Vikas -...

MECHANICAL ENGINEERING GATE : 2010

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Read t he fol l ow ing i nst r uct i ons car efu l l y.

1. All quest ions in this paper are of object ive type.

2. There are a total of 65 quest ions car rying 100 marks.

3. Quest ions 1 through 25 are 1-mark quest ions, quest ion 26 through 55 are 2-mark quest ions.

4. Quest ions 48 and 51 (2 pairs) common data quest ions and quest ion pairs (Q. 52 and Q.53) and (Q. 54 andQ.55) are l inked answer quest ions. The answer to the second quest ion of the above pair depends on theanswer to the first quest ion of the pair. I f the first quest ion in the l inked pair is wrongly answered or isun-at tempted, then the answer to the second quest ion in the pair wil l not be evaluated.

5. Quest ions 56 - 65 belong to general apt i tude (GA). Quest ions 56 - 60 wil l car ry 1-mark each, and quest ions61-65 wil l car ry 2-marks each. The GA quest ions wil l begin on a fresh page.

6. Un-at tempted quest ions wil l car ry zero marks.

7. Wrong answers will car ry NEGATIVE marks. For Q.1 to Q.25 and Q.56 - Q.60, 1/3 mark will be deducted foreach wrong answer. For Q. 26 to Q. 51, and Q.61 - Q.65, 2/3 mark wil l be deducted for each wrong answer.The quest ion pairs (Q. 52, Q. 53) and (Q. 54, Q. 55) are quest ions with l inked answers. There wil l be negat ivemarks only for wrong answer to the first quest ion of the l inked answer quest ion pair i .e. for Q. 52 and Q.54,2/3 mark wil l be deducted for each wrong answer. There is no negat ive marking for Q. 53 and Q.55.

8. Calculator (without data connectivity) is al lowed.

9. Char ts, graph sheets or tables are NOT al lowed.

GATE 2010 MECHANICAL ENGINEERING

Time Al l ow ed : 3 Hou r s M ax imum M ar k s : 150

GATE : 2010 MECHANICAL ENGINEERING

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Q.(1 – 25) car r y one mar k each .

1. The par abolic ar c y = x , 1 x 2 is r evolved

around the x-axis. The volume of the sol id ofrevolut ion is

(a)4

(b)2

(c)34

(d)32

2. The Blasius equat ion, 3

3

d fd

+ 2

22fd fd

= 0, is a

(a) second order nonlinear ordinary different ialequat ion

(b) thi rd or der nonlinear ordinary different ialequat ion

(c) t h i r d or der l i near or dinar y di f fer ent i alequat ion

(d) mixed or der nonlinear ordinary different ialequat ion

3. The value of the integral 2– 1

dxx

is

(a) –

(b) – /2

(c) /2

(d)

4. The modulus of the complex number 3 41 2

ii

is

(a) 5

(b) 5

(c) 1

5

(d)15

5. The funct ion y = 2 3x

(a) is cont inuous x R and differ ent iable

x R

(b) is cont inuous x R and differ ent iable

x R except at x = 3/2

(c) is cont inuous x R and differ ent iable

x R except at x = 2/3

(d) is cont inuous x R except at x = 3 anddifferent iable x R

6. Mobil ity of a statically indeterminate structure is

(a) – 1 (b) 0

(c) 1 (d) 2

7. There ar e two points P and Q on a planar r igidbody. The relat ive velocit y between the twopoints

(a) should always be along PQ

(b) can be or iented along any direct ion

(c) should always be perpendicular to PQ

(d) should be along QP when the body undergoespure t ranslat ion

8. The state of plane-str ess at a point is given byx = – 200MPa, y = 100 MPa and xy = 100 MPa.The maximum shear stress (in MPa) is

(a) 111.8

(b) 150.1

(c) 180.3

(d) 223.6

9. Whi ch of t he fol l ow i ng st at emen t s i sINCORRECT ?

(a) Grashof’s rule states that for a planar crank-rocker four bar mechanism, the sum of theshor test and longest l ink lengths cannot beless than the sum of the r emaining two l inklengths

