GATE Sample Paper for Mechanical Engineering

Sample Paper aglase .co

ME: Mechanical Engineering

Q.No. 1 Denoting L as liquid and M as solid in a phase-diagram with the subscripts representing different
phases, a eutectoid reaction is described by

(A) M₁ → M₂ + M₃

(B) L₁ → M₂ + M₃

(C) L₁ + M₂ → M₃

(D) M₁ + M₂ → M₃

Q.No. 2 A helical spring has spring constant k. If the wire diameter, spring diameter and the number of coils
are all doubled then the spring constant of the new spring becomes

(A) k/2

(B) k

(C) 8k

(D) 16k

Q.No. 3 Which one of the following statements about a phase diagram INCORRECT?

(A) It indicates the temperature at which different phases start to melt

(B) Relative amount or different phases can be found under given equilibrium conditions

(C) It gives information on transformation rates

(D) Solid solubility limits are depicted by it

Q.No. 4 A particle of unit mass is moving on a plane. Its trajectory. in polar coordinates. is given by
r(t) = t , θ(t) = t , where t is time. The kinetic energy of the particle at time t = 2 is
2

(A) 4

(B) 12

(C) 16

(D) 24

Q.No. 5 An urn contains four balls, each ball having equal probability of being white or black. Three black
balls are added to the urn. The probability that five balls in the urn are black is

(A) 2/7

(B) 3/8

(C) 1/2

(D) 5/7

Q.No. 6 Match the following sand mold casting defects with their respective causes.

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

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

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

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

Q.No. 7 The directional derivative Of (x,y,z) at point in the direction of vector ^ı − 2^ȷ + 2k^ is

(A) 3^ı − 3^ȷ − k^

(B) − 7

3

(C) 7

3

(D) 7

Q.No. 8 Which one of the following statements is true?

(A) Every group of order 12 has a non-trivial proper normal subgroup

(B) Some group of order 12 does not have a non-trivial proper normal subgroup

(C) Every group of order 12 has a subgroup of order 6

(D) Every group of order 12 has an element of order 12

Q.No. 9 In Materials Requirement Planning, if the inventory holding cost is very high and the setup cost is
zero, which one of the following lot sizing approaches should be used?

(A) Economic Order Quantity

(B) Lot-for-Lot

(C) Base Stock Level

(D) Fixed Period Quantity, for 2 periods

Q.No. 10 An epicyclic gear train is shown in the figure below. The number of teeth on the gears A, B and D
are 20, 30 and 20, respectively. Gear C has 80 teeth on the inner surface and 100 teeth on the outer surface. If
the carrier arm AB is fixed and the sun gear A rotates at 300 rpm in the clockwise direction, then the rpm of

D in the clockwise direction is

(A) 240

(B) -240

(C) 375

(D) -375

Q.No. 11 As per common design practice, the three types of hydraulic turbines, in descending order of flow
rate, are

(A) Kaplan, Francis, Pelton

(B) Pelton, Francis, Kaplan

(C) Francis, Kaplan, Pelton

(D) Pelton, Kaplan, Francis

Q.No. 12 The wall of a constant diameter pipe of length 1 m is heated uniformly with flux 𝑞” by wrapping a
heater coil around it. The flow at the inlet to the pipe is hydrodynamically fully developed. The fluid is
incompressible and the flow is assumed to be laminar and steady all through the pipe. The bulk temperature
of the fluid is equal to 0 ºC at the inlet and 50 ºC at the exit. The wall temperatures are measured at three
locations, P, Q and R, as shown in the figure. The flow thermally develops after some distance from the inlet.
The following measurements are made:

Among the locations P, Q
and R, the flow is thermally developed at

(A) P, Q and R

(B) P and Q only

(C) Q and R only

(D) R only

Q.No. 13 The function f (z) of complex variable z = x + i y, where i = √−1, is given as
f (z ) = (x − 3x y ) + iv(x , y) . For this function to be analytic, v(x,y) should be
3 2

(A) (3x y − y ) + constant
2 3

(B) (3x y − y ) + constant
2 2 3

(C) (x 3
− 3x
2
y) + constant

(D) (3x y − y ) + constant
2 3

Q.No. 14 A sample of 15 data is as follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The mode of the
data is

(A) 4

(B) 13

(C) 17

(D) 20

Q.No. 15 During a non-flow thermodynamic process (1-2) executed by a perfect gas, the heat interaction is
equal to the work interaction (Q₁-₂ = W₁-₂) when the process is

(A) Isentropic

(B) Polytropic

(C) Isothermal

(D) Adiabatic

Q.No. 16 The damping ratio for a viscously damped spring mass system, governed by the relationship
2

+ k x = F (t) is given by
d x dx
m + c 2
dt dt

(A) √ c

mk

(B) c

2√ k m

(C) c

√k m

(D) √ c

2m k

Q.No. 17 An attempt is made to pull a roller of weight W over a curb (step) by applying a horizontal force F
as shown in the figure.

