syl sampleqr

8/19/2019 Syl SampleQR

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Test Code : QR ( Short answer type ) 2005

M.Tech. in Quality, Reliability and Operations Research

The candidates applying for M.Tech. in Quality, Reliability and OperationsResearch will have to take two tests : Test MIII (obective type! in the forenoonsession and Test QR ( short answer type ! in the afternoon session.

"or Test MIII, see a different #ooklet. "or Test QR , refer to this #ooklet

ONLY.$f you are fro% Statistics / Mathematics Stream, you will be re&uired toNS!"R #RT I. $f you are fro% one of the "n$ineerin$ Streams, you will be re&uired toNS!"R #RT II.

$n #RT I, you will find in Test QR , a TOTL o% T"N &'() &uestions, dividedin T!O *roups : Statistics + #robability ' each carryin$ I-" &)&uestions. ou will be re&uired to answer a TOTL o% SI &0) &uestions ' taking T L"ST T!O &1) fro% each $roup.

$n #RT II, there will be si) groups *+*-, each containing three &uestions. ouwill be re&uired to answer fro% roup *+ (Mathe%atics! and fro% any threegroups fro% the re%aining five groups as per the instruction given within eachgroup.


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Syllabus

#RT I 2 STTISTI3S / MT4"MTI3S STR"M

Statistics 5S'6

/escriptive statistics for univariate, bivariate and %ultivariate data.

0tandard univariate probability distributions 1#ino%ial, 2oisson, 3or%al4 and

their fittings, properties of distributions, sa%pling distributions.

Theory of esti%ation and tests of statistical hypotheses.

Multiple linear regression and linear statistical %odels, 53O65.2rinciples of e)peri%ental designs and basic designs 17R/, R#/ 8 90/4.

*le%ents of nonpara%etric inference.

*le%ents of se&uential tests.

0a%ple surveys ' si%ple rando% sa%pling with and without replace%ent,

stratified and cluster sa%pling.

#robability 5S16

7lassical definition of probability and standard results on operations withevents, conditional probability and independence.

/istributions of discrete type 1#ernoulli, #ino%ial, Multino%ial,

ypergeo%etric, 2oisson, eo%etric and 3egative #ino%ial4 and

continuous type 1;nifor%, *)ponential, 3or%al, a%%a, #eta4 rando%

variables and their %o%ents.

#ivariate distributions (with special e%phasis on bivariate nor%al!, %arginal

and conditional distributions, correlation and regression.

Multivariate distributions, %arginal and conditional distributions,regression, independence, partial and %ultiple correlations.

Order statistics 1including distributions of e)tre%e values and of sa%ple

range for unifor% and e)ponential distributions4.

/istributions of functions of rando% variables.

Multivariate nor%al distribution 1density, %arginal and conditional

distributions, regression4.


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Syllabus

#RT II 2 "N*IN""RIN* STR"M

Mathematics 5"'6

*le%entary theory of e&uations, ine&ualities.

*le%entary set theory, functions and relations, %atrices, deter%inants,

solutions of linear e&uations.

Trigono%etry 1%ultiple and sub%ultiple angles, inverse circular functions,

identities, solutions of e&uations, properties of triangles4.

7oordinate geo%etry (two di%ensions! 1straight line, circle, parabola,

ellipse and hyperbola4, plane geo%etry, Mensuration.

0e&uences, series and their convergence and divergence, power series, li%it

and continuity of functions of one or %ore variables, differentiation and its

applications, %a)i%a and %ini%a, integration, definite integrals areas using

integrals, ordinary and partial differential e&uations (upto second order!,

co%ple) nu%bers and /e Moivre>s theore%.

"n$ineerin$ Mechanics 5"16

"orces in plane and space, analysis of trusses, bea%s, colu%ns, friction,

principles of strength of %aterials, workenergy principle, %o%ent of

inertia, plane %otion of rigid bodies, belt drivers, gearing.

"lectrical and "lectronics "n$ineerin$ 5"76

/.7. circuits, 57 circuits (+φ!, energy and power relationships,Transfor%er, /7 and 57 %achines, concepts of control theory and

applications.

