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1

Mechanical Engineering

Q. No. 1 to 25 Carry One Mark Each

1. A motor driving a solid circular steel shaft transmits 40kW of power at 500 rpm. If the

diameter of the shaft is 40 mm, the maximum shear stress in the shaft is ________MPa.

Key: 60 to 61

Exp: P=40kW N=500rpm

D= 40mm T=3

16T

d

3

3

P 60,000 40 60,000T 763.94N m

2 N 2 500

16 763.94 1060.79MPa

40

2. Consider the following partial differential equation for u(x,y) with the constant c > 1 :

u u

c 0y x

Solution of this equation is

(A) u x, y f x cy (B) u x, y f x cy

(C) u x, y f cx y (D) u x, y f cx y

Key: (B)

Exp: Given u u

c 0y x

uf ' x cy

x

uc f ' x cy

x

u u0

x y

u x, y f x cy

3. The following figure shows the velocity- time plot for a particle traveling along a straight line.

The distance covered by the particle from t = 0 to t= 5 s is __________m.

1 2 3 4 5

Time(s)

Velocity (m / s)

6

1

2

3

4




2

Key: 10 to 10

Exp: Area under V- T wave

1 2 3 4 5S a a a a a

1 1 4 4 21 1 1 1 1 2

2 2 2

0.5 1 2.5 6 10m

4. The damping ratio for a viscously damped spring mass system, governed by the relationship

2

2

d x dxm C kx F t ,

dt dt is given by

(A) c

mk (B)

c

2 km (C)

c

km (D)

c

2mk

Key: (B)

Exp: 2

x2

c

dx dx c cm c k F t ;

dt dt c 2 km

5. The differential equation 2

2

d y16y 0 for y x

dx with the two boundary conditions

x 0 x2

dy dy1and 1has

dx dx

(A) no solution (B) exactly two solutions

(C) exactly one solution (D) infinitely many solutions

Key: (A)

Exp:

2

2

x 0 x2

2

c 1 2 p

1 2

d y dy dy16y 0 1 1

dx dx dx

m 16 0

m 0 4i

y c cos4x c sin 4x and y 0

y c cos4x c sin 4x

1 2

2 2

y' x 4c sin 4x 4c cos4x

1y' 0 1 0 4c 1 c

4

2 2

2

1y ' 1 0 4c 1 c

2 4

1 1c and both not possible

4 4

Hence there is no solution

6. Metric thread of 0.8 mm pitch is to be cut on a lathe. Pitch of the lead screw is 1.5 mm. If the

spindle rotates at 1500 rpm, the speed of rotation of the lead screw (rpm) will be _________

Key: 800 to 800




3

Exp: Speed of rotation of lead screw 1500 0.8

800rpm1.5

7. The molar specific heat at constant volume of an ideal gas is equal to 2.5 times the universal

gas constant (8.314 J/mol.K). When the temperature increases by 100K, the change in molar

specific enthalpy is _______________ J/mol.

Key: 2908 to 2911

Exp: v v vC 2.5R where R 8.314J mol.K

T 100K

H ?

p

p v v

p v

H C T

C C R

[C 3.5R ]

So, H 3.5 8.314 100

[ H 2909.9J mol]

8. A particle of unit mass is moving on a plane. Its trajectory, in polar coordinates, is given by

r(t) = t2, t t, where t is time. The kinetic energy of the particle at time t = 2 is

(A) 4 (B) 12 (C) 16 (D) 24

Key: (C)

Exp: 2r = t ;θ= t

2

2

2

2

1K.E = mv ? at t 2 sec

2

m 1kg

drˆ ˆˆ ˆV r (t) r t 1(t) 2trdt

ˆ ˆV t (t) 2t(r)

at t 2s

ˆ ˆV 4(t) 4(r)

V 16 16 32

1 1K.E. mv 1 32 16

2 2

9. The Poisson‟s ratio for a perfectly incompressible linear elastic material is

(A) 1 (B) 0.5 (C) 0 (D) infinity

Key: (B)

10. A heat pump absorbs 10 kW of heat from outside environment at 250 K while absorbing 15

kW of work. It delivers the heat to a room that must be kept warm at 300K. The Coefficient of

Performance (COP) of the heat pump is ___________.

Key: 1.66 to 1.70




4

Exp:

11. Which one of the following is NOT a rotating machine?

(A) Centrifugal pump (B) Gear pump

(C) Jet pump (D) Vane pump

Key: (C)

Exp: In the given options all the pumps have rotating machine elements except Jet pump.

12. Consider the schematic of a riveted lap joint subjected to tensile load F, as shown below. Let d

be the diameter of the rivets, and Sf be the maximum permissible tensile stress in the plates.

What should be the minimum value for the thickness of the plates to guard against tensile

failure of the plates? Assume the plates to be identical.

(A) f

F

S W 2d (B)

f

F

S W (C)

f

F

S W d (D)

f

2F

S W

Key: (A)

Exp:

f f

f

F F FS S ; t

A t W 2d S w 2d

13. Water (density =1000 kg/m3) at ambient temperature flows through a horizontal pipe of

uniform cross section at the rate of 1 kg/s. If the pressure drop across the pipe is 100 kPa, the

minimum power required to pump the water across the pipe, in watts, is _______

Key: 100 to 100

Exp: given,

3

w

f

1000kg m

m 1kg s

P 10kPa

So,minimumpower require mgh

Pmg

g

1 100 1000100Watts

1000

300K

H.P

250K

H25kW Q

10kW

15kW

H

i/p

Q 25COP 1.66

W 15




5

14. For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter,

the average velocity in the fully developed region is constant. Which one of the following

statements about the average velocity in the developing region is TRUE?

(A) It increases until the flow is fully developed.

(B) It is constant and is equal to the average velocity in the fully developed region.

(C) It decreases until the flow is fully developed.

(D) It is constant but always lower than the average velocity in the fully developed region.

Key: (B)

Exp: The average velocity in pipe flow always be same either for developing flow or fully

developed flow.

15. Cylindrical pins of diameter 0.02015 mm are being produced on a machine. Statistical quality

control tests show a mean of 14.995 mm and standard deviation of 0.004mm. The process

capability index Cp is

(A) 0.833 (B) 1.667 (C) 3.333 (D) 3.750

Key: (B)

Exp: p

USL-LSL 15.02 14.98C 1.666

6σ 6 0.004

16. The product of Eigen values of the matrix P is

2 0 1

P 4 3 3

0 2 1

(A) -6 (B) 2 (C) 6 (D) -2

Key: (B)

Exp:

2 0 1

P 4 3 3

0 2 1

We know that, product of eigen values of P = determinant of P

2 3 6 0 1 8 6 8 2

17. Match the processes with their characteristics.

Process Characteristics

P : Electrical Discharge machining 1. No residual stress

Q : Ultrasonic machining 2. Machining of electrically conductive

materials

R : Chemical machining 3. Machining of glass

S : Ion Beam Machining 4. Nano-machining

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

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

Key: (A)




6

18. The Value of 3

x 0

x sin xlim is

x

(A) 0 (B) 3 (C) 1 (D) -1

Key: (D)

Exp: 3

2

x 0 x 0 x 0

x sin x sin xt t x t 0 1 1

x x

19. 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%

Key: (D)

Exp: m

2 1

1 2

12

V.I H A.V

1A

V

A V V

A V 2V

AA

2

By doubling welding speed, Area reduces by 50%

20. A six-face fair dice is rolled a large number of times. The mean value of the outcomes is

______.

Key: 3.5 to 3.5

Exp: The Probabilities corresponding to the outcomes are given below:

mean E x x.P x

1 1 1 1 1 11 2 3 4 5 66 6 6 6 6 6

1 211 2 3 4 5 6 3.5

6 6

21. Consider the two dimensional velocity field given by 1 1 2 2V 5 a x b y i 4 a x b y j,

1 1 2 2wher a , b , a and b are constants. Which one of the following conditions needs to be

satisfied for the flow to be incompressible?

(A) 1 1a b 0 (B) 1 2a b 0 (C) 2 2a b 0 (D) 2 1a b 0

Key: (B)

Exp: 1 1 2 2

1 2

Given V 5 a x b y i 4 a x b y j

u VFor Incompressible V 0; i.e., 0

x y

a b 0

Face 1 2 3 4 5 6

Probability 1/6 1/6 1/6 1/6 1/6 1/6




7

22. Consider a beam with circular cross-section of diameter d. The ratio of the second moment of

area about the neutral axis to the section modulus of the area is.