(b) Inversions of a mechanism are cr eated byfixing differ ent l inks one at a t ime

(c) Geneva mechan i sm i s an i n t er mi t t entmot ion device

(d) Gruebler ’s cr i ter ion assumes mobil i ty of aplanar mechanism to be one


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10. The natural frequency of a spr ing-mass systemon ear th is n. The natural frequency of thissystem on the moon (gmoon = gear th /6) is

(a) n

(b) 0.408 n

(c) 0.204 n

(d) 0.167 n

11. Tooth in ter fer ence in an ext er nal involut espurgear pair can be reduced by

(a) decreasing center distance between gear pair

(b) decreasing module

(c) decreasing pr essure angle

(d) increasing number of gear teeth

12. For the stabi l i ty of a float ing body, under theinfluence of gravity alone, which of the fol lowingis TRUE?

(a) Metacentre should be below centre of gravity

(b) Metacentre should be above centre of gravity

(c) Metacentre and centre of gravity must l ie onthe same hor izontal l ine

(d) Metacentre and centre of gravity must l ie onthe same ver t ical l ine

13. The maximum velocit y of a one-dimensionalincompr essible ful ly developed viscous flow,between two fixed paral lel plates, is 6 ms– 1. Themean velocity (in ms– 1) of the flow is

(a) 2 (b) 3

(c) 4 (d) 5

14. A phenomenon is modeled using n dimensionalvar iables with k pr imary dimensions. The numberof non-dimensional var iables is

(a) k (b) n

(c) n – k (d) n + k

15. A tur bo-char ged four -st r oke di r ect inject iondiesel engine has a displacement volume of0.0259 m3 (25.9 l i tres). The engine has an outputof 950 kW at 2200 rpm. The mean effect ivepressure (in MPa) is closest to

(a) 2

(b) 1

(c) 0.2

(d) 0.1

16. One ki logram of water at room temper ature isbrought into contact with a high temperaturethermal reservoir. The entr opy change of theuniverse is

(a) equal to entr opy change of the r eservoir

(b) equal to ent ropy change of water

(c) equal to zero

(d) always posit ive

17. A hydraulic turbine develops 1000 kW power fora head of 40 m. I f the head is reduced to 20 m,the power developed (in kW) is

(a) 177 (b) 354

(c) 500 (d) 707

18. The mater ial proper ty which depends only onthe basic crystal st ructure is

(a) fat igue strength

(b) work hardening

(c) fracture strength

(d) elast ic constant

19. I n a gat ing system, the rat io 1 : 2 : 4 represents

(a) spr ue base area : runner area : ingate area

(b) pour ing basin area : ingate area : runner area

(c) spr ue base area : ingate ar ea : cast ing area

(d) runner area : ingate area : cast ing area

20. A shaft has a dimension, 0.009

0.02535 . The respect ive

values of fundamental deviation and tolerance are

(a) – 0.025, ±0.008

(b) – 0.025, ±0.016

(c) – 0.009, ±0.008

(d) – 0.009, ±0.016

21. I n a CNC program block , N002 G02 G91 X40Z40..., G02 and G91 r efer to

(a) ci rcular inter polat ion in counter clockwisedirect ion and incr emental dimension

(b) ci rcular inter polat ion in counter clockwisedirect ion and absolute dimension

(c) cir cular interpolat ion in clockwise direct ionand incremental dimension

(d) cir cular interpolat ion in clockwise direct ionand absolute dimension


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22. The demand and forecast for February are 12000and 10275, respectively. Using single exponentialsmoothening method (smoothening coefficient= 0.25), for ecast for the month of M arch is

