressure drop of 700 psi is ________MMbbl (million barrels) (round off to two decimal places).(π = 3.14, 1 bbl = 5.615 ft3)
Q.48 A heavy oil reservoir with an initial oil recovery of 10% has the following properties.Confined area A = 1.5 acres, thickness of the reservoir h = 15 ft, effective porosity ϕ = 15%, irreducible water saturation Swr = 25%, oil formation volume factor Bo = 1.10 bbl/STB.An in-situ combustion test was conducted in the above reservoir. Oil recovery due to the combustion process at the well is observed to be 12000 bbl.The total (overall) oil recovery at the end of the in-situ combustion process is__________% (round off to nearest integer) of the original oil in place.(1 acre = 43560 ft2, 1 bbl = 5.615 ft3)
Q.49 A double acting duplex pump with a rod diameter of 2.5 inch and a stroke of 20 inch is to be operated at 60 strokes per minute for drilling down to 10000 ft. The flow rate is 600 gpm. If the volumetric efficiency of the pump is 80%, the liner size is  _________inch (round off to one decimal place).[1 gallon = 231 inch3]
Q.50 The fluid flow through an under-saturated oil reservoir is driven by solution gas drive mechanism. The reservoir parameters are as given below.Compressibility of water, 𝒄𝒘 = 𝟏 × 𝟏𝟎−𝟔 psi-1 Compressibility of formation, 𝒄𝒇 = 𝟏 × 𝟏𝟎−𝟓 psi-1 Connate water saturation, 𝑺𝐰𝐜 = 𝟎. 𝟐Initial reservoir pressure, 𝒑𝒊 = 𝟒𝟎𝟎𝟎 psiReservoir pressure at bubble-point, 𝒑𝐛 = 𝟑𝟎𝟎𝟎 psi Oil formation volume factor, 𝑩𝐨𝐢 = 𝟏. 𝟐𝟒 rb/STBFormation volume factor at bubble point pressure, 𝑩𝐨𝐛 = 𝟏. 𝟐𝟔 rb/STB.The percentage of oil recovered as a fraction of the Original Oil in Place (OOIP) is   ___________% (round off to one decimal place).
Q.51 During drilling, a well is damaged out to a radial distance of 5 ft from the periphery of the wellbore so that the permeability within the damaged zone is reduced to 1/50th of the undamaged effective permeability. After completion, the well is stimulated so that the permeability out to a radial distance of 15 ft from the periphery of the wellbore is increased to twenty times the permeability of the undamaged zone.The radial inflow equation for stabilized flow conditions under semi-steady state conditions is given by  where 𝒑𝐞 is effective pressure, 𝒑𝐰𝐟 is flowing bottom-hole pressure, 𝒒 is flow- rate, 𝝁 is viscosity, 𝒌𝐞 is average effective permeability, 𝒉 is reservoir thickness, 𝒓𝐞 is drainage radius, 𝒓𝐰 is wellbore radius and 𝑺 is skin factor.If 𝒓𝐰 = 𝟎. 𝟓 ft and 𝒓𝐞 = 𝟓𝟎𝟎 ft, then the increase in Productivity Index ratio  is                (round off to one decimal place).
   
Q.52 A depleted and shut-in oil reservoir originally contained 𝟐𝟓 × 𝟏𝟎𝟔 STB of oil with a formation volume factor of 1.35 res bbl/STB and a connate water saturation of 0.25. Cumulative oil production to date has been 𝟐. 𝟓 × 𝟏𝟎𝟔 STB of oil. The oil formation volume factor is now 1.25 res bbl/STB. Assuming no water influx, the gas saturation in the reservoir is   _______% (round off to one decimal place).
Q.53

Surface tension of liquid A in a capillary is being measured in the laboratory using capillary rise (refer the figure given below). The capillary radius (r) is 100 µm, the height of liquid column (h) is 10 cm and Ө = 38o. Density of air can be neglected. Assume liquid A to have the same density as water.Surface tension of liquid A at room temperature is ___________________dynes/cm (round off to one decimal place).

  
Q.54 Miscible displacement process is one of the EOR techniques. The performance of this process depends on fluid physical properties that affect flow behavior in a reservoir. Two of the important properties are density and viscosity.Consider the use of CO2 for one such process. The density of CO2 at the reservoir condition is         lb/ft3 (round off to one decimal place).Relevant data for this calculation are given below. Reservoir temperature = 300 oF (422 K) Reservoir pressure = 1470 psig (100 atm)Compressibility factor (z) at the reservoir condition = 0.5Values of Universal Gas Constant (R) in different units are listed below. Universal Gas Constant (R) = 8.314 m3.Pa.K-1mol-1= 10.731 psi. ft3.lb. mol-1 oR-1= 0.082 L. atm. K-1 mol-1
   
Q.55 In a counter current heat exchanger, the hot fluid enters at 175 oF and exits at 100 oF. The cold fluid enters at 75 oF and exits at 85 oF. For the calculation of heat transfer rate, consider the tube surface area (per unit length) to be 0.26 ft2/ft and a tube length of 40 ft. The overall heat transfer coefficient of the exchanger is 100 BTU/hr-ft2. The minimum number of tubes required in the exchanger for a heat duty of 𝟏𝟓 × 𝟏𝟎𝟓 BTU/hr is _______(round off to nearest integer).

Answer Key 

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

*MTA means Marks to All

GATE 2021 Metallurgical Engineering Previous Year Paper

GATE 2021 Metallurgical Engineering Previous Year Paper
General Aptitude (GA)

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

Q.1 Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P should not be seated adjacent to S. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A) 2
(B) 3
(C) 4
(D) 5
Q.2 Consider the following sentences:The number of candidates who appear for the GATE examination is staggering.A number of candidates from my class are appearing for the GATE examination.The number of candidates who appear for the GATE examination are staggering.A number of candidates from my class is appearing for the GATE examination.Which of the above sentences are grammatically CORRECT?
(A) (i) and (ii)
(B) (i) and (iii)
(C) (ii) and (iii)
(D) (ii) and (iv)
Q.3 A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM.The immediate next time that they will beep together is__________  
(A) 10.08 AM
(B) 10.42 AM
(C) 11.00 AM
(D) 10.00 PM
Q.4 If ⊕÷⊙= 2 ; ⊕÷ Δ = 3; ⊙ +Δ = 5; Δ ×⊗ =10,Then, the value of (⊗−⊕)2 , is:
(A) 0
(B) 1
(C) 4
(D) 16
Q.5 The front door of Mr. X’s house faces East. Mr. X leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another 50 m and stops. The direction of the point Mr. X is now located at with respect to the starting point is_______
(A) South-East
(B) North-East
(C) West
(D) North-West