(A) d

2 (B)

d

2

(C) d (D) d

Key: (A)

Exp:

4

3

dI I d64Z yy Z 2

d32

23. Saturated steam at 100°C condenses on the outside of a tube. Cold fluid enters the tube at 20°

C and exists at 50°C. The value of the Log Mean Temperature Difference (LMTD) is

________°C.

Key: 63.5 to 64

Exp: LMTD= o i

o i

80 50

n n 80 50

LMTD 63.82 C

24. In a metal forming operation when the material has just started yielding, the principal stresses

are 1 2 3180 MPa, 100 MPa, 0. Following Von Mises criterion, the yield stress is

________ MPa.

Key: 245 to 246

Exp:

As per Von-Mises criteria

2 2 2 2

1 2 1 3 5 1 m

2 2 2 2

m

m

2

180 100 100 0 0 180 2

245.76MPa

25. In the engineering stress-strain curve for mild steel, the Ultimate Tensile Strength (UTS) refers

to

(A) Yield stress (B) Proportional limit

(C) Maximum stress (D) Fracture stress.

Key: (C)

Q. No. 26 to 55 Carry Two Marks Each

26. A parametric curve defined by u u

x cos ,y sin2 2

in the range 0 u 1 is rotated

about the X – axis by 360 degrees. Area of the surface generated is.

(A) 2

(B) (C) 2 (D) 4

Key: (C)

100 C

50 C

i 50 C

o

100 C

80 C

20 C

T

S




8

Exp: u u

Given x cos , y sin 0 12 2

dx usin

d 2 2

dy ucos

dx 2 2

We know that surface area when the curve revolved about X- axis of a parametric curve is

2 22 21 1

0 0

1

1 122 2

0 0

0

dx dy u u u2 y du 2 sin sin cos du

du du 2 2 2 2 2

u1cos

u u 222 sin du 2 sin dx cos cos02 4 2 2 2

2

2 cos0 1 2

27. Assume that the surface roughness profile is triangular as shown schematically in the figure. If

the peak to valley height is 20 m, The central line average surface roughness Ra (in m ) is

(A) 5 (B) 6.67 (B) 10 (D) 20

Key: (A)

Exp: maxa

R 20R 5 m

4 4

28. 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

m4

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 systems is.

(A) 5k

M (B)

5k

2M (C)

3k

2M (D)

3k

M

Key: (B)

Exp: Max moment of inertia of Rod.




9

2

0 c

22

0

2 2 2

0

I I mr

ML 2L LI M

12 3 2

ML ML MLI

12 36 9

Mass moment of inertia of particular mass

2 2

particular

2 2 2

Total

0

2

2 2

n 2

M 2L MLI

4 3 9

ML ML 2(ML )I

9 9 9

M 0

L L 2L 2LK K I 0

3 3 3 3

5L.k

2ML 5L 5k9K 02ML9 9 2M

9

29. A point mass of 100 kg is dropped onto a massless elastic bar (cross-sectional area = 100 mm2,

length = 1m, Young‟s moduls = 100 GPa) from a height H of 10mm as shown (Figure is not to

scale). If g = 10m/s2, the maximum compression of the elastic bar is _______ mm.

Key: 1.50 to 1.52

Exp: Given that

2

2

m 100kg, g 10m sec , E 100GPa

H 10mm, L 1m 100mm,

A 100mm

From the given figure, we can say that this is case of Impact loading,

We know that, stress due to Impact load is

2

I.L

2 52

I.L

P P 2PhE

A A AL

P mg 100 10 1000N

1000 1000 2 1000 10 1 10151.7745 N mm

100 100 100 1000

I.L

5

L 151.7745 1000Compression 1.5177 mm

E 1 10

LL

3

0K K

m

m 100kg

H 10mm

L 1m

g




10

30. One kg of an ideal gas (gas constant, R = 400 J/kg.K; specific heat at constant volume,

c 1000J/kg.K) at 1 bar, and 300 K is contained in a sealed rigid cylinder. During an

adiabatic process, 100kJ of work is done on the system by a stirrer. The increase in entropy of

the system is _________ J/K.

Key: 286 to 288

Exp: Given that Vm 1kg, R 400 kJ kgK, C 1000 J kgK

1 1P 1bar, T 300K

Since the gas is contained in a sealed rigid cylinder, and given that adiabatic process is done to

the system, means no heat is transferred from/to the system, Q = 0

And we know from first law of thermo dynamics,

3

V 2 1 V 2 1

3

V 2 1

3

2

3

2

Q du W

0 mC T T 100 10 du mC T T

mC T T 100 10

1 1000 T 300 100 10

100 10T 300 400K

1000

We know that the first law of thermodynamics can be written as

V

V

V V

Tds mC dT pdV

dT P mRTds mC dv PV mRT P

T T V

dT mRT dT dVds mC dV mC mR

T rT T V

Entropy increase 2 22 1 V n

1 1

T VS S mC n mR

T V

Since the above process is a constant volume process that is 2 1V V

2 1

400S S 1 1000 n 287.6821J K

300

31. For an inline slider-crank mechanism, the lengths of the crank and connecting rod are 3m and

4m, respectively. At the instant when the connecting rod is perpendicular to the crank, if the

velocity of the slider is 1m/s, the magnitude of angular velocity (upto 3 decimal points

accuracy) of the crank is _________ radian/s.

Key: 0.26 to 0.27

Exp: Given that velocity of slider

BV 1m / sec

Length of crank (OA) = 3m

Length of connecting rod (AB) = 4m

From the configuration diagram

oA

sin 53.1305

Adiabatic wall, Q 0

still Q workgiven 100 kJ

A

O C5mm

4mm3mm

B




11

The velocity diagram for the above configuration diagram is

From velocity diagram o Oasin53.13

Ob

o

A OA

Oa 1 sin 53.130 0.8

V Oa OA 0.8

Angular velocity of crank OA

0.80.267 rad sec

3

32. In an epicyclic gear train, shown in the figure, the outer

ring gear is fixed, while the sun gear rotates

counterclockwise at 100rpm. Let the number of teeth on

the sun, planet and outer gears to be 50, 25, and 100,

respectively. The ratio of magnitudes of angular velocity

of the planet gear to the angular velocity of the carrier

arm is _________.

Key: 3 to 3

Exp: TS = 50

TP=25

TR=100

S.No Condition of motion Arm

Gear A Gear S Gear P Gear R

1 Arm is fixed gear S

with +1 revolution 0 +1

s

P

T

T s

R

T

T

2 Arm is fixed Gear S

with +x revolution 0 +x -x s

P

T

T s

R

Tx

T

3 Arm with + y

revolution y y y y

4 Total y x+y s

P

Ty x

T s

R

Ty x

T

S

R

N x y 100 ...(1)

50N y x 0

100

P

P

arm

y 0.5x ... 2

x 0.5x 100

100x 66.66rpm

1.5

y 33.34rpm

50N 33.34 66.66 99.99rpm

25

N 99.993 approx

N 33.32

O,c 1m sec

b

a

o53.130




12

33. Moist air is treated as an ideal gas mixture of water vapor and dry air (molecular weight of air

= 28.84 and molecular weight of water = 18). At a location, the total pressure is 100 kPa, the

temperature is 30°C and the relative humidity is 55%. Given that the saturation pressure of

water at 30°C is 4246 Pa, the mass of water vapor per kg of dry air is _____________ grams.

Key: 14.7 to 15.1

Exp: aM 28.84

w

s

v vv

s

v

v

M 18

P 100kPa, T 30 C, RH 55%,

P 4246Pa

P P0.55 P 2335.3Pa

P 4246

P 2335.30.622 0.622

P P 100000 2335.3

0.01487kg of vapour kg of D.A

14.87gmof vapour kg of D.A

34. Following data refers to the jobs (P, Q, R, S) which have arrived at a machine for scheduling.

The shortest possible average flow time is ___________ days.

Job Processing Time (days)

P 15

Q 9

R 22

S 12

Key: 31 (not matching with IIT key)

Exp: For shortest avg. flow time SPT rule is used

Job Sequence Processing Time In Out Flow Time

Q 9 0 9 9

S 12 9 21 21

P 15 21 36 36

R 22 36 58 58

Min Avg. Flow time 9 21 36 58

31days4

35. Two models, P and Q, of a product earn profits of Rs. 100 and Rs. 80 per piece, respectively.

Production times for P and Q are 5 hours and 3 hours, respectively, while the total production

time available is 150 hours. For a total batch size of 40, to maximize profit, the number of

units of P to be produced is ____________.

Key: 15 to 15

Exp: Let 1

2

x No. of units of P

x No. of units of Q




13

1 2

1 2

1 2

max. z 100 x 80x

5x 3x 150

x x 40

0,40

15,225

Z Rs.3200

Z Rs.3500 max.profit

So, for maximum profit, No. of units of P produced is 15 units.

