GATE: 1999(SECTION 75 Marks 1. Write in your answer book at the space provided for QUESTION - 1 thv -:':rlpd' ,r tih: most appropriate answer tu All the following twenty hve multiple choice sl,b'i1,',i\\'ns by writing only the alphabet A, B, C, or D th<1t curresponds to your choice of the (\11'>\\('1. Write this alphabet only in the ANSWER column, against the corresro:~dins :',,:1.i~,-";T]< ;[ (!H: suo .question, (,,', " 1 ,,- 2:5) Rank of the matrix given below is A) 1.1 r3 2 R l -G -4 12 (11) 1 2 (c) 3 1.2 For the function $ = ax2 y - y3 to represent tl,e \ clocitv potenti"i vi 11_k<l; il.tid, V ¢ should be equal to zero. In that case, the value .J1 'a' ha" to be (11) --1 (by 1 (c) -3 d (d) 1 Wi) 3 1.3 If the velocity Vedl)r in - dimcn"iol"d flow field is gi '.'en by (c) zero 1.4 Laplace transform b5)2 (a) (a + of (a + bt)2 where '0' (d) -4 x k and '/J' are ,:on~tants is given by (b) 2b2 53 1 (0 , a~ S +(5)2 a~ (c) 5 , -+-+52 2ab 2ab b2 (d) --+--+52 53 1.5 Following are the values of a tunction y(x) : y(-l) == 5, y(O), yil) ce- S dy dx at x 0= 0 as per Newton's (11) 0 (c) 2,0 central difference scheme is (b) 1.5 (d) 3,0 1.6 Analysis of variance is concerned with (a) determining change in a dependent variable per unit change in an independent variable (b) determining whether a qualitative factor affects the mean of an output variable (c) determining whether significant correlation exists between an output variable and an input variable (d) determining whether variances in two or more populations are significantly different. 1.7 A concentrated force, F is applied ( perpendicular to the plane of the figure) on the tip of the bent bar shown in Figure. The equivalent load at a section close to the fixed end is (a) ForceF (b) Force F and bending moment FL (c) Force F and twisting moment FL F L -1 (d) Force F,bending moment F L, and twisting moment FL to- L -1 T 1 L 1.8 Which theory of failure will you use for aluminium components under steady loading (a) Principal stress theory (b) Principal strain theory . (c) Strain energy theory (d) maximum shear stress theory 1.9 For the planar mechanism shown in Fig. 1.9,select the most appropriate choice for the motion of link 2 when link 4 is moved upwards. (a) Link 2 rotates clockwise -2 a2u 2 a2u at = c -ax 2 represents the equation for (a) (b) (c) (d) Vibration of a stretched string Motion of a projectile ir..a gravitational field Heat flow in thin rod Oscillation ofa simple pendulum 1.11 Bolts in the flanged end of pressure vessel are usually pre - tensioned Indicate which of the following statments (a) Pre ~.•ensioning t (b) Pre - tensioning (c) Pre - tensioning (d) Pre - tensioning in NOT TRUE? helps to seal the pressure vessel increases the fatigue life of the bolts reduces the maximum tensile stress in the bolts helps to reduce the effect of pressure pulsations in the pressure vessel 1.12 If 'p' is the guage pressure within a spherical droplet, then guage pressure within a bubble of the same fluid and of same size will (a) p 4 (b) E 2 (c) P (d) 2p 1.13 If velocity of water inside a smooth tube is doubled, the n turbulent flow heat transfer coefficient between the water and the tube will (a) remain unchanged (b) increase to double its value (c) increase but will not reach double its value (d) increase to more than double its value A Stirling