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

Q.6 Given below are two statements 1 and 2, and two conclusions I and II. Statement 1: All entrepreneurs are wealthy.Statement 2: All wealthy are risk seekers. Conclusion I: All risk seekers are wealthy.Conclusion II: Only some entrepreneurs are risk seekers.Based on the above statements and conclusions, which one of the following options is CORRECT?
(A) Only conclusion I is correct
(B) Only conclusion II is correct
(C) Neither conclusion I nor II is correct
(D) Both conclusions I and II are correct
Q.7 A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is  
(A) 3/16
(B) 45/236
(C) 1/4
(D) 3/4
Q.8
The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is______
(A) 1/8
(B) 1/6
(C) 1/4
(D) 1/2
Q.9 Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to________
(A) 1/4
(B) 1/8
(C) 1/16
(D) 1/32
Q.10 The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to weak demand.In relation to the first sentence above, what does the second sentence do?
(A) Restates an idea from the first sentence.
(B) Second sentence entirely contradicts the first sentence.
(C) The two statements are unrelated.
(D) States an effect of the first sentence.
Metallurgical Engineering (MT)

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

Q.1 For the matrix given below, the eigenvalues are: 
(A) 0, 2, 2
(B) 1, 1, 2
(C) 0, 1, 2
(D) 0, 1, 3
Q.2 Which one of the following is a homogeneous function of degree three?
(A) 𝑥3 + 2𝑥2𝑦2
(B) 𝑦2𝑥 + 2𝑦𝑥2
(C) 𝑦3 + 2𝑥2
(D) 𝑥𝑦2 + 3𝑥𝑦
Q.3 The divergence of a vector field ⃗𝑽→(𝒙, 𝒚, 𝒛), where its three components (𝑽𝒙 ,𝑽𝒚, 𝑽𝒛) are functions of x,y,z, is:
(A)
(B)
(C)
(D)
Q.4 Which one of the following is ‘center split’ defect in rolling operation?
Q.5 Single crystal turbine blades of nickel-based superalloys for aero-engines are manufactured using:
(A) Investment casting
(B) Die casting
(C) Squeeze casting
(D) Directional solidification
Q.6 Elements A and B have the same crystal structure. For a dilute solution of B in A, which one of the following is true?(Given: Δ𝑯𝒎𝒊𝒙 – Mixing enthalpy, aB – Activity of B and  XB – Mole fraction of B)
(A) 𝐼𝑓 Δ𝐻𝑚𝑖𝑥 = 0,  then 𝑎𝐵 < 𝑋𝐵
(B) 𝐼𝑓 Δ𝐻𝑚𝑖𝑥 = 0,   then 𝑎𝐵 > 𝑋𝐵
(C) 𝐼𝑓 Δ𝐻𝑚𝑖𝑥 > 0,   then 𝑎𝐵 < 𝑋𝐵
(D) 𝐼𝑓 Δ𝐻𝑚𝑖𝑥 < 0,   then 𝑎𝐵 < 𝑋𝐵
Q.7 For uniaxial tensile stress-strain behaviour of polycrystalline aluminium, which one of the following statements is FALSE?
(A) True stress is always higher than the engineering stress.
(B) At the ultimate tensile stress point on the true stress – strain curve, 𝑑σ/𝑑ε = 0
(C) Resilience is the area under the elastic region of the engineering stress – strain curve.
(D) Maximum true stress does not correspond to the maximum load.
Q.8 Which one of the following is FALSE for creep deformation?
(A) The minimum creep rate is obtained in the primary stage (stage I).
(B) Creep resistance decreases with decrease in grain size.
(C) Coble creep occurs via grain boundary diffusion.
(D) Nabarro-Herring creep occurs via lattice diffusion.
Q.9 Which one of the following elements alloyed with iron is a ferrite stabilizer?
(A) Nickel
(B) Manganese
(C) Carbon
(D) Silicon
Q.10 Which one of the following is the correct decreasing sequence of Quenching Power for quenchants used in heat treatment of steels?
(A) Oil > Water > Brine > Air
(B) Brine > Oil > Water > Air
(C) Brine > Water > Oil > Air
(D) Water > Brine > Oil > Air
Q.11 For a zeroth order chemical reaction, which one of the following is FALSE?
(A) Concentration versus time plot is a straight line.
(B) Increase in concentration of reacting species increases the rate of reaction.
(C) Half-life depends on the initial concentration and zero-order rate constant.
(D) Rate of reaction depends on temperature.
Q.12 Which one of the following elements oxidizes first in basic oxygen steel making process?
(A) Silicon
(B) Carbon
(C) Manganese
(D) Phosphorus
Q.13 Which one of the following is a hydrometallurgical operation?
(A) Roasting
(B) Leaching
(C) Zone refining
(D) Smelting

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

Q.14
Q.15 The grain size (X) of annealed specimens follows a symmetric distribution with a mean (µ) of 5 µm and a standard deviation (σ) of 0.5 µm. The percentage of specimens with grain size in the range 5 to 6 µm is expected to be:  ________(round off to nearest integer).Given: For the symmetric distribution: Probability (𝐗 ≤ µ + 𝟐σ) = 𝟎. 𝟗𝟖
Q.16 If 𝑬𝒐𝑵𝒊𝟐+/𝑵𝒊 − 0.25 V, the value of µ𝒐𝑵𝒊 𝟐+ (in J mol-1) at 298 K is:______(round off to nearest integer).Given: F = 96500 C mol-1
Q.17 Melting point of Cu is 1358 K and its enthalpy of melting is 13400 J mol-1. The value of free energy change (in J mol-1) for liquid to solid transformation at 1058 K is:  ________(round off to nearest integer).Assume: 𝑪𝑷𝒍𝒊𝒒𝒖𝒊𝒅 = 𝑪𝑷𝒔𝒐𝒍𝒊𝒅
Q.18 A body is subjected to a state of stress given by the following stress tensor: If yielding is predicted by the Tresca Criterion, the uniaxial tensile yield stress (in MPa) of the body should be less than or equal to:_______(round off to nearest integer).
Q.19 Consider homogeneous nucleation of a spherical solid in liquid. For a given undercooling, if surface energy of a nucleus increases by 20 %, the corresponding increase (in percent) in the critical radius of the nucleus is:_______(round off to nearest integer).
Q.20 If saturation magnetization of iron at room temperature is 1700 kA m-1, the magnetic moment (in A m2) per iron atom in the crystal is:   _____ × 10-23 (round off to 1 decimal place).(Given: Lattice parameter of iron at room temperature = 0.287 nm)
Q.21 In the X-ray diffraction pattern of a FCC crystal, the first reflection occurs at a Bragg angle (θ) of 30°. The Bragg angle (in degree) for the second reflection will be:    ____(round off to 1 decimal place).
Q.22 A 0.6 wt.% C steel sample is slowly cooled from 900 °C to room temperature. The fraction of proeutectoid ferrite in the microstructure is:______-(round off to 2 decimal places). Given: Eutectoid composition: 0.8 wt.% CMaximum solubility of carbon in α-Fe: 0.025 wt.% C
Q.23 If the degree of polymerization of polyethylene is 30000, the average molecular weight (in g mol-1) is:          (round off to nearest integer).(Given: Atomic weights of carbon and hydrogen are 12 and 1, respectively)
Q.24

Water flows over a plate of finite length. At 𝒙 = 𝒙𝟏 from the leading edge, the velocity of the flow is 𝑽𝒙 = 𝟎. 𝟓𝒚 – 𝟎. 𝟓𝒚𝟑. The thickness, δ (in meter) of the boundary layer at 𝒙 = 𝒙𝟏 is:  _________(round off to 2 decimal places).Given: V∞ is the free stream velocity.