36. Circular arc on a part profile is being machined on a vertical CNC milling machine. CNC part

program using metric units with absolute dimensions is listed below:

--------------------------------

N60 G01 X 30 Y 55 Z – 5 F 50

N70 G02 X 50 Y 35 R 20

N80 G01 Z 5

--------------------------------

The coordinates of the centre of the circular arc are :

(A) (30, 55) (B) (50, 55) (C) (50, 35) (D) (30, 35)

Key: (D)

Exp:

Centre of circular arc is (30, 35)

37. Two black surfaces, AB and BC, of lengths 5m and 6m,

respectively, are oriented as shown. Both surfaces extend

infinitely into the third dimension. Given that view factor

F12=0.5, T1=800K, T2=600K, Tsurrounding=300K and Stefan

Boltzmann constant, 8 2 45.67 10 W / m K , the heat

transfer rate from Surface 2 to the surrounding environment is

____________ kW.

Key: 13.7 to 13.9

Exp: Given that two black surfaces „AB‟ and „BC‟

Length of AB = 5m, BC = 6 m

And temperature of

Surface „1‟ o

BC 1T T 800 K

Temperature of surface „2‟ o

AB 2T T 600 K

Temperature of surroundings o

3T 300 K

2x

0,50

0,40

0,0

15,25

30,0 40,0

1x

y

x

20

30,56

30,35 50,35

20

B

Surface '1'Surroundings'3'

C

A




14

8 2 4

12

11 22

5.67 10 W / m K

F 0.5

W.K.T. F F 0, sin ce theyareflat surfaces

21 22 23

1 12 2 21

21

21

23 23

F F F 1

A F A F

6 0.5 5 F Assumeunit width forsurfaces

3F 0.6

5

0.6 0 F 1 F 1 0.6 0.4

Using resistance concept we can draw as follows

Since surfaces are black and area of surrounding is large we can write

b1 1 b2 2 b3 3

4 4

2 3b2 b323

2 232 23

8 4 4

E J ,E J ,E J

T TE EQ

11

A FA F

5.67 10 600 300 5 0.4

13.778kW / metre

38. Consider the matrix

1 0 1

2 2

P 0 1 0 .

1 0 1

2 2

Which one of the following statements about P is INCORRECT?

(A) Determinant of P is equal to 1.

(B) P is orthogonal.

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

(D) All Eigen values of P are real numbers

Key: ( D)

Exp:

1 0 1

2 2

P 0 1 0

1 0 1

2 2

T

1 1 1 1 1 1P 0 0 1

2 22 2 2 2

1 0 1 1 0 1

1 0 02 2 2 2

P.P 0 1 0 0 1 0 0 1 0

1 0 1 1 0 1 0 0 1

2 2 2 2

2 3

2 21

1

A F1 13

1

A F

b1 1E J1

b3 3E J2 23

1

A F

b2 2E J




15

P is an orthogonal matrix

(A) Is correct Inverse of P is its transpose only

(B) and (C) both are correct

(D) is incorrect

39. The Pressure ratio across a gas turbine (for air, specific heat at constant pressure,

pc 1040J / kg.K and ratio of specific heats, 1.4) is 10. If the inlet temperature to the

turbine is 1200K and the isentropic efficiency is 0.9, the gas temperature at turbine exit is

______ K.

Key: 675 to 684

Exp: p

2 1 3

is

C 1040J kg.K , r 1.4

P P 10, T 1200K

0.9

r 1

r0.4/1.43 3

4 4 4

4

1

3 4iso

3 4

1

4 3 iso 3 4

4

Isentropic Expansion,

P T 120010

P T T

T 621.5K

T T

T T

So, T T T T 1200 09 1200 621.5

T 679.38K

40. An initially stress-free massless elastic beam of length L and circular cross-section with

diameter d (d << L) is held fixed between two walls as shown. The beam material has Young‟s

modulus E and coefficient of thermal expansion .

If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to

buckle is proportional to

(A) d (B) d2 (C) d

3 (D) d

4

Key: (B)

Exp: 2

e

4

2

2

EIP EA T

1 IT

A

d1 64T

d4

T d

Ld

1

3

2

4

14

1P

2P

S

T




16

41. For the vector V 2yzi 3xz j 4xyk, the value of . V

is ____________

Key: 0 to 0

Exp:

V 2yzi 3xzj 4xyk

we know that . V 0 for any vector V

42. A 10 mm deep cylindrical cup with diameter of 15mm is drawn from a circular blank.

Neglecting the variation in the sheet thickness, the diameter (upto 2 decimal points accuracy)

of the blank is _________ mm.

Key: 28.71 to 28.73

Exp: 2 2D d 4dh 15 4 10 15 28.72 mm

43. A machine element has an ultimate strength u of 600 N/mm2, and endurance limit en of

250 N/mm2. The fatigue curve for the element on log-log plot is shown below. If the element

is to be designed for a finite of 10000 cycles, the maximum amplitude of a completely

reversed operating stress is _________ N/mm2.

Key: 370 to 390

Exp: u

en

600MPa

250MPa

N 10000 cycle

u u

max

log 0.8 log 250 log 0.8 log

3 6 3 4

log 480 log 250 log 480 log

3 1

3log 480 log 480 log 250 2log 480 log 250log

3 3

386.19MPa

44. A sprue in a sand mould has a top diameter of 20mm and height of 200mm. The velocity of

the molten metal at the entry of the sprue is 0.5m/s. Assume acceleration due to gravity as 9.8

m/s2 and neglect all losses. If the mould is well ventilated, the velocity (upto 3 decimal points

accuracy) of the molten metal at the bottom of the sprue is ________ m/s.

Key: 2.04 to 2.07

A

B

e

310 410 610

u(0.8 )




17

Exp: Apply Bernoulli‟s between (1) and (2)

22

2

2

V0.50.2

2g 2g

V 2.042 m s

45. Air contains 79% N2 and 21% O2 on a molar basis. Methane (CH4) is burned with 50% excess

air than required stoichiometrically. Assuming complete combustion of methane, the molar

percentage of N2 in the products is ________________

Key: 73 to 74

Exp: 4 2 2 2 2 2

79 79Stoichiometric reaction CH 2. O N 2H O CO 2 N

21 21

50% excess air

4 2 2 2 2 2 2

2

79 79CH 3. O N 2H O CO 3 N O

21 21

3 79

21%N 100 73.83%79

2 1 1 321

46. P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical

integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0

and x =1 for the curve. The difference between the two results will be.

(A) 0 (B) 0.25 (C) 0.5 (D) 1

Key: (A)

Exp: Let x 0 0.5 1

y 3 4 5

Trapezoidial rule

1

0

1

0

0.5 0.5f x dx 3 5 2 4 16 4

2 2

Simpsons rule

0.5 0.5f x dx 3 5 0 4 4 24 4

3 3

Difference 0

47. Heat is generated uniformly in a long solid cylindrical rod (diameter = 10mm) at the rate of

4×107 W/m

3. The thermal conductivity of the rod material is 25W/m.K. Under steady state

conditions, the temperature difference between the centre and the surface of the rod is

_________ °C.

Key: 10 to 10

2 2

1 1 2 21 2

P V P Vz z

2g 2g

200 mm

2

1




18

Exp: Given that, heat is generated uniformly i.e., 7 3g 4 10 W/ m

Diameter at the rod (d) = 10 mm

Thermal conductivity of the rod (K) = 25 W/mK

W.K.T for steady state, with internal heat generation, the conduction equation will be,

1 d dT gr. 0

r dr dr k

d dT grr 0

dr dr K

After Integration once

2

11

cdT gr dT grr. C (1)

dr 2K dr 2K r

After second Integration

2

1 2

grT r C n r C (2)

4K

Substituting boundary condition of dT

0at r 0in eq(1)dr

That gives 1C 0

2

2

grT r C ,

4k

substitute at r =0, T(r) = To

o 2 2 o

2

o

T o C C T

grT r T

4K

Substitute o S

10r r 5mm 0.005m andT r T

2

The above equation will become

2

oS o

grT T

4K

7 2

o

o s

4 10 0.005T T 10 C

4 25

d 10mm

3T

dT0

dr

Temperatureprofile

oT

7 3g 4 10 W / m

3K 25W / m K

ST




19

48. Two disks A and B with identical mass (m) and radius (R) are initially at rest. They roll down

from the top of identical inclined planes without slipping. Disk A has all of its mass

concentrated at the rim, while Disk B has its mass uniformly distributed. At the bottom of the

plane, the ratio of velocity of the center of disk A to the velocity of the center of disk B is.