cycle and a Carnot cycle operate between 50° C and 350° C. Their efficiencies are 11s and 11c respectively. In this case, which of the following statements is true? (a) 11s > 11c (b) 11s = 11c (c) 11s < 11c (d) The sign of ( 11s - 11J depends on the working fluids used A Carnot cycle refrigerator operates between 250 K and 300 K. Its coefficient of performance is (a) 6.0 (c) 1.2 (b) 5.0 (d) 0.8 I\~ ,/ In a pulverised-fuel-fired large power boiler, then heat transfer from the burning fuel to the walls of the furnace is (a) by conduction only (b) By convection only (c) by conduction and convection (d) predominantly by radiation A gas turbine power plant has a specific output of 350 kJl kg and an efficiency of 34%. A regenerator is installed and the efficiency increases to 51 % . The specific output will be closest to (a) 350 kJ/kg (b) 468 kJ/kg (c) 525 kJ/kg (d) 700 kJ/kg Kinematic viscosity of air at 20° C is given to be 1.6 x 10-5 m2/ s. Its kinematic viscosity at 70° C will be vary approximately (a) 2.2 x 10-5 m2/ s (b) 1.6 x 10-5 m2/ s 5 m2 / s (c) 1.2 x 10(d) 3.2 x 10-5 m2 / s Which of the following statements does NOT apply to the volumetric reciprocating air compressor? (a) It decreases with increase in inlet temperature (b) It increases with decrease in pressure ratio (c) It increases with decrease in clearance ratio (d) It decreases with increase in clearance to stroke ratio efficiency of a 1.20 Amtllent air dry- bulb temperature iuv;c;[ is 45° C and wet bulb temperature is 27° C. Select the cooled pos.,ibh- (ond":,sing temperawre from the following for an evaporatively (i;) 30° C (d) 48° C 1.21 Identify the sl ress - state in the FLANCE portion of a PARTIALLY DRAWN CYLINDRICAL CUP when deep - drawing without a blank holder (u) Td1sile in all three directions (i)) No ::.tressin the flange at all, because there is no blank-holder (cl Vi) T'nsile stress in one direction and compressive in the one other direction Compressive in two directions and tensile in the third direction 1.2.2 Which of the following materials requires the largest shrinkage allowance, while making a pillterll for casting? (0) (c) Aluminium Cast (b) Brass (d) Plain Carbon Steel Iron 1.23 In Electro-Discharge Machinino \EDM), the tool is made of (17) Copper \c) Cast Iron 1.24 Choose the correct statement: (0) (b) High Speed Steel (d) Plain Carbon Steel A fixture is used to guide the tool as weil as to locate and clamp the workpiece (b) A JIg is used to guide the tool as well as to locate and clamp the workpiece (c) jigs arc used on CNC machines to locate and clamp the workpiece and also to guide the lOl)l (d) No arrangement to guide the tool is provided in a jig. 1.25 The first algorithm for Linear Programming was given by (a) Bellman (c) (b) Dantzig (d) van Neumann Kulm 2. Write in your answer book at the space provided for QUESTION - 2 the correct or the most appropriate answers to ALL. The following twentyfive multiple choice subquestions, by writing one or more of the alphabets /\, B, C, or D which correspond (5) to your choice of the answer (5) Write the alphabets only in the ANSWER column, against the corresponding NUMBER of the sub - question. 3 [.5 3] are -3 (b) 5 (d) -4 2.1 The eigenvalues of the matrix (0) 6 (c) -3 x=l f 2xy dx is equal to 2 y l) (a) (b) (c) 3 8 1t 1t ......