  
Q.25 The vacancy concentration in a crystal doubles upon increasing the temperature from 27 °C to 127 °C. The enthalpy (in kJ mol-1) of vacancy formation is: ______ (round off to 2 decimal places).Given: R = 8.314 J mol-1 K-1

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

Q.26 The minimum value of y for the equation 𝒚 = 𝒙𝟐 − 𝟐𝒙 + 𝟒 is                       .
(A) 3
(B) 1
(C) 4
(D) 6
Q.27 Match the forming process (in Column I) with its name (in Column II): 
(A) P−1, Q−2, R−3, S−4
(B) P−3, Q−1, R−4, S−2
(C) P−3, Q−4, R−1, S−2
(D) P−1, Q−4, R−3, S−2
Q.28 Match the nondestructive technique (in Column I) with its underlying phenomenon (in Column II): 
(A) P−4, Q−3, R−2, S−1
(B) P−2, Q−1, R−3, S−4
(C) P−2, Q−1, R−4, S−3
(D) P−3, Q−2, R−1, S−4
Q.29 Number of degrees of freedom for the following reacting system is: M (s) + CO2 (g) = MO (s) + CO (g)
(A) 0
(B) 1
(C) 2
(D) 3
Q.30

The condition for getting the binary phase diagram of A-B (shown below) is:

Given: Δ𝑯𝒎𝒊𝒙s𝒐𝒍𝒊𝒅– Enthalpy of mixing of solid   Δ𝑯𝒎𝒊𝒙𝒍𝒊𝒒𝒖𝒊𝒅– Enthalpy of mixing of liquid
(A) Δ𝐻𝑚𝑖𝑥𝑠𝑜𝑙𝑖𝑑 = 0  and Δ𝐻𝑚𝑖𝑥𝑙𝑖𝑞𝑢𝑖𝑑 = 0
(B) Δ𝐻𝑚𝑖𝑥𝑠𝑜𝑙𝑖𝑑 ≪ 0  and Δ𝐻𝑚𝑖𝑥𝑙𝑖𝑞𝑢𝑖𝑑 = 0
(C) Δ𝐻𝑚𝑖𝑥𝑠𝑜𝑙𝑖𝑑 > 0 and Δ𝐻𝑚𝑖𝑥𝑙𝑖𝑞𝑢𝑖𝑑 = 0
(D) Δ𝐻𝑚𝑖𝑥𝑠𝑜𝑙𝑖𝑑 = 0  and Δ𝐻𝑚𝑖𝑥𝑙𝑖𝑞𝑢𝑖𝑑 ≪ 0
Q.31 In the absence of any external stress, which one of the following statements related to the interaction of point defect and a dislocation is FALSE:
(A) An oversized solute atom would preferentially migrate below the slip plane of an edge dislocation.
(B) A spherically symmetric point defect can interact with both the hydrostatic and shear stress fields of a dislocation.
(C) A point defect can locally modify the elastic modulus and thereby can change the interaction energy.
(D) Vacancies are attracted towards the compressive region of dislocation.
Q.32 A single crystal aluminium sample is subjected to uniaxial tension along[112] direction. If the applied tensile stress is 100 MPa and the critical resolved shear stress (CRSS) is 25 MPa, which one of the following slip systems will be activated?
(A) [1̅01](111)
(B) [1̅10](111)
(C) [101](111̅)
(D) [011](111̅)
Q.33

One-dimensional steady-state temperature distribution in two adjacent refractory blocks (with thermal conductivities, k1 and k2) of unit cross- sectional area are shown below. The temperature T1 and thermal contactresistance of the interface, respectively, are:

  
(A) 200 K, 0.5 K W-1
(B) 400 K, 1.0 K W-1
(C) 200 K, 0.25 K W-1
(D) 500 K, 0.5 K W-1
Q.34

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

   

(A)		 ∆𝑃𝑅 /2𝐿

(B)		 ∆𝑃𝑅/𝐿

(C)		 3∆𝑃𝑅/2𝐿

(D)		 2∆𝑃𝑅/𝐿
Q.35 Match the terms (in Column I) with the unit process (in Column II)
(A) P−2, Q−1, R−4, S−3
(B) P−4, Q−1, R−2, S−3
(C) P−4, Q−3, R−1, S−2
(D) P−2, Q−3, R−4, S−1
Q.36 A blast furnace uses hematite ore with 80% Fe2O3 and 20% gangue materials. It uses 600 kg coke per ton of hot metal. The coke contains 85% C and 15% ash. The composition of hot metal is 95.5% Fe and 4.5% C.The weight of iron ore used and slag produced per ton of hot metal respectively, are:Given: Atomic weight: O = 16, C = 12, N = 14, Fe = 56   All the compositions are in wt.%.1 ton = 1000 kgAssume that the gangue materials of the ore and ash content of coke form slag while Fe2O3 in the ore is consumed in making hot metal.
(A) 1705 kg, 431 kg
(B) 2131 kg, 546 kg
(C) 1705 kg, 331 kg
(D) 1500 kg, 431 kg

