USN O6MAT41

Max. Marks:100

(07 Marks)

(06 Marks)

isv:r'cos20_ rcose+2.(07 Marks)

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Fourth Semester B.E. Degree Examiii , Dec.201" 5lJan.20l6Engineering Mathematics - IV

Note: Answer any FIVE full questions, selectingatleast TI,VO questions from each part.

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PART _ A

I a. Use Taylor's series method upto 4th degree terms find y at the points x : 0.1 given that :

dv))f=^-+Y-'Y(o)=l'By Runge - Kutta method of order four, solve

y(0.4), taking step - length h:0.2.c. By using the Milne's predictor - Corrector method, find an approximate solution of the

.dv1 -equation #=*'-r, at the point x : A.4, given that y(0) : 1, y(0.1) : 0.9051,

Time: 3 hrs.

3a.h.

4a.

(06 Marks)

dv \/-Ythe equation ]l with y(0) - I, forox y+x

(07 Marks)

y(0.2) :0.8212, y(0.3) :0.7492. Apply corrector formula twice.

2 a. If (z) : u * iv is an analytic function, then prove that :

r^ -2 r^ t2

ffrtrr,l] .ffrEr,t

) =lrr,tl'

Find the analytic function (z) :u * iv, whose imaginary part

Find the bilinear transformation which maps the points zw : i, 0, -i, respectively. Hence find the image lzl < l.

State and prove Cauchy's integral formula. (06 Marks)

Expand f1z1= . -3-asLaurentseriesvalidfor:i) 1< lrl<2 11)lz- 1l> 1.(z-t)(2- z)(07 Marks)

State Cauchy's residue theorem. Using this theorem evaluate , t--1 , dz, where C isi z(l- 4z')

lzl: r-

Prove that Pn (x1=-l l.[.' - l)" ]. where n is a positive inreger.

lf-2n dx n '

Express (x) : 3x3 - x2 i 5x - 2 in terms of Legendre polynomials.

(06 Marks)

(07 Marhs)Obtain the series solution of Bessel's differential equation in the form y: AJn(x) + BJ,(x).

(07 Marks)I of2

b.

c.

PART _ B O6MAT4I

5 a. Fit a parabola y: a-f bx + cx2 by the method of least squares to the following data:

(07 Marks)While calculating the correlation coefficient between x and y from 25 pairs of observattons,

a person obtained the following values. Xxi : 125, Zxi2 : 650, Xy, : 100, tyi2 : 460,

Ixiyi : 508. It was later discovered that he had copied down the pairs (8, 12) and (6, 8) as (6,

14) and (8, 6) respectively. Obtain the correct value of the correlation coefficient. (07 Marks)

In a bolt factory there are four machines A, B, C, D manufacturing respectlely 20oh, l5oh,40Yo of the total production. Out of these 5oA, 40 ,3oh,2oA are defective. if a bolt is drawn at

random was found defective, what is the probability that it was manufactured by A or D.(06 Marks)

A pair of coins is tossed. A random variable X represents the number of heads turning up.

Find the discrete probability distribution for X. Find its mean and variance. (06 Marks)2o/o of the fuses manufactured by a firm and found to be defective. find the probability that a

box containing 200 fuses contains i) no defective fuses ii) 3 or more defective fuses iii) at

least one detective fuse. (07 Marks)c. The mean weight of 500 students at a certain college is 50 kgs and the standard deviation is

6 kgs. Assuming that the weights are normally distributed, find the number of studentsweighing i) between 40 and 50 kgs and ii) more than 60 kgs. (07 Marks)

Explain the following terms :

i) type I and type II errors ii) null h-vpothesis iii) level of significance. (06 Marks)A coin was tossed 400 times and the head turned up 21.6 times. Test the hypothesis that thecoin is unbiased at 5o/o level of confidence. (07 Marks)

c. A certain stimulus administered to each of 12 patients resulted in the following change inblood pressure '. 5,2,8, -1, 3,0, 6, ), l, 5,0, 4 (in appreciate units). Can it be concludedthat, on the whole, the stimulus will change the blood pressure? Use t6 65(1 l) : 2.20t.

(07 Marks)

8a. The joint probability distribution of two random variables X and Y is given below :

Find:i) Marginal distribution of X and Yii) Covariance of X and Y. (06 N{arks)Define a stochastic matrix. Find the fixed probability vector of the regular stochastic matrix.