3etwork analysis, = port network, trans%ission lines, ele%entary

electronics, analog and digital electronic circuits.

?


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Theromodynamics 5"86

9aws of ther%odyna%ics, internal energy, work and heat changes,

reversible changes, adiabatic changes, heat of for%ation, co%bustion,

reaction, solution and dilution, entropy and free energy and %a)i%u% work

function, reversible cycle and its efficiency, principles of internal and

e)ternal co%bustion engines.

"n$ineerin$ #roperties o% Metals 5"6

0tructures of %etals, tensile and torsional properties, hardness, i%pact

properties, fatigue, creep, different %echanis% of defor%ation.

"n$ineerin$ 9ra:in$ 5"06

7oncept of proection, point proection, line proection, plan, elevation,

sectional view (+

st

angle@?

rd

angle! of si%ple %echanical obects, iso%etricview, di%ensioning, sketch of %achine parts.

(;se of set s&uare, co%pass and diagonal scale should suffice!.

A


5/22

SM#L" Q;"STIONS

#RT I2STTISTI3S N9 #RO


6/22

is an appro)i%ately unbiased esti%ator of N for large N .

I.(a! 9et )+, )=, . . ., )n be a rando% sa%ple fro% the rectangular population withdensity

+ @ θ , H E ) E θ f()! D H otherwise

7onsider the critical region )(n! L H. for testing the hypothesis H : θ D +,where )(n! is the largest of )+, )=, . . ., )n.


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re=ised value of "ratio and draw your conclusions.

P. $f J+, J=, J? constitute a rando% sa%ple fro% a #ernoulli population with%ean p, show why 1J+ F =J= F ?J? 4 @ - is not a sufficient statistic for p.

. $f X and Y follow a trino%ial distribution with para%eters n, θ + and θ =, show

that

(a!+

=

+

!(!@(

θ

θ

−−

== xn

x X Y E ,

(b! =+

=+=

!+(

!+(!(

!@( θ

θ θ θ

−

−−−==

xn

x X Y V

"urther show, in standard notations,

(c!+

==+

=++ θ

θ θ

−=n

E V , (d!+

=+==+

+

!+(

θ

θ θ θ

−−−

=n

V E ,

(e! !+(!( == θ θ −= nY V

. 9ife distributions of two independent co%ponents of a %achine are known to be e)ponential with %eans µ and λ respectively. The %achine fails if at least one

of the co%ponents fails. 7o%pute the chance that the %achine will fail due tothe second co%ponent. Out of n independent prototypes of the %achine m of the% fail due to the second co%ponent. 0how that !(@ mnm − appro)i%atelyesti%ates the odds ratio µ λ θ = .

*RO;# S>1 2 #robability

+. (a! 5 coin is tossed an odd nu%ber of ti%es. $f the probability of getting%ore heads than tails in these tosses is e&ual to the probability of getting

%ore tails than heads then show that the coin is unbiased. (b! "or successful operation of a %achine, we need at least three co%ponents

(out of five! to be in working phase. Their respective chances of failureare P, A, =, and +=. To start with, all the co%ponents are inworking phase and the operation is initiated. 9ater it is observed that the%achine has stopped but the first co%ponent is found to be in working phase.


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(c! 5 lot contains =H ite%s in which there are = or ? defective ite%s with

probabilities H.A and H.- respectively. $te%s are tested one by one fro% the lot unless all the defective ite%s are tested.


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I.(a! 9et J+, J=, . . . , be a se&uence of independent and identically distributed;(H,+! variables. $f Cn D (!+@n , then show that Cn converges in probability toso%e constant 7. 5lso find 7.

(b! Three out of every four trucks on the road are followed by a car, while onlyone out of every five cars is followed by a truck.


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H otherwise

i! "ind c.ii! "ind the conditional e)pectation, *(J D )!, for H E ) E +.