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

Q.37 Consider the function (𝒙) = 𝒙 − 𝐜𝐨𝐬 𝒙. Using Newton-Raphson method, the estimated root of (𝒙) after the first iteration is:           (round off to 3 decimal places).Assume: Initial guess of the root = 0.5 radians.
Q.38 The work done by a force  ⃗𝑭→ = 𝟐𝒙𝒊 + 𝟑𝒚𝒋  along a straight line from point (0, 0) to (1, 2) is: ________(round off to nearest integer).
Q.39 A coin is tossed three times. Given that there are more heads than tails, the probability of getting exactly one tail is: __________(round off to 2 decimal places).
Q.40 A continuous fillet weld is made using a 3000 W welding machine. At a travel speed of 6 mm s-1, the cross-sectional area (in mm2) of the weld is:______(round off to nearest integer).Given: The unit energy required to melt the metal is 6 J mm-3.Heat transfer factor = 0.6 Melting factor = 0.5
Q.41 Liquid iron is cast into a spherical sand mold (6 cm radius) and a cubical sand mold (12 cm edge length). If solidification time is 60 minutes in the spherical casting, the time (in minutes) required to solidify in the cubical casting is: _________(round off to nearest integer).
Q. 42 True strain for 60% height reduction of a sample subjected to hot forging is: ______(round off to 2 decimal places).
Q.43 For the equilibrium reaction: 𝟐𝑪𝒖 (𝒔) + 𝑺𝑶𝟐(𝒈) = 𝑪𝒖𝟐𝑺 (𝒔) + 𝑶𝟐 (𝒈), thevalue of 𝒍𝒏at 973 K is:_____(round off to 2 decimal places).Given: 𝟐𝑪𝒖 (𝒔) + 𝟎. 𝟓𝑺𝟐(𝒈) = 𝑪𝒖𝟐𝑺 (𝒔)   ΔGo at 973 K = -100 kJ𝑺𝑶𝟐(𝒈) = 𝟎. 𝟓𝑺𝟐(𝒈) + 𝑶𝟐(𝒈)  ΔGo at 973 K = 292 kJ R = 8.314 J mol-1 K-1Assume: Cu and Cu2S are pure solids.
Q.44 One mole of an ideal gas at 10 atm. and 300 K undergoes reversible adiabatic expansion to a pressure of one atm. The work done (in Joule) by the gas is: ___________(round off to nearest integer).Given: R = 8.314 J mol-1 K-1; 1 atm. = 101325 Pa; Cp = 2.5R
Q.45 The figure shows the entropy versus temperature (S-T) plot of a reversible cycle of an engine. If T1 = 200 K and T2 = 600 K, the efficiency of the engine (in percent) is: ______(round off to 2 decimal places). 
Q. 46

Two dislocation lines parallel to z-axis lying in the x-z plane are shown in the figure. The glide force (in Newton) exerted by the edge dislocation on the screw dislocation is: __(round off to nearest integer).

For the edge, the shear stress component is given by:
Given: Shear modulus, G = 28 GPa Poisson’s ratio, ν = 0.3 Burgers vector, b = 0.29 nmDistance between the two dislocations = 0.5 nm
Q.47 In a material, a shear stress of 100 MPa is required to bow a dislocation line between precipitates with a spacing of 0.2 µm. If the spacing between the precipitates is increased to 0.5 µm, the shear stress (in MPa) to bow the dislocation would be: ____________(round off to nearest integer).
Q.48 A metal plate is in a state of plane strain (ε𝒛𝒛 = 𝟎) with σ𝒙𝒙 = σ𝒚𝒚≠ 𝟎 andτ𝒙𝒚 = τ𝒙𝒛 = τ𝒚𝒛 = 𝟎. If the Poisson’s ratio is 0.3, the ratio, σ𝒛𝒛/σ𝒙𝒙 is________(round off to 1 decimal place).
Q.49 An infinite metal plate has a central through-thickness crack of length 80/πmm. The maximum applied stress (in MPa) that the plate can sustain in mode I is: ________________(round off to nearest integer). Assume: Linear elastic fracture mechanics is validGiven: Fracture toughness, 𝑲𝑰𝑪= 20 MPa m1/2
Q.50

A hypothetical binary eutectic phase diagram of A – B is shown below. An alloy with 5 wt.% B solidifies with no convection. Assuming steady state, the critical temperature gradient (in K mm-1) required to maintain planar solidification front is: ___(round off to nearest integer).

Given: Diffusivity of B in liquid = 10-9 m2 s-1 Velocity of solidification front = 4 µm s-1
Q.51

A thick steel plate containing 0.1 wt.% C is carburized at 950 °C. The plate’s surface carbon concentration is maintained at 1.1 wt.% C. After 9 hours, the depth (in mm) below the surface at which the carbon concentration is 0.6 wt.% C will be: ________(round off to 2 decimal places).Given: Diffusivity of carbon in γ-Fe at 950 °C = 1.6 x 10-11 m2 s-1 Error function table:
Q.52 At 25 °C, iron corrodes in a deaerated acid of pH 3 with a corrosion current density of 4 µA cm-2. The corrosion potential (V) is:_____ (round off to 2 decimal places).Given: βc = 0.1 V per decade of current densityExchange current density of hydrogen on iron surface = 10-9 A cm-2 R = 8.314 J mol-1 K-1, F = 96500 C mol-1All potentials are with reference to standard hydrogen electrode.
Q.53 The radius of an interstitial atom which just fits (without distorting the structure) inside an octahedral void of a bcc-iron crystal (in nm) is:______(round off to 3 decimal places).Assume the radius of Fe atom to be 0.124 nm.
Q.54 Nickel undergoes isothermal oxidation at 800 K for a duration of 400 s resulting in a weight gain of 2 mg cm-2. The weight gain (mg cm-2) after a duration of 1600 s is:  _________(round off to nearest integer).Assume: Weight gain is proportional to square root of time.
Q.55 A solid sphere (0.5 m radius) is enclosed within a larger hollow sphere (1 m radius), as shown in figure. The radiation exchange takes place between the outer surface (surface 1) of the small sphere and the inner surface (surface 2) of the bigger sphere. The value of the view factor, F22 is:__________(round off to 2 decimal places).Given: View factor (Fij) is the fraction of the radiation leaving surface i that is intercepted by surface j.

Answer Key

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

GATE 2021 Mining Engineering Previous Year Paper

GATE 2021 Mining Engineering Previous Year Paper
General Aptitude  (GA)
Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1

Arun and Aparna are here.

Arun and Aparna is here.

Arun’s families is here.Arun’s family is here.

Which of the above sentences are grammatically CORRECT?
(A) (i) and (ii)
(B) (i) and (iv)
(C) (ii) and (iv)
(D) (iii) and (iv)
Q.2

The mirror image of the above text about the x-axis is
Q.3		 Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
(A) 1/36
(B) 1/12
(C) 1/8
(D) 1/4
Q.4		 ⊕ and ⊙ are two operators on numbers p and q such that𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p ⊕ q = px qThen, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=
(A) 40
(B) -26
(C) -33
(D) -40
Q.5		 Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
(A) 6
(B) 9
(C) 18
(D) 24

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

Q.6		 On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.The distance of point Q to point O, in the same units, should be            
(A) 3
(B) 4
(C) 5
(D) 6
Q.7		 The author said, “Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk on to the stage and start speaking. In my opinion, it is no less important for public speakers to rehearse their talks.”Based on the above passage, which one of the following is TRUE?
(A) The author is of the opinion that rehearsing is important for musicians, actors and public speakers.
(B) The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors.
(C) The author is of the opinion that rehearsing is more important only for musicians than public speakers.
(D) The author is of the opinion that rehearsal is more important for actors than musicians.
Q.8		 Some football players play cricket.All cricket players play hockey.Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
(A) No football player plays hockey.
(B) Some football players play hockey.
(C) All football players play hockey.
(D) All hockey players play football.
Q.9


In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.The probability that any point picked randomly within the square falls in the shaded area is_________
(A)
(B) 1/2
(C)
(D) 𝜋/4
Q.10		 In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum area of the triangle PQR possible, in cm2, is
(A) 18
(B) 24
(C) 48√3
(D) 144√3
Mining Engineering (MN
  1. – Q.19 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1		 Tricone roller bit is used with
(A) down-the-hole hammer.
(B) Jack hammer.
(C) rotary-percussive drill.
(D) rotary drill.
Q.2		 Resuing method of mining is practiced for
(A) thick vein deposit.
(B) massive shallow deposit.
(C) narrow vein deposit.
(D) massive deep-seated deposit.
Q.3		 The equipment used for both drop cut and terrace cut in surface mining is
(A) surface miner.
(B) shovel.
(C) dragline.
(D) bucket wheel excavator.