(A) 3

4 (B)

3

2 (C) 1 (D) 2

Key: (A)

Exp: 2

AI MR

2

B

MRI

2

T R T RA B

2 2 2 2

A A B B

22 2 2 2 2

A A B B

P.E K.E K.E K.E K.E

1 1 1 1MV I MV I

2 2 2 2

1 1 1 1 MRMV MR MV

2 2 2 2 2

22 2 2 BA A B

2 2 AA B

B

VV V V

2

V3 32V V

2 V 4

49. A block of length 200mm is machined by a slab milling cutter 34mm in diameter. The depth of

cut and table feed are set at 2mm and 18mm/minute, respectively. Considering the approach

and the over travel of the cutter to be same, the minimum estimated machining time per pass is

_____________ minutes.

Key: 12 to 12

Exp: Milling Time

22 2 (34 2) 2 2002 Dd d L12

f 18

Where, D Dia of cutter mm , d depth of cut mm

L length f feed mm min

50. A horizontal bar, fixed at one end (x = 0), has a length of 1 m, and cross-sectional area of

100mm2. Its elastic modulus varies along its length as given by E(x) = 100e

-x GPa, Where x is

the length coordinate (in m) along the axis of the bar. An axial tensile load of 10 kN is applied

at the free end (x=1). The axial displacement of the free end is _______ mm.

Key: 1.70 to 1.72

Exp: Given that

3 2

x 9x 5 x

6

P 10kN 10 10 N,A 100mm

100 e 10E x 100e Gpa 10 e

10

Change in the length of small strip

x

P dx

AE

m

H

A,B

x 0 x 1

L 1mm

xdx

P 10kN

Stripof lengthdx




20

Total change in the length of the bar L

x0

Pdx

AE

L L L 1

x x 1

x 5 x 5 5 5

0 0 0 0

3

5

P dx P dx P P Pe dx e dx e 1

A 100e A 10 e A 10 A 10 A 10

10 102.7183 1 1000 1.7183mm

100 10

51. Consider steady flow of an incompressible fluid through two long and straight pipes of

diameters d1 and d2 arranged in series. Both pipes are of equal length and the flow is turbulent

in both pipes. The friction factor for turbulent flow though pipes is of the form, nf K(Re)

where K and n are known positive constants and Re is the Reynolds number. Neglecting minor

losses, the ratio of the frictional pressure drop in pipe 1 to that in pipe 2, 1

2

P

P

, is given by

(A)

5 n

2

1

d

d

(B)

5

2

1

d

d

(C)

3 n

2

1

d

d

(D)

5 n

2

1

d

d

Key: (A)

Exp:

5 5n2 511 1 1 2 1 2 2

n5 2

2 1 2 2 2 1 12

ReP f L Q 12d g f d d 64......(1) f

P 12d g f L Q f d d ReRe

2

2

2

5n

1 1 2

n

2 2 1

5 n

1 2

2 1

V d 4QRe & Q AV d V V

4 d

4Q d 4QSo, Re

d d

P d d1Re From(1)

d P d d

P d

P d

52. The velocity profile inside the boundary layer for flow over a flat plate is given as

u ysin ,

U 2

where U is the free stream velocity and is the local boundary layer

thickness. If * is the local displacement thickness, the value of *

is

(A) 2

(B)

21

(C) 1+

2

(D) 0

Key: (B)

Exp: U y

sinU 2

*

0

displacement thickness

y1 U U dy 1 sin dy

2

2d1d

1L 2L

1 2L L




21

* 0

*

*

*

cos y 2

2

2cos 2 cos0

2

1 2

53. For a steady flow, the velocity field is 2V x 3y i 2xy j. The magnitude of the

acceleration of a particle at (1, -1) is

(A) 2 (B) 1 (C) 2 5 (D) 0

Key: (C)

Exp: 2V x 3y i 2xy j

54. Two cutting tools with tool life equations given below are being compared:

Tool 1: VT0.1

=150

Tool 2: VT0.3

=300

Where V is cutting speed in m/minute and T is tool life in minutes. The breakeven cutting

speed beyond which Tool 2 will have a higher tool life is ________ m/minute.

Key: 105 to 107

Exp: At Breakeven point

1 2

1 0.1 1 0.3

T T

150 300

V V

V 106.121 m min

2

2 3 3

x

2 2 2 2 2 2

y

x

2 2

y

2 2

x y

U x 3y

V 2xy

U Ua U V x 3y 2x 2xy 3 2x 6xy 6xy 2x

x y

V Va U V x 3y 2y 2xy 2x 2x y 6y 4x y 2x y 6y

x y

but x 1, y 1

a 2

a 2 1 1 6 1 6 2 4

a a a 4 16 20

a 2 5

2

2 3 3

x

2 2 2 2 2 2

y

x

2 2

y

2 2

x y

U x 3y

V 2xy

U Ua U V x 3y 2x 2xy 3 2x 6xy 6xy 2x

x y

V Va U V x 3y 2y 2xy 2x 2x y 6y 4x y 2x y 6y

x y

but x 1, y 1

a 2

a 2 1 1 6 1 6 2 4

a a a 4 16 20

a 2 5




22

55. A rectangular region in a solid is in a state of plane strain. The (x,y) coordinates of the corners

of the under deformed rectangle are given by P(0,0), Q (4,0), S (0,3). The rectangle is

subjected to uniform strains, xx yy xy0.001, 0.002, 0.003. The deformed length of the

elongated diagonal, up to three decimal places, is _________ units.

Key: 5.013 to 5.015

Exp: Given that

xx yy

xy

0.001, 0.002

r 0.003

Length of the diagonal 2 2PR 4 3 5m

To find the diagonal (PR) strain, the direction of the plane angle from the +ve x-axis will from

„R‟ towards „P‟

?

Where 1 o180 tan 3 / 4 216.87

o

xx yy xx yy xy

216.87

o

4 3

3

rcos2 sin 2

2 2 2

0.001 0.002 0.001 0.002 0.003cos 2 216.87 sin 2 216.87

2 2 2

0.0015 1.4 10 1.44 10

2.8 10

Elongation of the diagonal = 3

216.87 5 2.8 10 •5 0.014

Defined length of diagonal 5 0.014 5.014

S 0,3

xx

P 0,0

yy

Q 4,0

1 3tan

4

x

xx

R 4,3

yy

y

xyr5mm




23

General Aptitude

Q. No. 1 - 5 Carry One Mark Each

1. A right – angled cone (with base radius 5cm and height 12cm), as shown in the figure below,

is rolled on the ground keeping the point P fixed until the point Q (at the base of the cone, as

shown) touches the ground again.

By what angle (in radians) about P does the cone travel?

(A) 5

12

(B)

5

24

(C)

24

5

(D)

10

13

Key: (D)

Exp: 2 2L 5 12 13cm

Circumference of base circle = length of arc QR

(2 r) R (R Slant height of the Cone 13 cm)(2 5) 13

10

13

2. In a company with 100 employees, 45 earn Rs. 20,000 per month, 25 earn Rs. 30,000, 20 earn

Rs. 40,000,8 earn Rs. 60,000, and 2 earn Rs. 150,000. The median of the salaries is

(A) Rs. 20,000 (B) Rs.30,000 (C) Rs. 32,300 (D) Rs. 40,000

Key: (B)

Exp: All the values put either in ascending or descending order first.

Now number of observations equal to 100 [even]

The median of these values = Avg of two middle most observations.

th st50 observation 51 observation 30,000 30,000

30,0002 2

3. As the two speakers became increasingly agitated, the debate became __________.

(A) lukewarm (B) poetic (C) forgiving (D) heated

Key: (D)

4. P,Q, and R talk about S‟s car collection. P states that S has at least 3 cars. Q believes that S has

less than 3 cars. R indicates that to his knowledge, S has at least one Car. Only one of P, Q and

R is right the number cars owned by S is.

(A) 0 (B) 1 (C) 3 (D) Cannot be determined

Key: (A)

Exp: P States that S has atleast 3 cars, i.e., 3

Q believes that S has less than 3 cars, i.e., 3

R indicates that S has atleast one car 1

P‟s and Q‟s statements are exactly opposite in nature and R‟s statement is proportional to P‟s

statement.

R

R

L

P




24

From the given data, only one person statement is right as it mean that two persons statements

are wrong, i.e., P and R wrong when S has zero cars.

5. He was one of my best __________ and I felt his loss _________.

(A) friend, keenly (B) friends, keen (C) friend, keener (D) friends, keenly

Key: (D)

Q. No. 6- 10 Carry Two Marks Each

6. Two very famous sportsmen Mark and Steve happened to be brothers, and played for country K.

Mark teased James, an opponent from country E, “There is no way you are good enough to play

for your country.‟‟ James replied, “Maybe not, but at least I am the best player in my own family.”

Which one of the following can be inferred from this conversation?

(A) Mark was known to play better than James

(B) Steve was known to play better than Mark

(C) James and Steve were good friends

(D) James played better than Steve

Key: (B)

7. “Here, throughout the early 1820s, Stuart continued to fight his losing battle to allow his

sepoys to wear their caste-marks and their own choice of facial hair on parade, being again

reprimanded by the commander-in-chief. His retort that „A stronger instance than this of

European prejudice with relation to this country has never come under my observations‟ had

no effect on his superiors.”

According to this paragraph, which of the statements below is most accurate?