. 0uppose in a big hotel there are N roo%s with single occupancy and alsosuppose that there are N boarders. $n a dinner party to celebrate the%arriage anniversary of one of the boarders they start drinking alcohol totheir heartsK content and as a conse&uence they beco%e unable to identifytheir own roo%s.


11/22

*RO;# "?' 2 Mathematics

+(a! 9et f(x) be a polyno%ial in x and let a, b be two real nu%bers where a ≠ b.0how that if f(x) is divided by (x - a)(x - b) then the re%inder is

ab

a f b xb f a x

−−−− !(!(!(!(

.

(b! "ind if )7osy F y7os) D +.

=.(a! 9et 5 be the fi)ed point (H,A! and # be a %oving point (=t, H!. 9et M be the%idpoint of 5# and let the perpendicular bisector of 5# %eets the ya)isat R. "ind the e&uation of the locus of the %idpoint 2 of MR.

(b! $nside a s&uare 5#7/ with sides of length += c%, seg%ent 5* is drawnwhere * is the point on /7 such that /* D I c%.The perpendicular bisector of 5* is drawn and it intersects 5*, 5/ and#7 at the points M, 2 and Q respectively."ind the ratio 2M : MQ.

?(a! *valuate the value of ?.+@=.=P +@A. + +@. ...up to infinity.

(b! 0olve: y x

edx

yd ?==

=−

=

A.(a! 0how that

=

+!4=1(

?

+

=

+

+

+

====

+

++

++

+∞→termsnupto

nnn

Limn

.

(b! Test the convergence of the series ∞++++ OA

A

O?

?

O=

= AA??== x x x x . 5ssu%e

x L H and e)a%ine all possibilities.

I.(a! "ind the li%it of the following function as ) → H.

0in(!

++


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+==≤+byax

(b! $f the line a) F by F c DH is a nor%al to the curve )y D+ then show

that a⋅ b E H.

-.(a! $f ω is a co%ple) cube root of unity then show thata? F b? F c? ?abc D (a F b F c! (a F bω F cω=!(a F bω= F cω!.

(b! $f a D + F F F . . . . b D ) F F F . . . .c D F F F . . . .

then show that a? F b? F c? − ?abc D +.

P(a! 7able of a suspension bridge hangs in the for% of a parabola and is attachedto the supporting pillars =HH % apart. The lowest point of the cable is AH % belowthe point of suspension. "ind the angle between the cable and the supporting pillars. 0tate all the assu%ptions involved.

(b! 0how that

4.?I=I-A=M-MM1+=M

+M ++++= θ θ θ θ θ ososososos

(a! *valuate the following integrals directly and co%pare the%.

and a ,+≤ x b +≤ y

(b! /eter%ine ), y and B so that the ? ) ? %atri) with the following row vectorsis orthogonal : (+ @ U?, + @ U?, + @ U?!, (+@ U=, +@ U=, H!, (), y, B!.

*RO;# "?1 2 "n$ineerin$ Mechanics

∫∫ dxdy ∫∫ dxdy

+=


13/22

+.(a! The si%ple planar truss in the given "ig.+ consists of two straight twoforce

%e%bers 5# and #7 that are pinned together at #. The truss is loaded by adownward force of 2D+= V3 acting on the pin at #. /eter%ine the internala)ial forces "+ and "= in %e%bers 5# and #7 respectively. (3eglect theweight of the truss %e%bers!.

"ig. +

(b! /erive the e)pression for %o%ent of inertia $ of the shaded hollowrectangular section ("ig. =!.

"ig. =

=.(a! 5 turbine rotor weighs =H tonnes and has a radius of gyration of +.PI %eter when running at =HH rp%. $t is suddenly relieved of part of its load and itsspeed rises to =HI rp% in + sec. "ind the unbalanced unifor% turning%o%ent.

(b! 5n 5lu%iniu% thinwalled tube (radius@thickness D =H! is closed at eachend and pressuriBed by - M2a to cause plastic defor%ation. 3eglect theelastic strain and find the plastic strain in the circu%ferential (hoop!direction of the tube. The plastic stressstrain curve is given by σ D +PH(strain rate!H.=I.

+?