 

Q.4		 Surface miner does NOT have a
(A) differential gear for turning.
(B) tensioning arrangement for crawler.
(C) scraper plate behind the drum.
(D) pick cooling system.
 
Q.5		 Induced blasting enhances production in
(A) sublevel stoping.
(B) block caving.
(C) cut and fill mining.
(D) shrinkage stoping.
 
Q.6		 The measures of dispersion of a dataset are
(A) standard deviation, range and mode.
(B) standard deviation, range and interquartile range.
(C) variance, range and median.
(D) interquartile range, median and mode.
 
Q.7		 NONEL is used as down-the-hole initiator to
(A) avoid generation of air overpressure.
(B) provide precise delay.
(C) avoid deflagration of column charge.
(D) reduce ground vibration.
Q.8

The vectors ⃗𝒂→ and ⃗𝒃→ act in a plane as shown below. The magnitude of the vector⃗𝒄→ = ( → +) × (→ −)  is

               
(A) zero.
(B) half to the area bounded by the vectors → and.
(C) equal to the area bounded by the vectors → and.
(D) twice the area bounded by the vectors → and.
 
Q.9		 As per MMDR Act 1957, for the allocation of lease of minor minerals
(A) the State Government is authorised to give the permit.
(B) the Central Government is authorised to give the permit.
(C) the State Government is authorised to give permit but with the consent of Central Government.
(D) the Central Government is authorised to give permit but with the consent of State Government.
 
Q.10		 In photogrammetry, the ‘Tilt of a photograph’ refers to the angle between the
(A) lines joining the opposite fiducial marks of a photograph.
(B) normal to the plane of photograph and optical axis.
(C) vertical and the axis of the flight.
(D) vertical and optical axis of the camera.
Q.11		 The hydraulic sand stowing pipeline layout should be such that
(A) the geometric profile must coincide with the hydraulic gradient line.
(B) the hydraulic profile should always be below the hydraulic gradient line.
(C) the hydraulic profile should always be above the hydraulic gradient line.
(D) the geometric profile should always be above the hydraulic gradient line.
 
Q.12		 For a “positive definite” square matrix, the TRUE statement is
(A) the matrix is singular.
(B) all the eigen values of the matrix are greater than zero.
(C) all the eigen values of the matrix are zero.
(D) some of the eigen values can be less than zero.
 
Q.13		 The standard normal distribution is a
(A) non-parametric distribution.
(B) single parameter distribution.
(C) two-parameter distribution.
(D) three-parameter distribution.
 
Q.14		 Variance of the sum of two statistically independent random variables 𝑿and 𝒀, 𝝈𝟐 𝑿+𝒀, is
(A) 𝜎2𝑋 + 𝜎2𝑌.
(B) 𝜎2𝑋 + 𝜎2𝑌 + 2𝜎𝑋𝑌.
(C) 𝜎2𝑋 + 𝜎2𝑌 + 𝜎𝑋𝑌.
(D) 𝜎2𝑋 + 𝜎2𝑌− 2𝜎𝑋𝑌.
Q.15		 The difference between depreciation and amortization allowances in tax calculation is that
(A) depreciation is for a tangible asset applicable on its declared life; whereas amortization is for an intangible asset applicable on a specified period.
(B) depreciation is for an intangible asset applicable on its declared life; whereas amortization is for a tangible asset applicable on a specified period.
(C) depreciation is for a tangible asset applicable on a specified period; whereas amortization is for an intangible asset applicable on its declared life.
(D) depreciation is for an intangible asset applicable on a specified period; whereas amortization is for a tangible asset applicable on its declared life.
 
Q.16		 Owning cost of a machine does NOT include
(A) purchase price.
(B) insurance.
(C) interest.
(D) operating cost.
 
Q.17		 Folds are the structural features resulting from
(A) ductile deformation of earth crust.
(B) brittle deformation of earth crust.
(C) high impact tectonic stresses of earth crust.
(D) fracturing of earth crust.
Q.18

The CORRECT curve showing the relationship between vertical stress on a coal pillar and extraction ratio of abord and pillar panel in a horizontal seam is

  
(A) Curve A.
(B) Curve B.
(C) Curve C.
(D) Curve D.
 
Q.19		 Given impeller diameter D, speed of rotation n and air density ρ, for geometrically similar fans, the fan pressure is proportional to
(A) 𝑛𝐷2𝜌.
(B) 𝑛2𝐷2𝜌.
(C) 𝑛2𝐷5𝜌2.
(D) 𝑛3𝐷5𝜌.

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

 
Q.20		 A coal sample having moisture content of 8.0% has unit weight 15.6 kN/m3. The dry unit weight of the sample, in kN/m3 is_________ .[round off to 2 decimal places]
   
Q.21		 The value of the integral 𝑰 =computed using Simpson’s 1/3 rule with 2 sub intervals is  _________.[round off to 3 decimal places]
   
Q.22		 In the context of sound frequency analysis, the lower and upper frequencies of a 1/1 octave band are 710 Hz and 1420 Hz respectively. The corresponding centre frequency of the band in Hz, is__________ . [round off to the nearest integer]
   
Q.23		 In Battle Environmental Evaluation System (BEES) of Environmental Impact Assessment (EIA), “air pollution” has a Parameter Importance Unit (PIU) value of 52. The Environmental Quality (EQ) score of a project with respect to air pollution was 0.8 before the project implementation and it becomes 0.6 after the project implementation. The difference in the “Environmental Impact Unit (EIU)”, before and after the project implementation is  _________ .[round off to 2 decimal places]
   
Q.24		 A system consists of four components connected functionally in a parallel configuration. The reliability of the individual components is 0.40, 0.60, 0.50 and 0.40. The system reliability, is  ______ . [round off to 3 decimal places]
   
Q.25		 A vehicle is moving at a speed of 12 m/s on a level road. It applies emergency brakes and starts to skid without rolling in a straight path. The deceleration of the vehicle is constant after braking and it comes to rest at a distance of 15 m. Assuming, g = 10 m/s2, the coefficient of kinetic friction between the tyres and road is  __. [round off to 2 decimal places]

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

Q.26		 In a bord and pillar panel six shuttle cars, each of 10 tonne capacity, are deployed to transport coal produced by two continuous miners to a belt conveyor. Each shuttle car on an average carries 80% of its rated capacity and makes 7 round trips in an hour. The belt conveyor has a capacity such that the effective material cross section area is of 0.09 m2 and runs at a speed 1.1 m/s. The broken coal has a bulk density of 1.2 tonne/m3. The ratio between the production and the capacity of the belt conveyor, in percent is
(A) 65.46
(B) 71.42
(C) 78.56
(D) 82.46
Q.27		 With reference to the figure related to rock cutting by point attack tool, match the angle with corresponding name.
       