(A) Stuart‟s commander – in chief was moved by this demonstration of his prejudice.

(B) The Europeans were accommodating of the sepoys‟ desire to wear their caste – marks.

(C) Stuart‟s losing battle‟ refers to his inability to succeed in enabling sepoys to wear caste-

marks.

(D) The commander– in – Chief was exempt from the European preiudice that dictated how

the sepoys were to dress.

Key: (C)

8. The growth of bacteria (lactobacillus) in milk leads to curd formation. A minimum bacterial

population density of 0.8(in suitable units) is needed to form curd. In the graph below, the

population density of lactobacillus in 1 litre of milk is plotted as a function of time, at two

different temperatures, 25°C and 37°C.

Consider the following statements based on the data shown above:




25

(i) The growth in bacterial population stops earlier at 37°C as compared to 25°C

(ii) The time taken for curd formation at 25°C is twice the time taken at 37°C

Which one of the following options is correct?

(A) Only i (B) only ii (C) Both i and ii (D) Neither i nor ii

Key: (A)

Exp: From the graph, Statement (i) is correct,

The time taken for curd formation @25o C= 120 min,

the time taken for curd formation @ 37o C= 80 min, hence (ii) is incorrect.

9. Let S1 be the plane figure consisting of the points (x,y) given by the inequalities x 1 2 and

y 2 3. Let S2 be the plane figure given by the inequalities x y 2,y 1, and x 3 Let

S be the union of S1 and S2. The area of S is.

(A) 26 (B) 28 (C) 32 (D) 34

Key: (C)

10. What is the sum of the missing digits in the subtraction problem below?

5 _ _ _ _

4 8 _ 8 9

1 1 1 1


Key: (D)

|ME| GATE-2017-PAPER-II www.gateforum.com



1

Mechanical Engineering

Q. No. 1 to 25 Carry One Mark Each

1. A mass m of a perfect gas at pressure p1 and volume V1 undergoes an isothermal process. The

final pressure is p2 and volume is V2. The work done on the system is considered positive. If R

is the gas constant and T is the temperature, then the work done in the process is

(A) 21 1

1

Vp V ln

V (B) 1

1 1

2

pp V ln

p (C) 2

1

VRT ln

V (D) 2

1

pmRT ln

p

Key: (B)

Exp: Isothermal work done, W pdV

For isothermal, pV= C

2

1

1 1

V

1 1

V

2 11 1 1 1 1 1 2 2

1 2

p V pV C constant

dVso, W p V

V

V pW p V n p V n ( p V p V )

V p

2. Which one of the following statements is TRUE for the ultrasonic machining (USM) process?

(A) In USM, the tool vibrates at subsonic frequency.

(B) USM does not employ magnetostrictive transducer.

(C) USM is an excellent process for machining ductile materials.

(D) USM often uses a slurry comprising abrasive-particles and water.

Key: (D)

3. The standard deviation of linear dimensions P and Q are 3 m and 4 m, respectively. When

assembled, the standard deviation (in m ) of the resulting linear dimension (P+Q) is ________

Key: 5 to 5

Exp: Given that

Standard deviate of P is 3 m Variance of P is 9 m

Standard deviation of Q is 4 m Variance of Q is 16 m

Variance of P + Q = Var (P+Q) = Variance P + Variance Q

= 9 + 16= 25

Standard deviation of P + Q = + Variance = + 25 = 5

4. The emissive power of a blackbody is P. If its absolute temperature is doubled, the emissive

power becomes.

(A) 2P (B) 4P (C) 8P (D) 16P

Key: (D)

Exp: Emissive power of black body 4

bE T

Given 4

b 1E P T




2

4 4

b b2 2

now T 2T

E 2T 16 T E 16P

5. The state of stress at a point is x y z xz zx yz zy xy yx0 and 50MPa .

The maximum normal stress (in MPa) at that point is _________

Key: 49.9 to 50.1

Exp: It is a pure torsion case in 2D

xy yx 50MPa

xy yx 1

for pure Torsion,

So maximum Normal stress 1 50MPa

6. The determinant of a 2×2 matrix is 50. If one eigenvalue of the matrix is 10, the other

eigenvalue is ___________

Key: 5 to 5

Exp: Given that det of 2×2 Matrix is 50 and are Eigen Value is 10.

Other Eigen value is 5 det product of eigenvalues

7. Which one of the following statement is TRUE?

(A) Both Pelton and Francis turbines are impulse turbines.

(B) Francis turbine is a reaction turbine but Kaplan turbine is an impulse turbine.

(C) Francis turbine is an axial – flow reaction turbine.

(D) Kaplan turbine is an axial – flow reaction turbine.

Key: (D)

8. Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of

getting at least one head is ____________

Key: 0.75 to 0.75

Exp: Total No of outcomes when two coins are tossed is 4 and sample space

S HH,HT,TH,TT

Favorable out comes for existence of at least one head are HH, HT, TH.

Required probability = 3

0.754

9. A cantilever beam of length L and flexural modulus EI is subjected to a point load P at the free

end. The elastic strain energy stored in the beam due to bending (neglecting transverse shear)

(A) 2 3P L

6EI (B)

2 3P L

3EI (C)

3PL

3EI (D)

3PL

6EI

Key: (A)

xy0,

xy0,

1




3

Exp: x

L 2 2 2

x

0

2 3

M P.x

M dx P x dxU

2EI 2EI

P LU

6EI

10. It is desired to make a product having T-shaped cross-section from a rectangular aluminium

block. Which one of the following processes is expected to provide the highest strength of the

product?

(A) Welding (B) Casting (C) Metal Forming (D) Machining

Key: (C)

11. The heat loss from a fin is 6W. The effectiveness and efficiency of the fin are 3 and 0.75,

respectively. The heat loss (in W) from the fin, keeping the entire fin surface at base

temperature, is __________.

Key: 7.9 to 8.1

Exp: Given LossQ 6W

act loss

max

max

3, 0.75

Q Q

Q Heat loss keep entire fin as base temperature

6Q 8W

0.75

12. 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

Key: (D)

Exp: 12 / hour

For steady state finite queue length

So, 24 / hour

13. For the stability of a floating body the

(A) centre of buoyancy must coincide with the centre of gravity

(B) centre of buoyancy must be above the centre of gravity

(C) centre of gravity must be above the centre of buoyancy

(D) metacentre must be above the centre of gravity

Key: (D)

Exp: Stability of floating body is measure with the help of Meta center.

Floating body to be stable, Meta center must be above C.G

I

MG BG

,

For stability MG > 0

x xL

Px




4

14. The divergence of the vector yi xj __________

Key: 0 to 0

Exp: Let F yi xj

divergence of F y x 0x y

15. For a loaded cantilever beam of uniform cross-section, the bending moment (in N.mm) along

the length is M(x) = 5x2+10x, where x is the distance (in mm) measured from the free end of

the beam. The magnitude of shear force (in N) in the cross-section at x =10 mm is ________.

Key: 110 to 110

Exp:

2

x 10mm

dS.F (B.M)

dx

dS.F 5x 10x

dx

S.f 10x 10 S.F 10 10 10 110N

16. A sample of 15 data is a 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

Key: (C)

Exp: We know that mode is the value of the data which occurred most of

17 is mode

17. If a mass of moist air contained in a closed metallic vessel is heated, then its

(A) relative humidity decreases (B) relative humidity increases

(C) specific humidity increases (D) specific humidity decreases

Key: (A)

Exp: Given that mass of moist air contained in a closed metallic vessel is heated, means its specific

humidity is constant.

So, from the psychometric chart,

We can say that at constant specific humidity, as temperature increases relative humidity

decreases i.e., 2 1

So, Answer is (A)

1 2

12

Relative humidity

100%

DBT




5

18. In a slider-crank mechanism, the lengths of the crank and the connecting rod are 100mm and

160mm, respectively. The crank is rotating with an angular velocity of 10 radian/s counter-

clockwise. The magnitude of linear velocity (in m/s) of the piston at the instant corresponding

to the configuration shown in the figure is _____________

Key: 0.99 to 1.01

Exp: sin 2

V r sinn

o

if 90

sin180V r sin90 0.1 1 sin90 1m sec

2n

19. A machine component made of a ductile material is subjected to a variable loading with

min max50 MPa and 50 MPa. If the corrected endurance limit and the yield strength for

the material are '

e y100 MPa and 300MPa, the factor of safety is __________

Key: 1.99 to 2.01

Exp: max

min

y

e

50MPa

50MPa

300MPa

100MPa

max minmax

max minv

max v

yt e

50 500

2 2

50 5050

2 2

1

F.S

0 50 1

300 100 F.S

F.S 2

20. The crystal structure of aluminium is

(A) body-centred cubic (B) face-centred cubic

(C) close-packed hexagonal (D) body-centred tetragonal

Key: (B)

21. A steel bar is held by two fixed supports as shown in the figure and is subjected to an increases

of temperature T 100 C. and 200GPa, respectively, the magnitude of thermal stress (in

MPa) induced in the bar is __________.