(a) 431 (b) 9587

(c) 10706 (d) 11000

23. Lit t le’s law is a relat ionship between

(a) stock level and lead t ime in an inventorysystem

(b) wai t ing t ime and length of the queue in aqueuing system

(c) number of machines and job due dates in ascheduling problem

(d) uncer tainty in the act ivity t ime and projectcomplet ion t ime

24. Vehicl e manufact ur ing assembly l ine i s anexample of

(a) product layout (b) process layout

(c) manual layout (d) fixed layout

25. Simplex method of solving l inear pr ogrammingproblem uses

(a) al l the points in the feasible region

(b) only the corner points of the feasible region

(c) intermediate points wi thin the infeasibleregion

(d) only the inter ior points in the feasible region

Q.(26 – 55) car r y t w o mar k s each .

Not e : All length dimensions shown in the figures arein mm (unless otherwise specified). Figuresare not drawn to scale.

26. Torque exerted on a flywheel over a cycle is l istedin the table. Flywheel energy (in J per unit cycle)using Simpson’s rule is

Angle 0 60 120 180 240 300 360 (degree)

Torque 0 1066 – 323 0 323 – 355 0 (Nm)

(a) 542

(b) 993

(c) 1444

(d) 1986

27. One of the eigen vectors of the matr ix A = 2 2

1 3

is

(a)2

1

(b)2

1

(c)4

1

(d)1

1

28. Velocity vector of a flow field is given as V

= 2 x

y i – x2 z j . The vor t ici ty vector at (1, 1, 1) is

(a) 4 i – j

(b) 4 i – k(c) i – 4 j

(d) i – 4 k

29. The L aplace t r ansfor m of a funct ion f(t ) i s

2

1( 1)s s

. The funct ion f(t) is

(a) t – 1 + e– t

(b) t + 1 + e– t

(c) – 1 + e– t

(d) 2t + et

30. A box contains 2 washer s, 3 nuts and 4 bol ts.I tems are drawn from the box at random one ata t ime without replacement. The probabil i ty ofdrawing 2 washers first fol lowed by 3 nuts andsubsequent ly the 4 bolts is

(a) 2/315

(b) 1/630

(c) 1/1260

(d) 1/2520

31. A band brake having band-width of 80 mm, drumdiameter of 250 mm, coefficient of fr iction of 0.25and angle of wrap of 270 degrees is required toexert a fr iction torque of 1000 Nm. The maximumtension (in kN) developed in the band is

(a) 1.88

(b) 3.56

(c) 6.12

(d) 11.56


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32. A bracket (shown in figure) is r igidly mountedon wall using four r ivets. Each r ivet is 6 mm indiameter and has an effect ive length of 12 mm.

32 321000 N

20

75

25

100

12 40 12

.

Direct shear st ress (in MPa) in the most heavilyloaded r ivet is

(a) 4.4 (b) 8.8

(c) 17.6 (d) 35.2

33. A mass m at tached to a spr ing is subjected to aharmonic force as shown in figure. The amplitudeof the forced mot ion is observed to be 50 mm.The value of m (in kg) is

F(t ) = 100 cos(100 t ) Nmk = 3000 Nm – 1

(a) 0.1 (b) 1.0

(c) 0.3 (d) 0.5

34. For the epicycl ic gear arrangement shown in thef igur e, 2 = 100 r ad/s clockwise (CW) andarm = 80 rad/s counter clockwise (CCW). Theangular velocit y 5 (in rad/s ) is

N = Number of teeth for gear i

i

N = 20

N = 24

N = 32

N = 80

2

3

4

5

43

2

arm Shaft axis

(a) 0

(b) 70 CW

(c) 140 CCW

(d) 140 CW

35. A light ly loaded ful l journal bear ing has journaldiameter of 50 mm, bush bore of 50.05 mm andbush length of 20 mm. I f rotational speed of journalis 1200 rpm and average viscosity of liquid lubricantis 0.03 Pa s, the power loss (in W) wil l be