(A) P-2, Q-4, R-1, S-3.
(B) P-4, Q-2, R-1, S-3.
(C) P-2, Q-4, R-3, S-1.
(D) P-4, Q-2, R-3, S-1.
Q.28		 The pit bottom in a correlation survey is shown in the figure. Points C and D represent two suspended wires. The bearing of line CD is 𝟐𝟖𝟔𝐨𝟎𝟎′𝟎𝟎′′ and its length is 4.64 m. The angle CED is measured as 𝟎𝟎𝐨𝟎𝟎′𝟒𝟎′′. The length of line DE is 5.46 m. Considering the Weisbach triangle method, the bearing of the line CE is    
(A) 286o00’47”
(B) 285o59’12.9″
(C) 286o00’40”
(D) 285o00’47.1″
 
Q.29		 A dump truck moves up an incline of 50 with constant tractive force of 800 kN. The gross mass of the truck is 250 tonne and its rolling resistance is 545 kN. The acceleration due to gravity is 10 m2/s. The time required, in s, to reach a speed of 3.3 m/s from 1.0 m/s is
(A) 22.0
(B) 15.5
(C) 3.3
(D) 0.2
Q.30		 In a longwall panel, face is supported with shields of yield capacity 460 tonne per shield. The distance from the canopy tip to coal face is 0.15 m when the support is fully advanced. The depth of web is 0.60 m. The shields are set skin to skin at the face. Length of the canopy of the shield is 3.25 m and width 1.5 m. Setting capacity is 80% of the yield capacity. The setting resistance at the maximum and minimum span of the coal face, in tonne/m2, respectively are
(A) 61.33 and 72.15.
(B) 63.72 and 75.48.
(C) 76.66 and 90.19.
(D) 91.99 and 108.22.
 
Q.31		 A 10 ml sample of wastewater is diluted with water having no BOD, to fill a 300 ml BOD bottle. The initial DO of the diluted waste water is 9.0 mg/l. If the BOD5 of the waste water sample is 60 mg/l, the final DO of the diluted waste water in mg/l, is
(A) 5.0
(B) 6.0
(C) 7.0
(D) 8.0
   
Q.32

The Mohr circle of stress of a dry porous rock is shown in the figure. If the rock is fully saturated with a pore pressure 𝒑, then the Mohr circle takes the form of

       
(A)
(B)
(C)
(D)
Q.33		 The straight line shown depicts the failure criterion of a rock type. The values of stress at points A and B are as shown. The safety factor at the points A and B respectively are  
(A) 1.175 and 0.755
(B) 1.324 and 0.851
(C) 0.851 and 1.324
(D) 0.755 and 1.175
Q.34

Figure shown relates to the manufacture of roof bolts. With respect to the cost/revenue vs production level, match the appropriate trend line with corresponding description.

          
(A) (P)-(1); (R)-(3); (S)-(2).
(B) (P)-(2); (R)-(1); (S)-(3).
(C) (P)-(1); (R)-(2), (S)-(3).
(D) (P)-(3); (R)-(1); (S)-(2).
Q.35		 The value of 𝐥𝐢𝐦 (𝒙 √𝒙𝟐 + 𝒃𝟐 − √𝒙𝟒 + 𝒃𝟒 ) is      
(A) 0.
(B) b2/2
(C) ∞.
(D) 𝑏2.
Q.36		 In order to check whether iron ore is supplied to the specification of 62% Fe, a steel company has conducted a hypothesis test with the null hypothesis as H0: µFe = 62% and alternative hypothesis Ha: µFe < 62%. A random sample of 5 observations reveal the following grade values of the lot, 58%, 56%, 60%, 64%, 62%. The t-test statistic for the hypothesis is
(A) – 3.000
(B) 1.414
(C) – 1.414
(D) 3.000

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

 
Q.37

Production planning of a small quarry having 3 years of life is shown in the figure. The following information of revenue and cost data are available.

  

Selling price of ore = Rs. 1500/tonne Ore mining cost = Rs. 500/tonne Waste mining cost = Rs. 500/m3 Initial capital = Rs. 1000 million Discount rate = 10 %By neglecting depreciation, salvage value and corporate tax, the NPV of the mining project, in million Rs., is .____[round off to 2 decimal places]
   
Q.38

A triangular distributed load is applied on top of a beam as shown in the figure. The value of maximum bending  moment in kN-m is __________.[round off to 2 decimal places]

    
   
Q.39		 For a dumpy level, the bubble tube has sensitivity of 40″ for one division. While taking a staff reading at a distance of 60 m, the bubble is out of centre by 2 divisions. The error in staff reading in mm is  ________. [round off to one decimal place]
Q.40		 On an old plan of scale 1:1000, leasehold area of a mine is now measured as 802 cm2 using a planimeter. The plan is found to have shrunk, such that the original line of 10 cm is now measured as 9.8 cm on the plan. True lease hold mine area, in m2, is   _______. [round off to the nearest integer]
   
Q.41		 CO is released from a point source on a level ground at a rate of 25 g/s. The average wind speed is 5 m/s. The dispersion coefficients are 150 m and 200 m in horizontal and vertical directions, respectively, at a receiver station located on the ground along the downwind direction. Assuming the plume follows Gaussian dispersion model, the concentration of CO, in µg/m3, at the station is  _________ .[round off to 2 decimal places]
   
Q.42		 Assume that COVID-19 growth rate of number of infections per day (𝒄) in a certain population is represented by the following differential equation.𝟏𝟎𝟎dc/𝒅𝒕− 𝟕 𝒄 = 𝟎Where, 𝒕 stands for time in days. Time taken for the number of infections per day to double, in days, is  _______. [round off to the nearest integer]
   
Q.43		 Ore is hoisted from 620 m depth using a single skip of 7 tonne pay load. The skip winding system has constant acceleration/deceleration of 1 m/s2 and a constant speed of 10 m/s. The skip loading time and unloading time are 120 s and 60 s, respectively. Considering the overall utilization of the skip as 70 %, the maximum daily capacity of the winding system, in tonne, is  ___________ . [round off to the nearest integer]
   