6

Key: 218 to 222

Exp: Thermal Stress,

3 6

Th E T 200 10 11 10 100 220MPa

22. The Laplace transform of tte is

(A)

2

s

s 1 (B)

2

1

s 1 (C)

2

1

s 1 (D)

s

s 1

Key: (B)

Exp:

at

t

2

L e f t F s a1L te ;

s 1 where F s L f t

23. Consider a laminar flow at zero incidence over a flat plate. The shear stress at the wall is

denoted byw . The axial positions x1 and x2 on the plate are measured from the leading edge

in the direction of flow. If x2 > x1, then

(A) 1 2

w wx x0 (B)

1 2w wx x

0 (C) 1 2

w wx x (D)

1 2w wx x

Key: (C)

Exp: w w1 2

w

y 0 y 0 y 01 2

u u u

y y y

24. A mass m is attached to two identical springs having spring constant k as shown in the figure.

The natural frequency of this single degree of freedom system is

(A) 2k

m (B)

k

m

(C) k

2m (D)

4k

m

Key: (A)

Exp: Equivalent stiffness keq = k + k = 2k

Natural frequency, eq

n n

k 2k

m m

25. Given the atomic weight of Fe is 56 and that of C is 12, the weight percentage of carbon in

cementite (Fe3C) is _________.

Key: 6.3 to 7.0

Exp: Percentage of carbon by weight in cementite =12

100 6.67%56 3 12




7

Q. No. 26 to 55 Carry Two Marks Each

26. In an orthogonal machining with a tool of 9° orthogonal rake angle, the uncut chip thickness is

0.2mm. The chip thickness fluctuates between 0.25 mm and 0.4 mm. The ratio of the

maximum shear angle to the minimum shear angle during machining is ___________

Key: 1.45 to 1.53

Exp:

1

c

9

t 0.2mm

t 0.25mm to 0.4mm

1

c

o

c

o

c

max

min

tr costan , where r

1 r sin t

if t 0.25mm, r 0.8 & 42.08

if t 0.4mm, r 0.5 & 28.18

42.081.493

28.18

27. A cylindrical pin of 0.020

0.01025

mm diameter is electroplated. Plating thickness is 0.0052.0 mm.

Neglecting the gauge tolerance, the diameter (in mm, up to 3 decimal points accuracy) of the

GO ring gauge to inspect the plated pin is _________.

Key: 29.030 to 29.030

Exp: Diameter of GO end of ring gauge = 25.02 + 2(2.005) = 29.030mm

28. A helical compression spring made of wire of circular cross-section is subjected to a

compressive load. The maximum shear stress induced in the cross-section of the wire is 24

MPa. For the same compressive load, if both the wire diameter and the mean coil diameter are

doubled, the maximum shear stress (in MPa) induced in the cross-section of the wire is _____.

Key: 6 to 6

Exp:

max 3

1

1 2 2 1 2 1

331 1 2

3

2 2 1 2

2

2

8P.D

d

Given, 24MPa

P P , d 2d , D 2D

D d 24 1so, . 2

D d 2

24MPa

4

6MPa

29. In a counter-flow heat exchanger, water is heated at the rate of 1.5kg/s from 40°C to 80°C by

an oil entering at 120°C and leaving at 60°C. The specific heats of water and oil are

4.2kJ/kg.K and 2kJ/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

Key: (D)

Exp: Given counter flow Heat Exchanger




8

Water –

i

o

c

c

c

c

p

m 1.5kg / s

T 40 C

T 80 C

C 4.2KJ / kg.K

Oil - i

o

o

h

h

p

T 120 C

T 60 C

C 2kJ / kg.K

Overall Heat Transfer Coefficient = 400 W/m2K

Area (A) =?

c o i

c o i

c p c c

c p c c

i o

i o

3

2 2

Q UA LMTD

m C T T UA LMTD

m C T TA

Un( / )

1.5 4.2 80 40 1021.83m A 21.83m

40 20400

40n20

30. The rod PQ of length L = 2 m, and uniformly distributed mass of M = 10 kg, is released

from rest at the position shown in the figure. The ends slide along the frictionless faces OP and

OQ. Assume acceleration due to gravity, g = 10 m/s2. The mass moment of inertia of the rod

about its centre of mass and an axis perpendicular to the plane of the figure is (ML2/12). At

this instant, the magnitude of angular acceleration (in radian/s2) of the rod is ____________

Key: 7.25 to 7.75

Exp: 2M 10kg, g 10m sec

2

2

2

c

2

I

ML

12

M 2 1 2M M 2MI M

12 12 2 32

2MT I Mg 0.5

3

210 0.5

3

157.5rad sec

2

I

o

i

h

o

c

T 60 C20

T 40 C

i

o

h

i

c

T 120 C40

T 80 C

1m

Mg

1

2

1m

I

0.5




9

31. A steel plate, connected to a fixed channel using three identical bolts A, B and C, carries a

load of 6kN as shown in the figure. Considering the effect of direct load and moment, the

magnitude of resultant shear force (in kN) on bolt C is.

(A) 13 (B) 15 (C) 17 (D) 30

Key; (C)

Exp:

2 2 2 21 2

1 1

A C 1

6Pr imary shear 2kN

3

Secondaryshear 1500 kN mm

P e 1500M 0.3

r r 50 50

R R C r 0.3 50 15kN

Resultant shear at 'C' 2 15 17kN.

Resultant shear 15 2 17kN.

32. The volume and temperature of air (assumed to be an ideal gas) in a closed vessel is 2.87 m3

and 300K, respectively. The gauge pressure indicated by a manometer fitted to the wall of the

vessel is 0.5bar. If the gas constant of air is R = 287 J/kg. K and the atmospheric pressure is 1

bar, the mass of air (in kg) in the vessel is

(A) 1.67 (B) 3.33 (C) 5.00 (D) 6.66

Key: (C)

Exp: 3

gauge

atm

abs g atm

V 2.87m ; T 300K

P 0.5bar

R 287J / kg.K

P 1 bar

P P P 1.5bar

5

PV mRT

PV 1.5 10 2.87So, m 5kg

RT 287 300

33. For the laminar flow of water over a sphere, the drag coefficient CF is defined as

2 2

FC F / U D , where F is the drag force, is the fluid density, U is the fluid velocity and

D is the diameter of the sphere. The density of water is 1000 kg/m3. When the diameter of the

sphere is 100mm and the fluid velocity is 2m/s, the drag coefficient is 0.5. If water now flows

over another sphere of diameter 200mm under dynamically similar conditions, the drag force

(in N) on this sphere is _____________




10

Key: 19.9 to 20.1

Exp: Given that the condition is dynamic similarity, and in the given condition, Inertia and viscous

force plays major role, hence Reynold‟s number should be same for both model and prototype.

1 2

Re Re

In the first case: 3

1 1U 2m sec, D 100 mm, 1000 kg m

In the second case: 3

2 2U 2m sec, D 200mm, 1000 kg m

1 2

UD UD

Since same water is flowing over both sphere

1 2 1 2

1 1 2 2

2

2

,

U D U D

2 100 V 200

U 1m sec

So, Drage force in second case will be

22 2 2

2 F 2 2F C U D 0.5 1000 (1) 0.2 20N

34. A rod of length 20mm is stretched to make a rod of length 40 mm. Subsequently, it is

compressed to make a rod of final length 10mm. Consider the longitudinal tensile strain as

positive and compressive strain as negative. The total true longitudinal strain in the rod is

(A) –0.5 (B) –0.69 (C) –0.75 (D) –1.0

Key: (B)

Exp: i f

fTrue

i

20mm 10mm

truestrain,

10n n 0.69

20

35. A gear train shown in the figure consists of gears P, Q, R and S. Gear Q and gear R are

mounted on the same shaft. All the gears are mounted on parallel shafts and the number of

teeth of P, Q, R and S are 24, 45, 30 and 80, respectively. Gear P is rotating at 400 rpm. The

speed (in rpm) of the gear S is _________.

Key: 120 to 120

Exp: P Q R S

P

T 24, T 45, T 30, T 80

N 400rpm

P P Q QN .T N T




11

Q

Q Q S S

S

400 24N 213.33rpm

45

N T N T

213.33 45N 120rpm

80

36. In the Rankine cycle for a steam power plant the turbine entry and exit enthalpies are 2803 kJ/kg

and 1800 kJ/kg, respectively. The enthalpies of water at pump entry and exit are 121 kJ/kg and 124

kJ/kg, respectively. The specific steam consumption (in kg/k W.h) of the cycle is ______

Key: 3.5 to 3.7

Exp: Given that,

1 2

3 4

h 2803 kJ / kg, h 1800kJ / kg

h 121kJ / kg, h 124kJ / kg

T 1 2

p 4 3

net T P

net

W h h

2803 1800 1003kJ / kg

W h h

124 121 3kJ / kg

W W W 1000kJ / kg

3600Specific Steam consumption 3.6kg / kWh

W

37. A calorically perfect gas (specific heat at constant pressure 1000 J/kg.K) enters and leaves a gas

turbine with the same velocity. The temperatures of the gas at turbine entry and exit are 1100 K

and 400 K. respectively. The power produced is 4.6 MW and heat escapes at the rate of 300 kJ/s

through the turbine casing. The mass flow rate of the gas (in kg/s) through the turbine is.