[o 1 olttP=l 0 0 11. (07Ntarks)

lY, Y, ol

A psychologist makes the following assumptions concerning the behaviour of micesubjected to a particular feeding schedule. For any particular trial, S}oh of the mice that wentright on a previous experiment will go right on this trial, and 60oh of those mice that wentleft on the previous experiment will go right on this trial. Suppose 50% of the mice wentright on the first trial. Find the predictionofthe psychologist for the next twotrials. (07 Marks)

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c.

6a.

b.

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b.

b.

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3a.

4a.

a. Frove that div(curlA) = g.))

b. Find div F and curlF where

c. Show that the vector fr = 13*'+

suchthat p=grad$.

F = V(x3 + yt + z) -3xyz) .

-2yz)i+ (3y' -2zx) j+ (322 - 2xy)k

MATDIP4Ol

(07 Marks)

3), B: (i. 1, 1),

(07 Marks)

(06 &Iarks)

{07 Marks}

z -10 and3-t67(CI7 Marks)

(05 N{arks)

{07 Marks)

is irrotational and find $

Zx-y -z*6 =0 .

Find the image of the point (l , -2,3) in the plane 2x -t y -z:5.Find the shortest distance between the lines

x-15 _y -29 :r-5B -5

Find the constant 'a' so that the vectors 2i- j+ k, i + 2j-3k and 3i + aj + 5k are coplanar.

[- ]: + + +r -[*: ,r (o6Marks)

Frovethat la+b. b+c. c+ al:21 a,b.c l. (o7Marks)L]L-]

Find the unit normal vector to both the vectors 4i- j+3k and -2i+ j-zk. Find also the

sine of the angle between them. (07 Marks)

A particle moves along the curve x:t3 +1, y =t2,2:2t* 5 where t is the time. Find the

components of its velocity and acceleration at time t : 1 in the direction af 2i + 3j + 6k .

(06 Marks)

b. Find the angle between the surfaces x'+y2 +22 =9 and x=z1t +y'-3 at the point(2, -1,2). (07 tl{arks)

c. Find the.directional derivative of O = xy' +yz3 at the point (1, -2, -1) in tlie direction cf the

normal to the surface xlogz - yt = -4 at (- 1 , 2, 1 ). (07 Marks)

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Fourth Semester B.E. Degree Examina ec.2015|Jan.2016

Advanced Mathematics - llTime: 3 hrs. Max. Marks:100

Note: Answer any FIYE full questions.

I a. Find the direction cosines of the line which is perpendicular to tnie lines with directioncosines (3, -i, l)an(-3,2,4). (05Marks)

b. If cos c{,, cos p, cos y are the direction cosines of a line, then prove the following:i) sin2 cr +sin2 B +sin2 y = 2

i, cos 2cr + cos 2p + cos 2y = -1c. Find the projection of the line AB on the line CD where A: (1,2,

C : (0, 0, 1), D : (2,3,0).

2 a. Find the equation of the plane through (1, -2, 2\, (-3, l, -2) and perpendicular to the plane

I of2

(07 N{arks)

6 a. Find: Lb. Find: i)

c. Find: L

cos 2t

cos' t),at - cos

{cos t

L{e-'

Icosrcos 3t) .

ii) L{te-'sin 3t}.

MATDIP4Ol

(06 Marks)

(07 Marks)

(07 Marks)

(06 Marks)

(07 Marks)

(07 Marks)

II

I

I

I

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7 a. Find: [ {

4s+5(s-l)2(s+2)l'

Find: i) t-.-'J = t*2 I.'

[s' - 4s + 13.J

Find: L- J . ' I.Is'(s+1)J

8 a. Using Laprlace transforms, ,o1r. -d'{ -2+* y = e" with y(0) : 0, y'(0) : 1. (10 Marks)ox oxb. Using Laplace transformation method solve the differential equation y"+ 2y'-3y =siflt,

Y(0)=Y'(0)=0. (l,Marks)

b. ii)ri"'[x)]

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USN

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b

7a.b

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c.

5a.b.

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(lS Marks)(tS Marks)

(06 Marks)

(10 Marks)

(84 [4arks)(06 Mari<s)

(t0 Marks)

V//C ratio? (ls Marks)

(18 Marks)

GS 10262-(12 Marks)(0E Marks)

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Fourth Semester B.E. f)egree Examin .20115/Jan"2015

Goncrete Technology

Time: 3 hrs. Max" I\4arks:i00Note: l. Answer FIItE full qwestions, selecting

at least TWO questions f,rom ewch part.2. Use of IS 10262-2009 is permiwed..

PART _ Aa. Explain with flow chart, the cerrent manufactured by wet process. (tr0 Marks)b. List the different laboratory tests conducted on cement. Explain any one of them in detail.