(a) 37 (b) 74

(c) 118 (d) 237

36. For the configuration shown, the angular velocityof l ink AB is 10 r ad/s counter clockwise. Themagni t ude of t he r el at i ve sl i di ng veloci t y(in ms– 1) of slider B with respect to r igid l ink CDis

60A

B

D

C

AB = 250

AC = 500

BC = 250 3

(a) 0 (b) 0.86

(c) 1.25 (d) 2.50

37. A smooth pipe of diameter 200 mm carr ies water.The pressure in the pipe at sect ion S1 (elevat ion:10 m) is 50 kPa. At sect ion S2 (elevat ion: 12 m)the pressure is 20 kPa and velocity is 2 ms– 1.Density of water is 1000 kgm– 3 and accelerat iondue to gravity is 9.8 ms– 2. Which of the fol lowingis TRUE ?

(a) Flow is from S1 to S2 and head loss is 0.53 m

(b) Flow is from S2 to S1 and head loss is 0.53 m

(c) Flow is from S1 to S2 and head loss is 1.06 m

(d) Flow is from S2 to S1 and head loss is 1.06 m

38. Match the fol lowing

P: Compressible flow U: Reynolds number

Q: Free sur face flow V: Nussel t number

R: Boundary layer flow W: Weber number

S: Pipe flow X: Froude number

T: Heat convect ion Y: Match number

Z: Skin fr ict ion coefficient

(a) P - U; Q - X; R - V; S - Z; T - W

(b) P - W; Q - X; R - Z; S - U; T - V

(c) P - Y; Q - W; R - Z; S - U; T - X

(d) P - Y; Q - W; R - Z; S - U; T - V


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39. A mono-atomic ideal gas ( = 1.67, molecularweight = 40) is compressed adiabat ical ly from 0.1 MPa, 300K to 0.2 MPa. The universal gasconstant is 8.314 kJ kmol – 1 K – 1. The work ofcompression of the gas (in kJ kg– 1) is

(a) 29.7 (b) 19.9

(c) 13.3 (d) 0

40. Consider the fol lowing two processes;

a. A heat source at 1200 K loses 2500 kJ of heatto a sink at 800 K

b. A heat source at 800 K loses 2000 kJ of heatto a sink at 500 K

Which of the fol lowing statements is t rue?

(a) Process I is more ir reversible than Process I I

(b) Process I I is more irreversible than Process I

(c) I rreversibil ity associated in both the processesare equal

(d) Both the processes are reversible

41. A fin has 5 mm diameter and 100 mm length.The t her mal conduct ivi t y of f i n mater i al is400 Wm– 1 K – 1. One end of the fin is maintainedat 130C and its remaining sur face is exposed toambient ai r at 30C. I f t he convect ive heatt ransfer coefficient is 40 Wm– 2 K – 1, the heat loss(in W) from the fin is

(a) 0.08

(b) 5.0

(c) 7.0

(d) 7.8

42. A moist air sample has dry bulb temper ature of30C and specific humidity of 11.5 g water vapourper kg dry ai r. Assume molecular weight of airas 28.93. I f the saturat ion vapour pressure ofwater at 30C is 4.24 kPa and the total pressureis 90 kPa, then the relat ive (in %) of ai r sampleis

(a) 50.5

(b) 38.5

(c) 56.5

(d) 68.5

43. Two pipes of inner diameter 100 mm and outerdiameter 110 mm each ar e joined by flash-but twelding using 30 V power supply. At the interface,1 mm of mater ial melts from each pipe whichhas a resistance of 42.4 . I f the unit melt energyis 64.4 MJ m– 3, t hen t ime required for welding(in s) is

(a) 1 (b) 5

(c) 10 (d) 20

44. For tool A, Taylor ’s tool l i fe exponent (n) is 0.45and constant (K) is 90. Similar ly for tool B,n = 0.3 and K = 60. The cut t ing speed (in m/min)above which tool A wi l l have a higher tool l i fethan tool B is