•••. x 2 1C (d) 4 the variation 2--> In a flow field in x, y-plane, ~ \0 of velocity with time t is given by v=- tx- + yt)i (a) (c) - 2 dx and v =- (x + Y )i The acceleration of the particle in this field, occupying point (1,1) at time t =- 1 will be -+ 2 1 31 + (x":+4x) -+y=-xti-8 dx (b) (d) i7 51 dZy . dy , The above equation is a (a) partial differential equation (b) non-linear differential equation (c) non-homogeneous differential,equation (d) ordinary differential equation We wish to solve .12 - 2 = 0 by Newton Raphson technique. Let the initial guess b Xo =- 1.0 Subsequent estimate of x (i.e. Xl) will be (a) 1.414 (b) 1.5 2.0 (d) none of these Four arbitrary points ("\'1f\)' (x2, Y2)' (.13' Y3)' (.14' Y4) are given in the X, y - plane. Using the method of least squares, if, regressing y upon X gives the fitted line y =- ax + b; and regressing y upon x gives the fitted line y =: ax + b; and regressing x upon y gives the fitted line x = elf + d, then (a) the two fitted lines must coincide (b) the two fitted lines need not comcide (c) it is possible that ae = a 1 (d) a must be -(e) (' A thin walled cylindrical vessel of welllthickncs:, t and dIameter d ISHlled with gas to a gauge pressure of p. The maximum shear stress un 'he vessel wall WIll then be ~) ~ pd t pd (M ~ pd 2t pd ~ & 2.8 A lean elastic beam of given flexural rigidity, EI, is loaded by a single force F as shown in Figure. How many boundary conditions are necessary to determine the deflected centre line of the beam? (a) 5 (b) 4 (c) 3 UndefLected potition F (d) 2 2.9 As shown in Figure. a person A is standing at the centre of a rotating platform facing person B who is riding a bicycle, heading East. The relevant speeds and distances are shown in given figure. person a bicycle, heading East. At the instant under consideration, what is the apparent velocity of B as seen by A? (a) 3 mls heading East (b) 3 ml s heading West (c) 8 mls heading East T SII'\ 1 (d) '13 m I shead ing East 2.10 For the audio cassette mechanism shown in Figure given below where is the instantaneous centre of rotation (point) of the two spools? (a) Point P lies to the left of both the spools but at infinity along the line joining A and H (b) Point P lies in between the two spools on the line joining A and H, such that PH = 2 AP (c) Point P lies to the right of both the spools on the line joining A and H, such that AH = HP (d) Point P lies at the intersection of the line joining B and C and the line joining G and F 2.11 With regard to belt drives with given pulley diameters, centre distance and coefficient of friction between the pulley and the belt materials, which of the statement below are FALSE? (a) A crossed flat belt configuration can transmit more power than an open flat belt configuration (b) A "V"belt has greater power transmission capacity than an open flat belt (c) Power transmission is greater when belt tension is higher due to centrifugal effects than the same belt drive when centrifugal afects are absent. (d) Power transmission is the greatest just before the point of slipping is reached 2.12 Under repeated load ing a material has the stress-strain following statements is true? curve sho\yn in Figure, which of the Fig. The smaller the shaded area, the better the material damping (b) The larger the shaded area, the better the material damping (c) Material damping is an