(tr0 Marks)

a. Explain the role/effect of fine and coarse aggregates in concrete. (10 Marks)b. Write a note on "mechanical properties" of coarse aggregates. Explain the test for

detennination of aggregate impact value as per 152386 part IV - 1903. (10 Vlarks)

a. Define the term "Workability." List the factors affecting workability of concrete andmethods of measurement of workability. (10 Marks)

b. List the various stages/process of production of quaiity concrete. Explain in brief. (10 Marks)

. Discuss the role of chemical admixtures and mineral admixtures in cernent concrete.

Write short notes on: (i) Rice husk ash, (ii) Air entering agents

PART _ BList the factors influencing strength of concrete.What is the relation between:i) Compressive strength and tensile strength of concrete as per IS456-2000ii) Cube strength and cylindrical strength of concrete. {04 Marks)Explain in brief the principles of flexural/modulus of rupture testing of concrete under third-point loading rnethod.

Mention the different modulii of elasticity of concrete.What are the factors affecting shrinkage and creep of concrete?Write short notes on:i) plastic shrinkageii) drying shrinkage

Define the term durability. What is its significance and its impact onWrite shoft notes on:i) Sulphate attack and its controlii) Freezing and thawing and its remedial measures

a. Discuss the steps involved in Indian standard method of concrete mix design1 982).

b. List out the variables in proportioning of concrete mix design.

*****

USN 10cv43,/

Fourth Semester B.E. Degree fxi\fid ec.20l5lJan.20l6

HLo>Es

oXHX

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; Ei b. ql;ffi the maximum slope and maximum deflection for the beam shown in Fig.Q2(b).

H E ffinjusate beam method. Take E : 2xl0s N/rnm2, I : 40x 106 mma. (10 Marks)EE n&E F 3 . ffitate Castigliano's First and second theorems. (04 Marks)

$E - *ilS Using Castigliano's theorem compute the deflection at the mid-point of simply supported

E *. A[-*- ' beam of span 'L' and flexural rigidity EI, carrying a point load 'W' at the midspan.

ISO c. Determine the vertical and horizontal deflection at 'c' of the beam shown i,, li!.tli:i6 {- Take E :200 GPa and I: 80x106 mma. (10 Marks)-.: c.i

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.=Nd+g6*otr-c g)

StructuralAnalysis-l 4\\

rime: 3 hrs. M*. r"rffFirdo

Note: Answer any FIWfull questions, selecting il $.atleast I"WO questionsfrom each part. d* \

xvJPART - A t,\\

a)rI a. Define : (i) Degree of static indeterminacy and ii) Degree of qqffi (05 Marks)

b. Derive an expression for strain energy due to bending. ^t _, u (06 Marks)

c. Determine the degree of static and kinematic indeterm&@s for the structures shown inFig.Ql(c) (0e Marks)

BTE A 2 a. For the beam loaded ,, ,fuffiffi Fig.Q2(a), determine the slope and deflection at the free

S E end. Take E :204x10o*ffi&r' Use moment area method. (10 Marks)(Bcl€ i ,.LxtEE -e h lalsr;t 4* , =*, ,,F;t'* I a-*'ouu'; E /-*,v'E; * B t

Fig.Q2(b)

Fig.Ql(c)

Fig.Q3(c)

I of2

10Cv43.l

4 a. Using strain energy method determine reaction at B for the beam shown in Fig.Qa(a). DrawBMD and SFD. (10 Mcrks)

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A

Fig.Qa(a) Fig.Qa@) dlrb. Analyse the fixed beam shown in Fig.Qa@) using strain energy method and dgv/BMD.(10 Marks)

PART - B5 a. A three hinged symmetrical parabolic arch has a span of 36m

b. A cable is suspended between two points A and B,A cable is suspended between two points A and B, ffigr aliartof 6 m. It supports a UDL of intensity 20lcTrtim. Coffie:i) The length of the cables. f1....ir) Maximum and minimum tension in the

W (08 Marks)

For the beam shown in Fig.Q6(a), cpanptrte the reaction at B by consistent deformationmethod. Draw BMD and SFD.

^L.$ (10 Marks)

I 6ok+. r gokv

n I i'"- n [l --'*rt\'nf- o* {--tra -f ?-- * - 'rf 6ro C zt

FieffiMi Fie.e6(b)b. Analyse the fixeffim shown in Fig.Q6(b) using consistent deformation method.