(A) 6.14 (B) 7.00 (C) 7.50 (D) 8.00

Key: (B)

Exp: Given that,

p

1

2 L

in out

C 1000J / kgK

T 1100K, P 4.6MW

T 400K, Q 300kJ / s

E E

2 2

1 21 1 2 2 L

V Vh gz h gz Q P

2 2

1 2 1 2

1 2 L

V V , Z Z

So, h h Q P

p 1 2 L

3 6

mC T T Q P

300 10 4.6 10m

1000 1100 400

m 7kg / sec

4

32

1

T

S

P

LQ

21




12

38. Three masses are connected to a rotating shaft supported on bearings A and B as shown in the

figure. The system is in a space where the gravitational effect is absent. Neglect the mass of

shaft and rods connecting the masses. For m1 = 10kg, m2 = 5kg and m3 = 2.5 kg and for a shaft

angular speed of 1000 radian/s, the magnitude of the bearing reaction (in N) at location B is

_________.

Key: 0 to 0

Exp: 1 1

2 2

3 3

m 10kg, r 0.1m

m 5kg, r 0.2m

m 2.5kg, r 0.4m

x 1 1 1 2 2 2 3 3 3

o o o

o o

y 1 1 1 2 2 2 3 3 3

A B

F m r cos m r cos m r cos

10 0.1 cos0 5 0.2 cos120 2.5 0.4 cos240

1 0.5 0.5 0

F m r sin m r sin m r sin 0 5 0.2 sin120 2.5 0.4sin 240 0

R R 0kN

39. A strip of 120 mm width and 8mm thickness is rolled between two 300 mm-diameter rolls to

get a strip of 120 mm width and 7.2 mm thickness. The speed of the strip at the exit is 30

m/min. There is no front or back tension. Assuming uniform roll pressure of 200 MPa in the

roll bite and 100% mechanical efficiency, the minimum total power (in kW) required to drive

the two rolls is _________.

Key: 9.4 to 9.8

Exp: Width = 120mm

Initial thickness to = 8mm

Diameter of Roller = 300mm

Radius of Roller = 150mm

Final thickness = 7.2mm

i fh t t 8 7.2 0.8 mm

Power require to drive one roller

p pP T. F L .A.L . Here, PA L b

2

pP L .b phere, L R h

6 3

VP R h .b.

R

30200 10 0.8 10 0.12 4.8 kW

60

So, power require to drive 2 roller = 2P 2 4.8 kW 9.6 kW




13

40. A product made in two factories p and Q, is transported to two destinations, R and S. The per

unit costs of transportation (in Rupees) from factories to destinations are as per the following

matrix:

Destination

Factory R S

P 10 7

Q 3 4

Factory P produces 7 units and factory Q produces 9 units of the product. Each destination

requires 8 units. If the north-west corner method provides the total transportation cost as X (in

Rupees) and the optimized (the minimum) total transportation cost Y (in Rupees), then (X-Y),

in Rupees, is

(A) 0 (B) 15 (C) 35 (D) 105

Key: Answer is not matched with IIT Key

Exp:

By North West corner Rule

Total cost, x = 10×7+7×1+4×8 = Rs.105

By VAM

Total minimum Cost, y = 3×8+4×1+7×7=Rs.77

So, x–y = Rs.28

41. One kg of an ideal gas (gas constant R = 287 J/kg.K) undergoes an irreversible process from

state-1 (1 bar, 300 K) to state -2 (2 bar, 300 K). The change in specific entropy (s2 – s1) of the

gas (in J/kg. K) in the process is ___________

Key: –201 to –197

Exp: Given

1 2

1 2

m 1kg, R 287J / kg.K

P 1bar, P 2bar

T 300K, T 300K

22 1

1

Same Temperature

P 2S S R n 287 n 198.93J / kgKP 1

10 7

3 4

R S

P 7

Q 9

8 8

10 7

3 4

R S

P 7

Q 1 8

1 2

10 71

3 42

V 6 V 7

u 0 7 17

u 3 9 18 1

8 8

7 3




14

42. A 60 mm-diameter water jet strikes a plate containing a hole of 40mm diameter as shown in

the figure. Part of the jet passes through the hole horizontally, and the remaining is deflected

vertically. The density of water is 1000 kg/m3. If velocities are as indicated in the figure, the

magnitude of horizontal force (in N) required to hold the plate is _________

Key: 627 to 629

Exp: Force in X–direction = Rate of change of momentum

i f x directionP P

1 21 2

x direction

1 1 2 2 1 2

2

1 2

2 2

m V m V

V A V A V V V

V A A

1000 20 20 0.06 0.044

628.32N

43. The arrangement shown in the figure measures the

velocity V of a gas of density 1 kg/m3 flowing

through a pipe. The acceleration due to gravity is

9.81 m/s2. If the manometric fluid is water (density

1000 kg/m3) and the velocity V is 20 m/s, the

differential head h (in mm) between the two arms of

the manometer is ____________

Key: 19 to 21

Exp: Given

3 2

g

3

m

1kg / m , g 9.81m / s

1000kg / m , V 20m / s

m

2

m

V 2gH Where H h 1

VSo, h 1

2g

2

20h 1000 1

2 9.81

h 0.0204m 2.04cm h 20.4mm

1D 6cm

120m / s V

2D 4cm

220m / s V




15

44. A metal ball of diameter 60mm is initially at 220 °C. The ball is suddenly cooled by an air jet

of 20°C. The heat transfer coefficient is 200 W/m2.K and 9000kg/m

3, respectively. The ball

temperature (in °C) after 90 seconds will be approximately.

(A) 141 (B) 163 (C) 189 (D) 210

Key: (A)

Exp: D = 60mm = 0.06m

2 3

i

p

T 220 C, h 200W / m K, 9000kg / m

T 20 C, C 400J / kgK, K 400W / mK

t 90sec T ?

p

h.A 200 90 3.t.VC 0.03 9000 400i

f f

f

T T 220 20e e

T T T 20

T 141.3 C

45. A single – plate clutch has a friction disc with inner and outer radii of 20mm and 40 mm,

respectively. The friction lining in the disc is made in such a way that the coefficient of

friction varies radially as 0.01r, where r is in mm. The clutch needs to transmit a friction

torque of 18.85kN.mm. As per uniform pressure theory, the pressure (in MPa) on the disc is

_________

Key: 0.49 to 0.51

Exp:

i o

2 2 3

r 20mm, r 40mm, 0.01r

T 18.85kN mm

p ?

P p2 rdr

T p2 r dr p 0.01r 2 r dr 0.0628r .dr

4040 43

20 20

4 43

rT 0.0628p r dr 0.0628p

4

40 2018.85 10 0.0628p

4 4

p 0.5MPa

46. The surface integral S

F.ndS over the surface S of the sphere 2 2 2x y z 9, where

F=(x+y) i+ (x+z) j+(y+z) k and n is the unit outward surface normal, yields ______.

Key: 225 to 227

Exp:

F x y i x z J y z k

divF x y x z y z 1 0 1 2x y z

By divergence theorem,

^

2 2 2

S V

F.ndS divFdV where V is volume of given surface of sphere x y z 9

V

4 272dV 2V 2 72 226.1947

3

dr

r

ir

0r




16

47. Block 2 slides outward on link 1 at a uniform velocity of 6 m/s as shown in the figure. Link 1

is rotating at a constant angular velocity of 20 radian/s counterclockwise. The magnitude of

the total acceleration (in m/s2) of point P of the block with respect to fixed point O is

________

Key: 243 to 244

Exp: Acceleration of the block,

v 6m / sec, 20rad / sec

2 22 2 2

r cr

2

cr

22

r

2 2 2

r

a a a r 2v

r OP,

a 2v 2 6 20 240m / s

a r 20 0.1 40

a 240 40 243.31m / sec

48. During the turning of a 20mm-diameter steel bar at a spindle speed of 400 rpm, a tool life of

20 minute is obtained. When the same bar is turned at 200 rpm, the tool life becomes 60

minute. Assume that Taylor‟s tool life equation is valid. When the bar is turned at 300 rpm, the

tool life (in minute) is approximately.