independent material property and does not depend on this curve (d) None of these 2.13 Consider the system of two wagons shown in Figure. The natural frequencies of this system are (a) (a) 0, 5 III (b) -,- JkJ2I Jk 2111 (c) ff Jk 1/1 2111 (d) 0, 2.14 Water flows through a vertical contraction from a pipe of diameter d to another of diameter d/2 ( see Figure). The flow velocity at the inlet to the contraction is 2ml s and pressure 200 kN 1m2. If the height of the contraction measures 2m ,then pressure at the exit of the contraction will be very nearly (a) 168 kN/m2 (b) 192 kN/m2 (c) 150 kN / m2 (d) 174 kN/m2 2.15 An aeroplane is cruising at a speed of 800 kmph at an altitude, where the air temperature 0° C. The flight Mach number at this speed is nearly (a) 1.5 (b) 0.254 (c) 0.67 (d) 2.04 is ',l, ~ (' nrL'Fose': Ic' c·,'11a 1 iHm didIl1d~>r \vlrC with enamel paint (k 0:: 0.1 \V I mK) to increase heat _11> 1 r!1 \(' (I!r-ll i,' hecll tt ,1I1sfercoeffic.ient is 100 W 1m2 K, then optimum thickness of IIIlc1Ila.' t p.-1Ull :"ilt\ 'iuld b{~' (II) CU~; ;;:111 (b) 0.5 mm ;" I:; l ,,;,-,:,'! \"';:'i (c) ).1 '7 1\" 1 illl: ",,',,1,,-'" dw rn:jdYIC':<"lCcy~tC'lll executes (d) :2 nun Choose the correct statement (s) frorn il process, the iUUCi" II:g (b) (L; (Ii) (1/) No hC<l1 is (lclP"IU1Cd r"u work is done Nt> lllclS:,iiuws across the boundary of the system Nu (hel"',.il reacliun takes piace within the system 2.18 Th(' sikn('{"r \ If an internal (II) reduces 1l00:5e (/J) (c) combustion engine (BSFC) decn'.,sc mClle'di,e: brake specific fuel consumption f/:TC (d) has no effect on ;Is efficiency 2.n Select slaleJ1i€~,~S lrum List II matching the processes in List I. Enter your answer as D,C if the co, (cll di(lICe fur( I) i" (Gi and that for (2) is (C) List II (A) Dry bulb temperature dc'",,' - point temperature (B) f)nv- point temperature increases, but decreases. increases and List I (I) (",.Lng ilnd d('lnllludification 12) Ch,'micc,j ,1cJ-T' Idification dry - bulb temperature remains uncharged (C) Dry - bulb and wet - bulb temperahlres decrease (D) Dry - bulb temperature decreases, but, dew - point temperature increases 2,20 ('PJ'tt;'rt Sek'cl statements from List II matching the processes cboic'2 for (l) is (A) and that for (2) is (B) List [ (11 intere. Y'iing (7) in List I. Enter your answer as A,B if the List II compression (A) No hedt trc1J1sfer during compression (8) Reduces low pressure compressor work (C) Heat rejection during compression (D) Reduces high pressure compressor work out of the Isothermal :,:,21 for hutt -wdding 40 mm thick steel plate's. when the expected quantity welding oelS of such jobs is 5000 process pel month over" period of 10 year,:" choose the best suitable folloWlllg c1\'aihblc alternatives. \[1) (t'I ~';ublller~ed ,lie wdding hl"clrnn hpaIn w,;,ldinr, (b) O}\.y - du:::iyknl' (d) MIC welding welding 2.22 Wlut i, clpprOXlrnate percentage change is the life, t, of a tool with zero rake angle used in orthogol1,d cutting when its clearance angle, C'I., is changed from 100 to 70 ? (Hint: Flank wear rilte is proportional to cot a) (1/) 30'''., increase (b) 30%, decrease (e) 70'Y., increase (d) 70% decrease 2.23 Suppose X is a normal random variable with mean 0 and variance 4. Then the mean of the absolute value of X is (a) - 1 2.fi (b) .fin 2.fi ..r; 2 (d) ..r; In computing Wilson's economic lot size for an item, by mistake the demand rate estimate used was 40% higher than the tree demand rate. Due to this error in the lot size computation, the total cost of setups plus inventory holding per unit time. would rise above the true optimwn by approximately (a) 1.4'X, (b) 6.3% (c) 18.3% (d) 8.7% At a production machine, parts arrive according to a poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 minutes. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine! in queue)? (a) (c) 0.0247 0.0173 (b) 0.0576 (d) 0.082 Answer any FlFI'EEN questions. Each question carries 5 marks. 3. Consider the signbo<lrd mounting shown in Fig. given. The wind load acting perpendicular to the planl' of the figure is F = 100 N. We wish to limit the deflection, due to bending, at point A of the hollow cylindrical pole of outer diameter 150 mm to 5 mm. Find the wall thickness --4 1m L--l.._ for the pole. Assume E 2.0 x 1011 N 1m2. ~ (5) cc 4. The peak bending stress at critical section of a component varies between 100 MN/m2 and 300 MN 1m2. The ultimate tensile strength of the material is 700 MN I m2"yield point in tension is 500 MN 1m2, endurance limit for reversed bending is 350 MN 1m2. Findthidactor of safety. . (5) 5. A proper base isolation is to be designed for mounting a sensitive instrument as shown 1.11 Figure. At the point of mounting, the base vibration due to other disturbances is indicated in the figure. If the permitted absolute displacement amplitude on the rigid mounting pad is 0.001 mm, find the stiffness of the spring. Assume that the total mass of the mounting pad and the instrument is 50 kg. (5) _Jll(l) Movnt1ner pad J y(tloo.u'n(,oontlmm 21. The lives of two tools, A and B,governed by the equations vt0125 == 2.5 and vt0250 == 7 respectively in certain machining operation where v is the cutting speed in mls and t is the tool life in seconds. Find out the speed v* at which both the tools will have the same life. Also calculate the corresponding tool life t*. (a) (b) If you have to machine at a cutting speed of 1 ml s, then which one of these tools will you choose in order to have less frequent tool changes? 22. The bore diameter, 0, of a plain ring having a height of 35 mm was measured using two spherical balls, each of diameter d == 25.000 mm. The ring was placed on a surface table and then both the balls were placed inside the ring. In this position, the height, h, of the top of the upper ball from the surface table was found to be 42.000 mm. (a) Derive the expression for 0 in terms of d and h. (b) What is the bore diameter, 0, of the ring? I ANSWERS I 1.1 (a) 1.9 (b) 1.17 (a) 1.25 (b) 2.1 (a, d) 2.9 (d) 2.17 (a, b, c) 2.24 (c) 2.2 (a) 2.10 (d) 2.18 (a) 2.25 (c) 2.3 (d) 2.11 (c) 2.19 (c, d) 2.4 (d) 2.12 (a) 2.20 (c, b) 2.5 (b) 2.13 (c) 2.21~ 2.6 (d) 2.14 (c) 2.22 (b) 2.7 (d) 2.15 (c) 2.23 (b) 2.8 (d) 2.16 (b) 1.2 (d) 1. 10 (a) 1.18 (a) 1.3 (d) 1.11 (c~ 1. 19 (a) 1.4 (c) 1. 