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?.(a! 5 unifor% ladder I % long and +A kg %ass is placed against a vertical wall

at an angle IHH

to the horiBontal ground. The coefficient of friction between ladder and wall is H.= and between ladder and ground in H.I.7alculate how far up the ladder a %an of -? kg. can cli%b before the ladder shifts.

(b! /eter%ine the dia%eter of a steel shaft rotating at an angular velocity of ?HHrp% trans%itting IHH 2. The allowable stress D HH kg@c%=. The allowableangle of twist D H.IH per %, D W +HI kg@c%= .


15/22

ii! The e)tension of the tie rod under load ( *D =HA 3@%= !

iii!The %ini%u% dia%eter of the tie rod if a factor of safety of =.I isapplied to the proof stress

(b! "ind the width of the belt necessary to trans%it ++.=I V< power to a pulley of dia%eter ?HH%% when the pulley %akes +-HH rp%. 5ssu%e thecoefficient of friction between the belt and the pulley is H.== and angle of contact is =+Ho. Ma)i%u% tension in the belt will not e)ceed +H3@%%width.

*RO;# "?7 2 "lectrical and "lectronics "n$ineerin$

+.(a! 5 centrifugal pu%p, which is geardriven by a /7 %otor, delivers +H kg of water per %inute to a tank of height ++ %eter above the level of the pu%p./raw the block diagra% of the overall arrange%ent. /eter%ine input power across the gearing and current taken by the %otor operated at ==Hvolt provided the efficiency of the pu%p, gearing and %otor respectively bePH, PH and H only. (Take g D . %s =!.

(b! The r%s value of a sinusoidal alternating voltage at a fre&uency of IH B is+IIvolt. $f at t D H it crosses the Bero a)is in a positive direction, deter%inethe ti%e taken to attain the first instantaneous value of +II volt. ow %uchti%e it takes to fall fro% the %a)i%u% peak value to its halfN *)plain withsuitable wavefor% .

=.(a! On fullload unity power factor test, a %eter having specification of =?I 6and I5 %akes -H revolutions in - %inutes, but its nor%al speed is I=Hrevolution@V


16/22

through that branch of the circuit containing capacitor. (5ll

resistances@reactances are in oh%s!.

"ig. I

(b! "ig. -

Refer "ig. -. "ind the e)pression for V H.


17/22

(b! $n the =port network given below, the para%eters at two parts are related

by the e&uations,

6+ D 56= #$= $+ D 76= /$=

i!"ind e)pressions for 5,#,7 and /ii! 0how that 5/ #7 D +iii!


18/22

*RO;# "?8 @ Thermodynamics

+.(a! $n a ther%odyna%ic syste% of a perfect gas, let ; D f (6,T! where ;, 6 andT refer to internal energy, volu%e of a gra%%olecule of the substance andte%perature (in absolute scale! respectively. 5n a%ount of heat δQ isadded so that the volu%e e)pands by δ6 against a pressure 2. 2rove that:

7 p ' 76 D ! "

"

V

V

# !

+

δ

δ

δ

δ

where 7 p and 76 stand for specific heat at constant pressure and specific

heat at constant volu%e respectively.

(b! H.+I cu.%. of air at a pressure of +.H- kg@c% = is co%pressed to a volu%e of H.HH cu.%. at ?-+ kg@c%=. 7alculate (i! the &uantity of heat reected, (ii!change in internal energy if the process of co%pression is a! 5diabatic b!2olytropic with n D +.?.

=.(a! 5 co%pression ignition engine has a stroke of = c% and a cylinder dia%eter of + c%. The clearance volu%e is API c%?. The fuel inectiontakes place at constant pressure for A.I of the stroke. "ind the air standard efficiency of the engine assu%ing that it works on diesel cycle. $f the fuel inection takes place at +H of the stroke, find the loss in air standard efficiency.