The coefficient Of static friction between the roller and the ground (including the
edge Of the step) is g. Identify the correct free body diagram (FBD) Of the roller when the roller is just about
to climb over the step.

(A)

(B)

(C)

(D)

Q.No. 18 A car having weight W is moving in the direction as shown in the figure. The center of gravity
(CG) of the car is located at height h from the ground, midway between the front and rear wheels. The
distance between the front and rear wheels is l. The acceleration of the car is a, and acceleration due to
gravity is g. The reactions on the front wheels R and rear wheels R are given by
f r

(A) R f = Rr =
W

2
−
W

g
(
h

l
)a

(B) R f =
W

2
+
W

g
(
h

l
)a ;R r =
W

2
−
W

g
(
h

l
)a

(C) R f =
W

2
−
W

g
(
h

l
)a ;R r =
W

2
+
W

g
(
h

l
)a

(D) R f = Rr =
W

2
+
W

g
(
h

l
)a

Q.No. 19 The stress state at a point in a material under plane stress condition is equi-biaxial tension with a
magnitude of 10 MPa. If one unit on the σ - τ plane is 1 MPa, the Mohr's circle representation of the state-of-
stress is given by

(A) a circle with a radius equal to principal stress and its center at the origin of the σ - τ plane

(B) a point on the σ axis at a distance of 10 units from the origin

(C) a circle with a radius of 10 units on the σ - τ plane

(D) a point on the τ axis at a distance of 10 units from the origin

Q.No. 20 During a non-flow thermodynamic process (1-2) executed by a perfect gas, the heat interaction is
equal to the work interaction (Q₁-₂ = W₁-₂) when the process is

(A) Isentropic

(B) Polytropic

(C) Isothermal

(D) Adiabatic

Q.No. 21 A thin uniform rigid bar of length L and mass M is hinged at point O, located at a distance of L/3
from one of its ends. The bar is further supported using springs. each of stiffness k. located at the two ends.
A particle of mass m = M/4 is fixed at one end of the bar. as shown in the figure. For small rotations of the

bar about O. the natural frequency of the system is

(A) √ 5k

M

(B) √ 5k

2M

(C) √ 3k

2M

(D) √ 3k

M

Q.No. 22 The lengths of a large stock of titanium rods follow a normal distribution with a mean (𝜇) of 440
mm and a standard deviation (𝜎) of 1 mm. What is the percentage of rods whose lengths lie between 438 mm
and 441 mm?

(A) 81.85%

(B) 68.4%

(C) 99.75%

(D) 86.64%

Q.No. 23 A gas is heated in a duct as it flows over a resistance heater. Consider a 101 kW electric heating
system. The gas enters the heating section of the duct at 100 kPa and 27 ºC with a volume flow rate of 15
m3/s. If heat is lost from the gas in the duct to the surroundings at a rate of 51 kW, the exit temperature of the
gas is (Assume constant pressure, ideal gas, negligible change in kinetic and potential energies and constant
specific heat; Cp = 1 kJ/kg·K; R = 0.5 kJ/kg·K.)

(A) 32 ºC

(B) 37 ºC

(C) 53 ºC

(D) 76 ºC

Q.No. 24 For a single server with Poisson arrival and exponential service time. the arrival rate is 12 per hour.
Which one of the following service rates will provide a steady state finite queue length?

(A) 6 per hour

(B) 10 per hour

(C) 12 per hour

(D) 24 per hour

Q.No. 25 The type of weld represented by the shaded region in the figure is

(A) groove

(B) spot

(C) fillet

(D) plug

Q.No. 26 A cantilever Of length l, and flexural rigidity El, stiffened by a spring Of stiffness k, is loaded by a
transverse force P, as shown.

The
transverse deflection under the load is
3

(A) Pl

3E I
[
3E I

3E I +2k l
3
]

3 3

(B) Pl

3E I
[
6E I −k l

6E I
]

3 3

(C) Pl

3E I
[
3E I −k l

3E I
]

3

(D) Pl

3E I
[
3E I

3E I +k l
3
]

Q.No. 27 The forecast for the monthly demand Of a product is given in the table below.

The forecast is made by using the
exponential smoothing method. The exponential smoothing coefficient used in forecasting the demand is

(A) 0.10

(B) 0.40

(C) 0.50

(D) 1.00

Q.No. 28 A frictionless gear train is shown in the figure. The leftmost 12-teeth gear is given a torque of 100
N-m. The output torque from the 60-teeth gear on the right in N-m is

(A) 5

(B) 20

(C) 500

(D) 2000

Q.No. 29 Consider an ideal vapor compression refrigeration cycle. If the throttling process is replaced by an
isentropic expansion process, keeping all the other processes unchanged, which one of the following
statements is true for the modified cycle?