,N

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(10 Marks)-#7 Analyse the*ffifruous beam shown in Fig.Q7 by Clapeyron's theorem. Draw SFD & BMD.(20 Marks)d*f \ t-,^.65P -3"\$lo

A two hinged parabolic arch has a span of 32m and a rise of 8m. A uniformly distributed load of1 kN/m coverc 8 m horimntal length of the left side of the arch. If I : Io sec0, where 0 is theinclination of the arch section to the horizontal, and Io is the moment of inertia at the crown.Find the horizontal thrust at hinged and bending moment at 8 m from the left hinge. Also findthe normal thrust and radial shear at this'section.

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rise of 6m. Thearch carries a UDL of intensity 30 lcIt{/m over left half a concentrated load of60 kN at 9 m from right support. Compute the bending normal thrust and radialshear at 9 m from left support. *' &-"/ (12 Marks)

afrart horizontally and a central dipdip

6a.

qs-^

(10 Marks)

Fig.Q6(b)

Fig.Q7

(20 Marks)

USN

Time: 3 hrs.

theodolite not in adjustment.c. List the errors eliminated by repetition method.

(rf i'- {

a note on:

d*yry rurarrilurlu lsrts. rr,, JuDlcn!tii'-" Explain the method of determining the

ffi" aounactic lens. ii) Subtense diaphragm.

The instrument is fitted with anallactic lens and the constant isPQ and reduced level of Q, that of P being 321.500 M.

100. Compute the length of(10 Marks)

t0cY44

(08 Marks)(04 Marks)

(10 Marks)' r ri c4q,ii*fg, ltl*ifl*, ,r tlri

(04 Marks)(06 Marks)following

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Fourth Semester B.E. Degree Examination, Dec.20l5 I Jan.20l6Surweying - ll

,f,r^r3fl1

Max. Marks:ffi"

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ad

c).i tttGl- '- n),.JJ1B ,rl1E =_1 *1=::.J :It I

=h-oo llcoo

.=N(d+

ETotr-c c)

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bd-!doo13o0cEx>xF3

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Note: Answer FIVE fall questions, selectingat least TWO questionsfro* each part.

P.{RT _ AI a. Distinguish between:

D Plunging and swinging of the telescope.ii) Clamp screw and tangent screw.

. ii| Transitting and line oicollimation.

"',;jj'-* b;l '.With neat sketches, explain measurement of a straight line by a theodolite in adjustment and

it-,,;*;..:.*i

i,iffi@r.

2 a. What are the permanent adjustments of a theodolite? Explain the spire test. (08 Marks)b. Explain the procedure to lay off an angle with a greater precision than the least count of the

instrument. (08 Marks)What is an'index error'? WJry it isis ne.ffiary to be adjusted?

"lri.(04 Marks)

c. In order to ascertain t

3 a. Derive the expression for the -|rorizintal distance, vertical distance and the elevation of anelevated object by double pla4qmethod, when the base is inaccessible. (07 Marks)

b. List the advantages of total$, ion over the conventional surveying instruments. (03 Marks)In order to ascertain the'..devation of the top (Q) of the signal on a hill, observations weremade from two instnriYrrjnt stations P and R at a horizontal distance 100 M apart, the stations\,.P and R being leqi,i,ith Q. The angles of elevation of Q at P and R were 28'42' and 18o6'

respectivestaff readings upon the bench mark of elevation 287.280 M wereM and 3.750 M when the instrument was at P and R, the telescope beingine the elevation of the foot of the signal if the height of the signal aboveho

itsr

3M.

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C)

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tir r.. ri!! - t,;ftr' .

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.E .i' ^ ,,^qb.'" Explam the method of determining the constant of a tachometer, in the field.E E v\* p. A tachometer is setup at an intermediate point on a traverse course PQ ariate point on a traverse course PQ and the

observatiatrons were made on a vertrcally held stattStaff station Vertical ansle Staff intercept (M) Axial Hair Reading (M)

P +9o36, 2.3s0 2.r0so +606' 2.055 r.895

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PART _ B "I

5 a. Define degree of a curve. Establish the relationship between degree of curve and its radius.

b. Calculate the ordinates at 10 M fof a circulat'"cunq'ffih80 M and the versed sine is 4 M. ,r

c. The chainage at the point of irt..r..tibn of the tangents to a raiiway to*.1sl:angle between them is 150'. Calculate allthe data necessary for curve setting out agffibfradius 250 M by the deflection angle method. The peg intervals may be taken as ffiffi.Alsoapply the arithmetic check.

*e*W Marks)

Distinguish between compound curve and reverse curve. {*--* *

(0+ Marks)

Derive the relationships between various elements of a reverse curv6"&$vdeen two parallel

| ;-: jgl . ..tJ;rr .; I

6a.b.

c.