(b! 5 diesel engine has a co%pression ratio +A to + and the fuel supply is cutoff at H.H of the stroke. $f the relative efficiency is H.I=, esti%ate theweight of fuel of a calorific value +HAHH k.cal per kg that would be

re&uired per horsepower.?.(a! 7alculate the change in entropy of saturated stea% at a given pressure such

that the boiling point D +I=.- o7 and the latent heat at this te%perature DIH?.- cal@g%. 1;se 9og e +.I- D H.AAI.4

(b! /raw the pv and T−Φ diagra%s for a diesel cycle in which + kg of air at+ kg @ c%= and H H7 is co%pressed through a ratio of +A to +. eat is thenadded until the volu%e is +.P ti%es the volu%e at the end of co%pression,

+


19/22

after which the air e)pands adiabatically to its original volu%e. Take

7v D H.+- and γ D +.A+.

A.(a! The appro)i%ated e&uation for adiabatic flow of super heated stea%

through a noBBle is given by pvn D constant. 0how that

p= @ p+ D (= @ (nF+!! n @ (n+!

where p+ D pressure of stea% at entry Y p= D pressure of stea% at throatand p= @ p+ is the critical pressure ratio.

(b! The dry saturated stea% is e)panded in a noBBle fro% pressure of +H bar to pressure of A bar. $f the e)pansion is super saturated, find the degree of under cooling.

I. Three rods, one %ade of glass (k D +.H


20/22

%echanical properties of the steel are σu D +HH M2a, σH D +H+H M2a and

σe D I+H M2a. /eter%ine the bar dia%eter to give infinite fatigue life based on a safety factor of =.I.

= (a! The stress intensity for a partialthrough thickness flaw is given by V D!=@( t a$e%a ΠΠ where, a is the depth of penetration of the flaw

through a wall thickness t. $f the flaw is I %% deep in a wall += %% thick,

deter%ine whether the wall will support a stress of +P= Mpa if it is %ade

fro% a special 5lalloy. 1"racture toughness (V $7! of %aterial independent of crack length, geo%etry or loading syste% D =A M2a %.4

= (b! 5 =I %% dia%eter bar is subected to an a)ial tensile load of +HH k3. ;nder

action of this load a =HH %% gauge length is found to e)tend H.+ )+H? %%.

/eter%ine the %odulus of elasticity of the bar %aterial

?.(a!


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+.(a! 5 triangular pris% of AH%% base and -H%% height is lying on 2 on one

of its rectangular faces with its a)is inclined at AIo to 62. $t is sectioned bya horiBontal plane at a distance of +I%% above the ground. /raw its top

view.

(b! Two balls are vertically erected to +c% and ?H c% respectively above the

flat ground. These balls are away fro% a ? c% thick wall (on the ground!

by += c% and =+ c% respectively but on either side of the wall. The

distance between the balls, %easured along the ground and parallel to the

wall is =P c%. /eter%ine their appro)i%ate distance .

=. (a! 0ketch the profile of a s&uare thread, knuckle thread and a whiteworththread showing all relevant di%ensions in ter%s of the pitch.

(b! 0ketch:

i! single riveted lap oint,ii! double riveted lap oint chainriveting,iii! double riveted lap oint BigBagriveting, and

iv! single cover single riveted butt oint.

?.(a! /raw the iso%etric view of an octahedron erected vertically up on one of its vertices. (/istinct free hand sketch only.!

(b! /ia%eter of a hollow spherical %etallic ball with si) tiny holes is +H c%.The holes are of sa%e siBe and are placed on both sides of the three principal a)es of the ball. ou are given si) cylindrical rods with conical

top as shown in "ig. P. *nter the si) rods through these si) holes so that the

conical tops of the rods ust touch the centre of the ball. 3ow draw any twoviews of the whole arrange%ent in third angle proection. 1ou %ayconsider dia%eter of the hole is ust enough to allow the rod to enter.4

=+


22/22

"ig. : 0ectional 6iew of the Rod

A. 5 parallelepiped of di%ension +HH×-H×H is truncated by a plane which passesthrough I, AI and -I unit distance on the associated edges fro% the nearesttop point of the obect. /raw the iso%etric view of the truncated solid obect.$n third angle proection %ethod, draw its plan. (5ll di%ensions are in %%!.

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