(a) 26.7 (b) 42.5

(c) 80.7 (d) 142.9

45. A taper hole is inspected using a CMM , with aprobe of 2 mm diameter. At a height , Z = 10 mmfr om the bot tom, 5 points are touched and adiameter of circle (not compensated for pr obesize) is obtained as 20 mm. Similar ly, a 40 mmdiameter is obtained at a height Z = 40 mm. Thesmaller diameter (in mm) of hole at Z = 0 is

Z = 40

Z = 20

Z = 0

(a) 13.334 (b) 15.334

(c) 15.442 (d) 15.542

46. Annual demand for window frames is 10000. Eachframe costs Rs. 200 and or der ing cost is Rs. 300per order. I nventory holding cost is Rs. 40 perframe per year. The supplier is wil l ing to offer2% discount i f the order quant ity is 1000 or more,and 4% i f or der quant i ty is 2000 or more. I f thetotal cost is to be minimized, the retai ler should

(a) order 200 frames every t ime

(b) accept 2% discount

(c) accept 4% discount

(d) order Economic Order Quant ity


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47. The project act ivit ies, pr ecedence r elat ionshipsand durat ions are descr ibed in the table. Thecr i t ical path of the pr oject is

Act ivi ty Precedence Durat ion (in days)

P - 3 Q - 4 R P 5 S Q 5 T R, S 7 U R, S 5 V T 2 W U 10

(a) P-R-T-V

(b) Q-S-T-V

(c) P-R-U-W

(d) Q-S-U-W

Common Dat a Quest i onsCom mon Dat a for Q. (48 – 49) :

I n a steam power plant operat ing on the Rank inecycle, steam enter s the turbine at 4 M Pa, 350C andexi ts at a pressure of 15 kPa. Then i t enter s thecondenser and exits as saturated water. Next, a pumpfeeds back the water to the boi ler. The adiabat icefficiency of the tur bine is 90%. The thermodynamicstates of water and steam are given in table.

State h(kJkg– 1) s(kJkg– 1 K – 1) v(m3 kg– 1)

Steam 4MPa, 350°C 3092.5 6.5821 0.06645

h f hg sf sg vf vg Water: 15 KPa 225.94 2599.1 0.7549 8.0085 0.001014 10.02

h is specific enthalpy, s is specific entropy and v t hespecific volume; subscr ipts f and g denote saturatedliquid state and saturated vapour state.

48. The net work output (kJ kg– 1) of the cycle is

(a) 498 (b) 775

(c) 860 (d) 957

49. Heat supplied (kJ kg– 1) to the cycle is

(a) 2372

(b) 2576

(c) 2863

(d) 3092

Com mon Dat a for Q. (50 – 51) :

Four jobs are to be pr ocessed on a machine as perdata l isted in the table.

Job Processing t ime (in days)

Due date

1 4 6 2 7 9 3 2 19 4 8 17

50. I f the Ear l iest Due Date (EDD) rule is used tosequence the jobs, the number of jobs delayed is

(a) 1 (b) 2

(c) 3 (d) 4

51. Using the Shor test Processing Time (SPT) rule,total tar diness is

(a) 0

(b) 2

(c) 6

(d) 8

L ink ed Answ er Quest i onsStatement for L ink ed Answer Q. (52 – 53) :

A massless beam has a loading pat tern as shown inthe figure. The beam is of rectangular cross-sect ionwith a width of 30 mm and height of 100 mm.

2000 2000

B C

3000 N m – 1

A

52. The maximum bending moment occurs at

(a) Locat ion B

(b) 2675 mm to the r ight of A

(c) 2500 mm to the r ight of A

(d) 3225 mm to the r ight of A

53. The maximum magni tude of bending str ess (inMPa) is given by

(a) 60.0

(b) 67.5

(c) 200.0

(d) 225.0


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Statement for L ink ed Answer Q. (54 – 55) :

I n shear cutt ing operat ion, a sheet of 5 mm thicknessis cut along a length of 200 mm. The cut t ing blade is400 mm long (see figure) and zer o-shear (S = 0) isprovided on the edge. The ult imate shear st rengthof the sheet is 100 MPa and penetrat ion to thicknessrat io is 0.2. Neglect fr ict ion.