Q.44		 In an analysis of fragmented blast muck, the mean fragment size is found to be 60 cm with uniformity index of 1.25. Considering Rosin-Ramler equation, the cumulative mass fraction, in percent, to pass the grizzly screen size of 100 cm is  _______. [round off to 2 decimal places]
Q.45		 A single-acting reciprocating ram pump, while running at 120 rpm, delivers water at a rate of 10 litres per second. Considering the ram diameter is 150 mm and stroke length is 300 mm, the volumetric efficiency of the pump, in percent is _______ .[round off to one decimal place]
Q.46		 In a sand stowing arrangement, the slurry has a sand concentration of 35% by volume. The specific gravity of sand grain is 2.6. The concentration of sand by weight, in percent, in the slurry is ___________ . [round off to one decimal place]
Q.47		 In a surface mine, third bench from the pit bottom is blasted, as shown in the figure. The width, height and slope angle of each bench are 8 m, 6 m, and 800, respectively. A fly rock is projected at an angle of 450 with the horizontal with initial velocity, v. If the acceleration due to gravity is 10 m/s2 then the minimum velocity (v) in m/s required for the fly rock to reach just beyond toe of the pit slope is  _________  .[round off to 2 decimal places]      
Q.49

A random variable X is defined by

 

The value of (𝑿𝟐) is   _______. [round off to one decimal place]
Q.51		 Rate of fuel consumption 𝒇𝒄 (litres per hour) of a truck varies with truck speed 𝒙, (kmph) as given below𝒇c = 𝟐𝟎 + 𝒙𝟐/𝟓𝟎The fuel price is Rs. 70 per litre. Other costs amount to Rs. 500 per hour. If the truck travels 100 km from a coal mine to a thermal plant, the speed of the truck, in kmph, that minimizes the total cost is   _________. [round off to one decimal place]
Q.54		 A coal mine operating in three shifts produces 400 tonnes of coal per day with a face OMS of 1.0 from panel A, and 200 tonnes of coal with face OMS of 1.0 from panel B. The panel A and panel B are in parallel with resistance0.6 Ns-2m-8 and 0.5 Ns-2m-8, respectively. If the panels are supplied with minimum permissible quantity as per CMR 2018, the requisite regulator resistance to meet the conditions in Ns-2m-8 is   _______. [round off to 2 decimal places]
   
Q.55

A set of three steel bars of equal cross-sectional area of 0.01 m2 are loaded, as shown in the figure. The elastic modulus of steel is 200 GPa. The overall change of length of the complete set of bars, in mm, is  _______ . [round off to 3 decimal places]

   

Answer Key 

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

GATE 2021 Mechanical Engineering Section-II Previous Year Paper

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 = 𝑎𝟐𝒑, where 𝑎 ∈ 𝕽 and 𝑎 ∉ {−𝟏, 𝟎, 𝟏}. Based on the given information, the eigen value of 𝑨𝟐 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 = −𝒚𝟐𝒕, 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.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 𝒓𝟏 and 𝒓𝟐 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.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 ∫𝟓.𝟐𝟒 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 𝒄𝒑 =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 m3 contains 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: 𝑻𝒔𝒂𝒕 =143.61 °C, 𝒗𝒇 = 0.001084 m3/kg, 𝒗𝒈 = 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) 𝑄𝑇𝑟
(B) 𝑄−1𝑟
(C) 𝑟/𝑟𝑇𝑟
(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) 2𝑦𝑒𝑦 = 𝑒𝑥 + 𝑒
(B) 𝑦2𝑒𝑦 = 𝑒𝑥
(C) 𝑦𝑒𝑦 = 𝑒𝑥
(D) (1 + 𝑦)𝑒𝑦 = 2𝑒𝑥
Q.29

A factory produces 𝒎 (𝒊 = 𝟏, 𝟐, … , 𝒎) products, each of which requires processing on 𝒏 (𝒋 = 𝟏, 𝟐, … , 𝒏) workstations. Let 𝒂𝒊𝒋 be the amount of processing time that one unit of the 𝒊𝒕𝒉 product requires on the 𝒋𝒕𝒉 workstation. Let the revenue from selling one unit of the 𝒊𝒕𝒉product be 𝒓𝒊 and 𝒉𝒊 be the holding cost per unit per time period for the 𝒊𝒕𝒉 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 𝒅𝒊𝒕, and the capacity of the 𝒋𝒕𝒉 workstation in time period 𝒕 is 𝒄𝒋𝒕. Consider the aggregate planning formulation below, with decision variables 𝑺𝒊𝒕 (amount of product 𝒊 sold in time period 𝒕), 𝑿𝒊𝒕 (amount of product 𝒊 manufactured in time period 𝒕) and 𝑰𝒊𝒕 (amount of product 𝒊 held in inventory at the end of time period 𝒕.

𝑺𝒊𝒕 ≥ 𝒅𝒊𝒕       ∀  𝒊, 𝒕< 𝒄𝒂𝒑𝒂𝒄𝒊𝒕𝒚 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕 >< 𝒊𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝒃𝒂𝒍𝒂𝒏𝒄𝒆 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕 >𝑿𝒊𝒕, 𝑺𝒊𝒕, 𝑰𝒊𝒕 ≥ 𝟎; 𝑰𝒊𝟎 = 𝟎

The capacity constraints and inventory balance constraints for this formulation are
(A) 𝑚∑ 𝑎𝑖𝑗𝑋𝑖𝑡 ≤ 𝑐𝑗𝑡 ∀ 𝑗, 𝑡 and 𝐼𝑖𝑡 = 𝐼𝑖,𝑡−1 + 𝑋𝑖𝑡 − 𝑆𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(B) 𝑚∑ 𝑎𝑖𝑗𝑋𝑖𝑡 ≤ 𝑐𝑗𝑡 ∀ 𝑖, 𝑡 and 𝐼𝑖𝑡 = 𝐼𝑖,𝑡−1 + 𝑋𝑖𝑡 − 𝑑𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(C) 𝑚∑ 𝑎𝑖𝑗𝑋𝑖𝑡 ≤ 𝑑𝑖𝑡   ∀ 𝑖, 𝑡 and 𝐼𝑖𝑡 = 𝐼𝑖,𝑡−1 + 𝑋𝑖𝑡 − 𝑆𝑖𝑡 ∀ 𝑖, 𝑡𝑖
(D) 𝑚∑ 𝑎𝑖𝑗𝑋𝑖𝑡 ≤ 𝑑𝑖𝑡   ∀  𝑖, 𝑡 and  𝐼𝑖𝑡 = 𝐼𝑖,𝑡−1 + 𝑆𝑖𝑡 − 𝑋𝑖𝑡 ∀ 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.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.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.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.35 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.35 Find the positive real root of 𝒙𝟑 − 𝒙 − 𝟑 = 𝟎 using Newton-Raphson method. If the starting guess (𝒙𝟎) is 2, the numerical value of the root after two iterations (𝒙𝟐) 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, 𝒔𝟎 = 0.1 mm/revolution and chip velocity, Vf = 20 m/min, the shear strength, 𝑟𝒔 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.46

Water flows out from a large tank of cross-sectional area 𝑨𝒕 = 𝟏 m2 through a small rounded orifice of cross-sectional area 𝑨𝒐 = 𝟏 cm2, located at 𝒚 = 𝟎. Initially the water level, measured from 𝒚 = 𝟎, is 𝑯 = 𝟏 m. The acceleration due to gravity is 9.8 m/s2.