12 (d) 1.20 (d) 1.5 (b) 1.13 (c) 1.21 (b) 1.6 (d) 1.14 (c) 1.22 (a) 1.7 (c) 1.15 (b) 1.23 (a) 1.8 (d) 1.16 (c) 1.24 (b) . •• 1.1 Since the valfue of I~ a~ I EXPLANATIONS ~:61 =0 I 1.2 .. Rank of the matrix is less than 2. Given, 4> = axZy _ y3 -'" = axz -3yz and -z = - 6y Oy Oy alj> - aZIj> aZIj> =2 ax axy and - z ax = Zay Now, or or aZIj> aZIj> -z +-z =0 ax ay Zay - 6y = 0 =3 1.3 Given, ii = 2xyi +(2l-xz)] a a ax 2xy k ~ = i(O) ::-j(O) +k [ : (2yz - x) Z - ~ (2XY)] o (-2x -2x)k = -4xk. Z + bZtZ + Zabt. 1.4 (a + bt)Z = a Laplace transform 1 of 1 = s of tn Laplace transform = -'-I S ~ +1 .. Laplace transform of (a+ bt)Z y(l)-y(-l) =- aZ S + 2bz -3 S +-z 2ab S 1-(-1) 8-5 = -2- = 1.5 1.6 Analysis of variance is used in comparing two or more populations, e.g. Different types of manures for yielding a single crop. 1.14 Efficiency 11 of Stirling cycle is equal to Carnot cycle only when regenerative arrangement is employed. Coefficient of performance (C.O.P) = T 1 T _ T z = 250 300 _ 250 =5 Viscosity of gas increases with increasing temperature. A-A.I = IS-A. 3 = 3 -3-A. I (5-A.) (-3-A.)-9 - [15 + SA.- 3A.- A.2] - 9 A.2-9-15-2A. = A.2- 2A.- 24 = 0 2 A. - 6A.+ 4A.- 24 = 0 A.(A.-6)+4(A.-6) = 0 (A.-6)(A.+4)= 0 A. = 6, or -4. 2.2 Area of half circle = nr2 2 = -- n . 12 2 =2 n 4n2 n 4.12 2 Moment of area about y-axis = Area x = -2 x -= -3 3n 3n 2.3 v = (x2 + yt)i v = x2 + yt, At t = 1, v(l,l) = 1+1x 1=2 - au ax = 2x=2xl=2 ' -=y=l-=t=l at By au au ax u- au ax + v- + w- + - au By au az au at (2 x 2 + 0 + 0 + 1) i = 5i 2 _ _ 2.5 X -2-0,x1-xOf(xo)=f(l) f(xo) j'(x ) o _ .!.. _ -1+ 2 -1.5 = 12- 2 =-1 j'(x) = 2x f'(xo) = 2xo = 2(1) = 2. 2.6 y=ax + b x=cy +d From equation (ii), x-d=cy or 1 c From equations (i) and (iii), ........(i) ...... (ii) y= -x- d a = - and b = - d. c 1 2.7 Maximum shear stress P1 - P2 2 pd_pd = 2t 4t =- pd 8t 2 ., pd P1 = -2t andp d2y M = EI dx2 2 =- pd 4t Since the above equation is second order differential equa~ion, therefore two boundary conditions are needed to solve y. Relative velocity = 8 - (-1 x 5) = 13 m/ s towards East. From continuity equation, At VI == A2 V2 "4 2" or V2 == 8m/s. Applying Bernoulli's theorem, - 1t (d)2 x v2 P1 w + V1 -2 g 2 +Z1 (2)2 200 x 1000 + --+0 2x 9.81 9810 -+---+2 w 2x 9.81 P2 (8)2 P2 = 150.38 kN/m2 v M = C = 800 kmph 800 x 1000 C = 3600 x 331.2 ,=0.67 C = JYRT = .J1.4 x 0.287 x 273 x 1000 = 331.2 m/ s. 2.16 Critical radius of insulation, rc = h k 0.1 = 100 = 0.001 = 1 mm. Radius of min = 0.5 mm . . , Thickness of coating = 1 - 0.5 = 0.5 mm. 2.17 For an isolated system, Q = a, W = 0 and mass does not flow through the system. 2.22 Wear rate is proportional to tany, i.e. Wear rate oc tan y Where, y = Clearance angle When clearance angle is decreased from 10° to 7°,then tan 10° - tan 7° percentage change in t = ----x 100 = 30.37 % decrease. tan 10° Mean of the absolute value of X = 0 V; = 2 V; = 02 [2 [2 2J2 ,J; o = 4 2. EOQ=-- ~2RCo CC Inventory cost = ~2RCoCc Cost rise = ~2 x 1.4RCoCc = 1.183 ~2RCoCc 1.183-1 1 2.25 Given, A. 0.35/min., = Il = 2' /min 1 = 0.5 min. p = !: = Il 0.35 =0.7 0.5 Probability of 'n' numbers in the system, Pn (t) = Pn (1 - p) = (0.7)8(1- 0.7) = 0.0173. SECTIONB 3. Given F = 100 N, do = 150 mm 0= 5 mm, . E = 2 x 1011N/m2 Thickness of pole = ? The system of signboard mounting can be considered as a cantilever loaded at A i.e, W = 100 N and also having anticlockwise moment of M = 100 x 1 = 100 Nm at the free end. Deflection of a cantilever having concentrated load at the free end, Wz3 MI2 Y = 3EI + 2EI 100 x 53 0.005 = 3 x 2 x 10 x I 1= 0.0000053 11 A 1 s'" 100 x 52 + 2 x 2 x 10 11 xI 1t 64 (d40 - d4) = 0.0000053 i 0.0000053 x 64 ----1£ (0.15)4- dt di = 0.1415 m di-do Wall thickness of pole = -2- = 150-141.5 2 = 4.25 mm