(A) 25 (B) 32 (C) 40 (D) 50

Key: (B)

Exp: Taylors Tool life equation,

n

n n

1 1 2 2

n

2 1

1 2

n

3 3

n

3 1

1 3

0.63

3

3

3

VT C

V T V T

T V

T V

60 400

20 200

n 0.63

at N 300rpm, T ?

T V

T V

T 400

20 300

T 31.57 min utes

T 32min utes




17

49. Consider the matrix A = 50 70

70 80

whose eigenvectors corresponding to eigenvalues 1and 2

are 2

1 2

1

70 80x and x .

50 70

respectively. The value of T

1 2x x is_________

Key: 0 to 0

Exp: 50 70

A70 80

Eigen vectors are 2

1 2

1

70 80X ; X

50 70

Where 1 2, Eigen values of A

2T

1 2 1 2 1

2 1 1 2

80X X 70 50 70 80 50 70

70

70 5600 70 3500 70 9100

70 130 9100 9100 9100 0

1 2sum of eigen values

Trace 50 80 130

50. The radius of gyration of a compound pendulum about the point of suspension is 100mm. The

distance between the point of suspension and the centre of mass is 250mm. Considering the

acceleration due to gravity as 9.81 m/s2, the natural frequency (in radian/s) of the compound

pendulum is _________.

Key: 15 to 16

Exp: k 100mm 0.1m

L 250mm 0.250m

g 9.81m / sec

22

n 2

I mk m 0.1

mgL m 9.81 0.250W 15.66rad / sec

I m 0.1

51. Consider the differential equation 3y'' x 27y x 0 with initial conditions y(0) = 0 and

y'(0) 2000. The value of y at x = 1 is ________.

Key: 93 to 95

Exp:

2 2

c 1 2 p

c 1 2

1 1

3y '' x 27y x 0, y 0 0, y ' 0 2000

Auxillaryequation, 3m 27 0 m 9 0 m 0 3i

y c cos3x c sin3x and y 0

y c cos3x c sin3x

y 0 0 c 0 0 c 0

2

2

y c sin3x

y' 3c cos3x




18

2 2

2000y' 0 2000 2000 3c c

3

2000 2000y sin3x, y 1 sin3 94.08

3 3

52. If f(z) = (x2+ay

2) + ibxy is a complex analytic function of z = x + iy, where i = 1, then

(A) a = –1, b= –1 (B) a = –1, b = 2 (C) a = 1, b = 2 (D) a = 2, b = 2

Key: (B)

Exp: Given 2 2f z x ay i bxy is analytic

2 2

u x,

x y y x

value u x ay , bxy

u2x by

x x

u2ay bx

y y

Clearly for b = 2 and a= –1 above Cauchy-Riemann equations holds

53. A project starts with activity A and ends with activity F. The precedence relation and durations

of the activities are as per the following table:

Activity Immediate Predecessor Duration

(days)

A - 4

B A 3

C A 7

D B 14

E C 4

F D,E 9

The minimum project completion time (in days) is ______

Key: 30 to 30

Exp:

Minimum project completion Time = Length of longest path = 4+3+14+9=30 Days.

54. Maximize Z = 5x1+3x2

Subject to

x1+2x2 10,

x1–x2 8,

x1, x2 0.

In the starting Simplex tableau, x1 and x2 are non-basic variables and the value of Z is zero.

The value of Z in the next Simplex tableau is _________.

1 2

3

4

5 6F

9

A

4C, 7

B3 14

D

4

E




19

Key: 40 to 40

Exp:

max

st

nd

26 2Z at ,

3 3

1 Table at 0,0 Z 0

2 Table at 8,0 Z 40

55. The principal stresses at a point in a critical section of a machine component are

1 2 360MPa, 5MPa and 40 MPa. For the material of the component, the tensile

yield strength is y 200 MPa. According to the maximum shear stress theory, the factor of

safety is

(A) 1.67 (B) 2.00 (C) 3.60 (D) 4.00

Key: (B)

Exp:

y1 3 w1 3

2 2 FOS

20060 40

FOS

FOS 2

General Aptitude

Q. No. 1 - 5 Carry One Mark Each

1. If you choose plan P, you will have to _______ plan Q, as these two are mutually _________.

(A) forgo, exclusive (B) forget, inclusive

(C) accept, exhaustive (D) adopt, intrusive

Key: (A)

2. P looks at Q while Q looks at R. P is married, R is not. The number of people in which a

married person is looking at an unmarried person is


Key: (B)

3. If a and b are integers and a – b is even, which of the following must always be even?

(A) ab (B) a2 + b

2 + 1 (C) a

2 + b + 1 (D) ab – b

Key: (D)

Exp: According to the given relation of a–b = even, there is a possibility of odd-odd (or) even-even

is equal to even. From the options, Option (D) is correct. Since, odd × odd–odd (or) even × even–even → is always even number.

Alternate Method:

Let a – b =2K a b 2K

2 2ab b b 2K b b b 2Kb b b b 2Kb b b 1 2Kb

Even; since 2Kb always even & b (b-1) also even, if with b is odd or even.

0,0

0,5

2x

0, 8

8,0

10,0

26 2,

3 3

1x




20

4. A couple has 2 children. The probability that both children are boys if the older one is a boy is

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

Key: (C)

Exp: No.of Favourablecases

ProbabilityTotal No.of Possible cases

The probability that both children are boys if the order one is a boy = 1

2

5. The ways in which this game can be played __________ potentially infinite.

(A) is (B) is being (C) are (D) are being

Key: (C)

Q. No. 6- 10 Carry Two Marks Each

6. “If you are looking for a history of India, or for an account of the rise and fall of the British

Raj, or for the reason of the cleaving of the subcontinent into two mutually antagonistic parts

and the effects this mutilation will have in the respective sections, and ultimately on Asia, you

will not find it in these pages; for though I have spent a lifetime in the country, I lived too near

the seat of events, and was too intimately associated with the actors, to get the perspective

needed for the impartial recording of these matters.”

Which of the following closest in meaning to „cleaving‟?

(A) Deteriorating (B) Arguing (C) Departing (D) Splitting

Key: (D)

7. There are 4 women P, Q, R, S, and 5 men V, W, X, Y, Z in a group. We are required to form

pairs each consisting of one woman and one man. P is not to be paired with Z, and Y must

necessarily be paired with someone. In how many ways can 4 such pairs be formed?

(A) 74 (B) 76 (C) 78 (D) 80

Key: (C)

Exp: If P is paired with y; they Q has 4 choices

R has 3 choices

S has 2 choices

Total 24 choices

(or)

If Q is paired with y; then P has 3 choices

R has 3 choices

S has 2 choices

Total 18 choices

(or)

If R is paired with y; then P has 3 choices

Q has 3 choices

S has 2 choices

Total 18 choices




21

(or)

If S is paired with y; then P has 3 choices

Q has 3 choices

S has 2 choices

Total 18 choices

Total number of ways = 24+18+18+18=78

8. In the graph below, the concentration of a particular pollutant in a lake is plotted over

(alternate) days of a month in winter (average temperature 10°C) and a month in summer

(average temperature 30°C).

Consider the following statements based on the data shown above:

(i) Over the given months, the difference between the maximum and the minimum pollutant

concentrations is the same in both winter and summer.

(ii) There are at last four days in the summer month such that the pollutant concentrations on

those days are within 1 ppm of the pollutant concentrations on the corresponding days in

the winter month.

Which one of the following options is correct?

(A) Only i (B) Only ii (C) Both i and ii (D) Neither i nor ii

Key: (B)

Exp: The difference between the maximum and the minimum pollutant concentrations

(i) in winter = 8-0=8 ppm ,

(ii) in summer = 10.5-1.5=9 ppm

(i) is false & (ii) is correct from the graph.

9. All people in a certain island are either „Knights‟ or „Knaves‟ and each person knows every

other person‟s identity. Knights NEVER lie, and knaves ALWAYS lie.

P says “Both of us are knights”. Q says “None of us are knaves”.

Which one of the following can be logically inferred from the above?

(A) Both P and Q are knights

(B) P is a knight; Q is a knave

(C) Both P and Q are knaves

(D) The identities of P, Q cannot be determined

Key: (D)




22

10. X bullocks and Y tractors take 8 days to plough a field. If we halve the number of bullocks and

double the number of tractors, it takes 5days to plough the same field. How many days will it

take X bullocks alone to plough the field?

(A) 30 (B) 35 (C) 40 (D) 45

Key: (A)

Exp: Given Number of days required that X bullocks and Y tractors to plough a field = 8 days

(1) i.e, X Y 8D 8X 8Y 1day Number of days required that

Xbullocks and 2y tractors

2

to plough field 5 2

X 5i.e, 2Y 5D X 10Y 1D

2 2

From (1) & (2); we have

5X 11X8X 8Y 10Y Y

2 4

15XFrom 1 X Y 8D 8D X 30days

4

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