Neglecting any losses, the time taken by water in the tank to reach a level of 𝒚 = 𝑯/𝟒 is      seconds (round off to one decimal place).
Q.47

Consider the open feed water heater (FWH) shown in the figure given below:

Specific enthalpy of steam at location 2 is 2624 kJ/kg, specific enthalpy of water at location 5 is 226.7 kJ/kg and specific enthalpy of saturated water at location 6 is 708.6 kJ/kg. If the mass flow rate of water entering the open feed water heater (at location 5) is 100 kg/s then the mass flow rate of steam at location 2 will be       kg/s (round off to one decimal place).
Q.48

A high velocity water jet of cross section area = 0.01 m2 and velocity = 35 m/s enters a pipe filled with stagnant water. The diameter of the pipe is 0.32 m. This high velocity water jet entrains additional water from the pipe and the total water leaves the pipe with a velocity 6m/s as shown in the figure.

The flow rate of entrained water is  __________litres/s (round off to two decimal places).
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.50

An adiabatic vortex tube, shown in the figure given below is supplied with 5 kg/s of air (inlet 1) at 500 kPa and 300 K. Two separate streams of air are leaving the device from outlets 2 and 3. Hot air leaves the device at a rate of 3 kg/s from outlet 2 at 100 kPa and 340 K, and 2 kg/s of cold air stream is leaving the device from outlet 3 at 100 kPa and 240 K.

Assume constant specific heat of air is 1005 J/kg.K and gas constant is 287 J/kg.K. There is no work transfer across the boundary of this device. The rate of entropy generation is   ________kW/K (round off to one decimal place).
Q.51

A block of negligible mass rests on a surface that is inclined at 300 to the horizontal plane as shown in the figure. When a vertical force of 900 N and a horizontal force of 750 N are applied, the block is just about to slide.

The coefficient of static friction between the block and surface is  __________(round off to two decimal places).
Q.52

The wheels and axle system lying on a rough surface is shown in the figure.

Each wheel has diameter 0.8 m and mass 1 kg. Assume that the mass of the wheel is concentrated at rim and neglect the mass of the spokes. The diameter of axle is 0.2 m and its mass is 1.5 kg. Neglect the moment of inertia of the axle and assume g = 9.8 m/s2. An effort of 10 N is applied on the axle in the horizontal direction shown at mid span of the axle. Assume that the wheels move on a horizontal surface without slip. The acceleration of the wheel axle system in horizontal direction is __________𝐦/𝐬𝟐 (round off to one decimal place).
Q.53 A cantilever beam with a uniform flexural rigidity (EI = 200 x 106 N.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).
Q.55

The figure shows the relationship between fatigue strength (S) and fatigue life (N) of a material. The fatigue strength of the material for a life of 1000 cycles is 450 MPa, while its fatigue strength for a life of 106 cycles is 150 MPa.

The life of a cylindrical shaft made of this material subjected to an alternating stress of 200 MPa will then be     cycles (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     

Latest Current Affairs 03 February 2022

NATIONAL NEWS

Sovereign Green Bonds announced in Budget 2022. 

Union Minister Nirmala Sitharaman announced during the Budget 2022 presentation that Sovereign Green Bonds will be a major part of the Central Government’s Borrowing Programme 2022-23. Green Bonds are financial instruments that are issued by any sovereign entity

Statue of Equality to be inaugurated in Hyderabad on February 5

Prime Minister Narendra Modi will be inaugurating a 216-feet tall ‘Statue of Equality in Hyderabad on February 5, 2022. The statue commemorates 11th-century Bhakti Saint Sri Ramanujacharya.

India first to qualify for four consecutive U19 World Cup finals

India has become the first-ever team to qualify for four consecutive ICC U19 World Cup finals. India reserved its final berth after defeating Australia by 96 runs in the semi-finals. 

ISRO targets to launch Chandrayaan 3 in August 2022

Union Minister Jitendra Singh informed the Lok Sabha that the Chandrayaan 3 mission is scheduled for launch in August 2022. The mission will include a lander and a rover, similar to the Chandrayaan 2 mission, which had failed to soft-land its lander module on the lunar surface. 

India to face England in U19 World Cup final on February 5th India will be facing another unbeaten team, England in the ICC U19 World Cup final on February 5, 2022. The ICC U19 World Cup final will be played on February 5th at Sir Vivian Richards Stadium, North Sound in Antigua. 


IPL 2022 to be held in Mumbai and Pune, playoffs likely in Ahmedabad

BCCI president Sourav Ganguly confirmed recently that IPL 2022 will be held in Mumbai and Pune in Maharashtra. The playoffs are likely to be held in Ahmedabad but a formal decision is likely to be taken at a later stage. BCCI President Sourav Ganguly said that the final decision on the venues for the knockout stages will be taken later. 

Yash Dhull third Indian to score century in ICC U19 World Cup

India U19 skipper Yash Dhull has become the third Indian to score a century in the ICC U19 Cricket World Cup after Virat Kohli and Unmukht Chand. Yash Dhull’s century knock came in India’s semi-final match against Australia in ICC U19 World Cup 2022. Dhull along with deputy skipper Shaik Rasheed put up an incredible 204-run partnership for the third wicket. This was the second biggest partnership of the tournament. The Indian U19 skipper said after the win that the plan was for him and Rasheed to bat till the end and it worked. 


Indian diplomat to skip Beijing Winter Olympics 2022 over Galwan soldier row

Indian diplomat in Beijing has decided to boycott Beijing Winter Olympics 2022 opening and closing ceremony after China chose a soldier, who fought in the Galwan Valley clash against Indian troops in 2020, as a torchbearer. India’s foreign ministry called it “regrettable” while announcing the diplomat’s decision. The Ministry said that it indeed regrettable that the Chinese side has chosen to politicise an event like the Olympics. Indian media channel Doordarshan said that it will not live telecast the opening or closing ceremony. Only one Indian athlete will be participating in the Winter Olympics in Beijing. 

INTERNATIONAL NEWS 

NASA’s Artemis 1 mission to Moon delayed till March

 NASA’s ‘Artemis 1’ mission, which aims to return humans to the Moon, has been delayed until March. The mission was scheduled for launch earlier in February. NASA informed that the ‘Space Launch System’ rocket and Orion spacecraft will not be rolled out for testing “no earlier than March 2022”. NASA has added additional time to complete closeout activities inside the Vehicle Assembly building before rolling the integrated rocket and spacecraft out for the first time.

NASA to retire ISS in January 2031

NASA is planning to retire the International Space Station (ISS) in January 2031 and has laid out a detailed transition plan. The space agency is planning to open the ISS for commercial activities in its final decade of operational life. NASA plans to “deorbit” the ISS and crash it into the Pacific Ocean in January 2031. Before its retirement, NASA is looking forward to a decade of results from research and technology development aboard the space station. 

 

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