(A) Coefficient of performance is higher than that of the original cycle.

(B) Coefficient of performance is lower than that of the original cycle.

(C) Coefficient of performance is the same as that of the original cycle.

(D) Refrigerating effect is lower than that of the original cycle.

Q.No. 30 A slender rod of length ,L diameter d ( L >>d ) and thermal conductivity 1k is joined with another
rod of identical dimensions, but of thermal conductivity k₂ , to form a composite cylindrical rod of length 2L
. The heat transfer in radial direction and contact resistance are negligible. The effective thermal conductivity
of the composite rod is

(A) k₁+ k₂

(B) √k k 1 2

(C) k1k2

k 1 +k 2

(D) 2k 1 k 2

k 1 +k 2

1 1
0
⎡ √2 √2 ⎤

Q.No. 31 Consider the matrix P = 0 1 0 which one of the following statements about P is
−1 1
⎣ 0 ⎦
√2 √2

INCORRECT?

(A) Determinant of P is equal to 1.

(B) P is orthogonal.

(C) Inverse of P is equal to its transpose.

(D) All eigenvalues of P are real numbers.

Q.No. 32 F(z) is a function of the complex variable z = x + iy given by 𝐹(𝑧) = 𝑖 𝑧 + 𝑘 𝑅𝑒(𝑧) + 𝑖 𝐼𝑚(𝑧). For
what value of k will F(z) satisfy the Cauchy-Riemann equations?

(A) 0

(B) 1

(C) -1

(D) y

Q.No. 33 In a rigid body in plane motion, the point R is accelerating with respect to point P at 10∠180°
m/s². If the instantaneous acceleration of point Q is zero, the acceleration (in m/s²) of point R is

(A) 8∠233°

(B) 10∠225°

(C) 10∠217°

(D) 8∠217°

Q.No. 34 In an arc welding process. welding speed is doubled. Assuming all other process parameters to be
constant, the cross sectional area of the weld bead will

(A) increase by 25 %

(B) increase by 50 %

(C) reduce by 25 %

(D) reduce by 50 %

Q.No. 35 Let f : C → C be an entire function with f(0) = 1, f(1) = 2 and f'(0) = 0. If there exists M > 0 such
that |f (z )| ≤ M f or all z ∈ C, then f (2) =
′′

(A) 2

(B) 5

(C) 2 + 5i

(D) 5 + 2i

Q.No. 36 For a linear programming problem, which one of the following statements is FALSE?

(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign

(B) Both primal and its dual can be infeasible

(C) If primal is unbounded, then its dual is infeasible

(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual can differ

Q.No. 37 The equation of motion for a spring-mass system excited by a harmonic force is
M ẍ + K x = F cos(ωt) Where M is the mass,K is the spring stiffness, F is the force amplitude and is the

angular frequency of excitation. Resonance occurs when is equal to

(A) √ M

K

(B) 1

2π
√
K

M

(C) 2π√ K

M

(D) √ M
K

Q.No. 38 For a position vector r→ = x ^i + y^j + z k the norm of the vector can be defined as
→| = √x + y + z . Given a function ϕ = ln |r→| its gradient ▽Ф is
|r
2 2 2

(A) r→
→
(B) r

→
|r |

(C) →→→r

r ⋅r

→
(D) r

→
|r |
3

Q.No. 39 A plane-strain compression (forging) of a block is shown in the figure. The strain in the z-direction
is zero. The yield strength (Sy) in uniaxial tension/compression of the material of the block is 300 MPa and it
follows the Tresca (maximum shear stress) criterion. Assume that the entire block has started yielding. At a
point where 𝜎z=40 MPa (compressive) and 𝜏xy the stress component 𝜎y is

(A) 340 MPa (compressive)

(B) 340 MPa (tensile)

(C) 260 MPa (compressive)

(D) 260 MPa (tensile)

Q.No. 40 For an ideal gas with constant properties undergoing a quasi-static process, which one of the
following represents the change of entropy (△s) from state 1 to 2?