400

S

54. Assuming for ce vs displacement curve to berectangular, the work done (in J) is

(a) 100 (b) 200

(c) 250 (d) 300

55. A shear of 20 mm (S = 20 mm) is now providedon the blade. Assuming force vs displacementcur ve to be t rapezoidal , t he maximum force(in kN) exer ted is

(a) 5

(b) 10

(c) 20

(d) 40

Gen er al Apt i t u de (GA) Qu est i on s

Q. (56 – 60) car r y on e mar k each .

56. 25 persons are in a room. 15 of them play hockey,17 of them play football and 10 of them play bothhockey and football . Then the number of personsplaying neither hockey nor football is

(a) 2 (b) 17

(c) 13 (d) 3

57. Choose the most appropr iate word from theopt ions given below to complete the fol lowingsentence:

I f we manage t o ou r nat ur alresources, we would leave a bet ter planet forour chi ldr en.

(a) uphold

(b) restrain

(c) cher ish

(d) conserve

58. The quest ion below consists of a pai r of relatedwords fol lowed by four pairs of words. Select thepai r that best expr esses the r elat ion in theor iginal pair.

Unemployed : Wor k er

(a) fal low : land (b) unaware : sleeper

(c) wit : jester (d) renovated : house

59. Which of the fol lowing opt ions is the closest inmeaning to the wor d below:

Ci r cu i t ous

(a) cycl ic (b) indirect

(c) confusing (d) crooked

60. Choose the most appropr iate word from theopt ions given below to complete the fol lowingsentence:

H i s r at her casual r em ar k s onpol i t ics his lack of ser iousnessabout the subject .

(a) masked (b) bel ied

(c) betrayed (d) suppressed

Q. (61 – 65) car r y t w o mar k s each .

61. Har i (H ), Gi ta (G), I r fan (I ) and Saira (S) aresibl ings (i .e., brothers and sisters). Al l were bornon I st January. The age difference between anytwo successive sibl ings (that is born one afteranother) is less than 3 years. Given the fol lowingfacts:

I . Har i’s age + Gita’s age > I r fan’s age + Saira’sage

I I . The age difference between Gita and Saira is1 year. However, Gita is not the oldest andSai ra is not the youngest

I I I .There are no twins.

In what order wer e they bor n (oldest fir st )?

(a) HSIG (b) SGHI

(c) IGSH (d) IHSG

62. 5 ski l led wor kers can build a wall in 20 days; 8semi-skil led workers can build a wall in 25 days;10 unski l led workers can build a wall in 30 days.I f a team has 2 sk i l led, 6 semi-sk i l led and 5unskil led workers, how long wil l i t take to buildthe wal l?

(a) 20 days (b) 18 days

(c) 16 days (d) 15 days


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63. Modern war fare has changed from large ,scaleclashes of ar mies to suppr ession of civ i l ianpopulat ions. Chemical agents that do their worksi lent ly appear to be suited to such war fare; andr egr et fu l l y, t her e exi st peopl e i n mi l i t ar yestablishments who think that chemical agentsare useful tools for their cause.

Which of the fol lowing statements best sums upthe meaning of the above passage:

(a) Modern war fare has resulted in civi l st r i fe

(b) Chemical agent s ar e usefu l i n moder nwarfare

(c) Use of chemical agents in war fare would beundesirable

(d) People in mil i tary establishments l ike to usechemical agents in war

64. Given digits 2, 2, 3, 3, 3, 4, 4, 4, 4 how manydist inct 4 digit number s greater than 3000 canbe formed?

(a) 50

(b) 51

(c) 52

(d) 54

65. I f 137 + 276 = 435 how much is 731 + 672?

(a) 534

(b) 1403

(c) 1623

(d) 1513

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