(A) Δs = C ln(p
T2

T1
) − R ln(
P2

P1
)

(B) Δs = C V ln(
T2

T1
) − Cp ln(
V2

V1
)

(C) Δs = C ln(p
T2

T1
) − CV ln(
P2

P1
)

(D) Δs = C V ln(
T2

T1
) + R ln(
V1

V2
)

Q.No. 41 The preferred option for holding an odd-shaped workpiece in a centre lathe is

(A) live and dead centres

(B) three jaw chuck

(C) lathe dog

(D) four jaw chuck

Q.No. 42 In a counter-flow heat exchanger. water is heated at the rate of 1.5 kg/s from 40℃ to 80℃ by an
oil entering at 120℃ and leaving at 60℃. The specific heats of water and oil are 4.2 kJ/kg•K and2 kJ/kg•K.
respectively. The overall heat transfer coefficient is 400 W/m²•K. The required heat transfer surface area (in
m²) is

(A) 0.104

(B) 0.022

(C) 10.4

(D) 21.84

Q.No. 43 As per common design practice, the three types of hydraulic turbines, in descending order of flow
rate, are

(A) Kaplan, Francis, Pelton

(B) Pelton, Francis, Kaplan

(C) Francis, Kaplan, Pelton

(D) Pelton, Kaplan, Francis

Q.No. 44 A parabola x =y² with 0 ≤ 1 ≤x is shown in the figure. The volume of the solid of rotation obtained

by rotating the shaded area by 360° around the x-axis is

(A) π/4

(B) π/2

(C) π

(D) 2π

Q.No. 45 Froude number is the ratio of

(A) buoyancy forces to viscous forces

(B) inertia forces to viscous forces

(C) buoyancy forces to inertia forces

(D) inertia forces to gravity forces

Q.No. 46 If a mass of moist air contained in a closed metallic vessel is heated, then it’s

(A) relative humidity decreases

(B) specific humidity increases

(C) relative humidity increases

(D) specific humidity decreases

Q.No. 47 If the characteristic curves of the partial differential equation x u xx + 2x
2
ux y = ux − 1 are μ(x, y)
= c1 and ν(x, y) = c2, where c1 and c2 are constants, then

(A) μ(x , y) = x 2
− y, ν(x , y) = y

(B) μ(x , y) = x 2
+ y, ν(x , y) = y

(C) μ(x , y) = x 2
+ y, ν(x , y) = x
2

(D) μ(x , y) = x 2
− y, ν(x , y) = x
2

Q.No. 48 Let X₁ and X₂ be two independent exponentially distributed random variables with means 0.5 and
0.25, respectively. Then Y = min (X₁, X₂) is

(A) exponentially distributed with mean 1/6

(B) exponentially distributed with mean 2

(C) normally distributed with mean 3/4

(D) normally distributed with mean 1/6

1.4
Q.No. 49 The evaluation of the definite integral ∫−1
x |x |dx by using Simpson's 1/3rd (one-third) rule with
step size h = 0.6 yields

(A) 0.914

(B) 1.248

(C) 0.581

(D) 0.592

Q.No. 50 The rotor of a turbojet engine of an aircraft has a mass 180 kg and polar moment of inertia 10 kg-
m² about the rotor axis. The rotor rotates at a constant speed of 1100 rad/s in the clockwise direction when
viewed from the front of the aircraft. The aircraft while flying at a speed of 800 km per hour takes a turn with
a radius of 1.5 km to the left. The gyroscopic moment exerted by the rotor on the aircraft structure and the
direction of motion of the nosewhen the aircraft turns, are

(A) 1629.6 N·m and the nose goes up

(B) 1629.6 N·m and the nose goes down

(C) 162.9 N·m and the nose goes up

(D) 162.9 N·m and the nose goes down

Q.No. 51 Match the following non-dimensional numbers with the corresponding definitions:

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

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

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

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

Q.No. 52 Which one of the following welding methods provides the highest heat flux (W/mm2)?

(A) Oxy-acetylene gas welding

(B) Tungsten inert gas welding

(C) Plasma arc welding

(D) Laser beam welding

Q.No. 53 The Fourier cosine series for an even function f(x) is given by f (x ) = a + ∑0
∞

n=1
a n cos(nx )

The value of the coefficient a₂ for the function f(x) = cos²(x) in [0,π] is

(A) -0.5

(B) 0.0

(C) 0.5

(D) 0

Q.No. 54 For an Oldham coupling used between two shafts, which among the following statements are
correct? I. Torsional load is transferred along shaft axis. II. A velocity ratio of 1:2 between shafts is obtained
without using gears. III. Bending load is transferred transverse to shaft axis. IV. Rotation is transferred along
shaft axis.

(A) I and III

(B) I and IV

(C) II and III

(D) II and IV

Q.No. 55 The position vector OP of point P (20, 10) is rotated anti-clockwise in X-Y plane by an angle θ=
30° such that point P occupies position Q, as shown in the figure. The coordinates (x, y) of Q are

(A) (13.40, 22.32)

(B) (22.32, 8.26)

(C) (12.32, 18.66)

(D) (18.66, 12.32)