7a.b.

c.

straights. u%m *

(08 Marks)

Two parallel railway lines are to be connected by a reverse culqei &ch section having tfiesame radius. If the lines are lzll.4 apart and the maximur" d@{re betrry9gg,!g"ngdg!J_".S.f*;measured parallel to the straights is 48 M, find the maxirpqln a\iowable'ihAifs:Tf65ffi'radii are to be different, calculate the radius of the secondrffich if that of the frst branch is60 M. Also calculate the length of both the branches. "., &

(08 Mark'i) i{::ry

What is a transition curve? Explain the requirenffiqof a transition curve. (06 Marks)Why are vertical curves provided on highwaysrl,Di$t ttre different types of vertical curves. '':

\# (04 Marks)A transition curve is required for a (gular curve of 200 M radius, the gauge being 1.5 Mand maximum super elevation is to 0.15 M. The transition is designed for a

velocity such that no lateral imposed on the rails and the rate of gain of radialacceleration is 30 cmlsec3. Calq ihe required length of curve and the design speed.

(10 Marks)

8 a. What ts "Zero circle" of a planimeter? Explain any one method of finding its area.(06 Marks)b. Calculate the area of a figure from the following using a planimeter with the point inside the

figure.Initial reading : 9.918Final reading= 4.254Constant M and C respectively are 100 cm2 and23.52L The zero mark of the disc crossedthe inde*..once in anticlockwise direction. (04 Marks)Tfre #h'ir enclosed by the contours in a lake are as follows:enclosed b the contours in a lake are as foContour (M) 270 275 280 285 290Area (M") 2050 8400 16300 24600 31s00

Calculate the volume of water between the contour 270}l4 and290 M by the trapezoidal andprismoidal rule. (10 Marks)

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c.

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2 of2

USN IOCV45

Max" Marks: l0O

(06 frAarles)

(09 Marks)

({}4 Marks)

{02 Ntarks}

Fourth Semester B.E. Degree Examinatiod 015/Jan.2016

Tirne: 3 hrs.

4a.b.

Hydraulics and Hydraulic Machines

Note: 7. Answer FIVE fuU questions, selectingat least TWO questions from each XsarL

2. Missing data may suitably be sssuvned"

1a.Ltr.

c.

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a,F

A 2.5ro ship model was tested in fresh water p : 1000kg/rm3 and measureffrents indicatedthat there was resistance of 45N when the rrodel was moved at Zrnls. Workout the veiocityof 40m prototype. Also calculate the force required to drive the prototype at this speed

throngh sea water ( p : 1025k g ^\. {{}5 Marks)

2 a. With neat sketches differentiate between flow through pipes and flow through open channelswith examples. {SS N[arks)

b. Derive an expression for the discharge through an open channel using Manning's forrrulia"(06 Mark-s)

c. An earthen channel ,,rrith a base width 2m and side slope lF{ to 2V carries waier rvith a

depth of ira. The bed slope is I in625. Calcuiate the discharge if n:0.03. Also calculateaverage shear stress at the channel boundary. (06 Marks)

3 a. Define specific eneigy. Draw specific energy curve, and then derive expressions f'or criticaldepth, critical velocity and rninimum specific energy. (tr0 klarks)

b. Derive the expression for sequent depth of hydraulic jump interrns of Froude number befcrea hydraulic jump in a rectangular channel flow. {CI6 &tarks}

c. A horizontal rectangular channel 4rn wide carries a discharge of i6rni3/s. Deterrnine whethera jump may occur at an initial depth of 0.5m or not. If'a jurnp occurs, deterraine the sequent

PART _ ADeflrne the dimensional homogeneity. Give an example.Briefly explain geornetric, kinematic and dynamic similarities.

detpth to this initiai depth. Also determine the energy loss in the Jump.

State impulse rnomentum equation.

c.

Shou, that in case of jet striking the flat plates mounted on wheels. tkre eff,rciency will be

rnaximum when the tangential velocity of wheel is half that of jet and maximum efficiencyis only 50%. (10 i\[arks).A T5mm diameter jet having a velocity of 30m/s strikes a flat pla.te, the norrnai of which is

inclined at 45o to the axis of the jet. Find the normal force exeded on the plate.i) When the plate is stationary ii) when plate is moving with velocity of i5m/s in thedirection ofjet away ftom the jet.Also determine the power and efficiency of the system when the plate is moving. ($E ivlarks]

PART _ B5 a. Show that when the jet of water sffiking symmetrical moving curved vane at tire centre, the

8rnaximum efficiency for semicircular vane is _ (1+ cos0).

27'

1 of2

(1t) k[arks)

r trCIcv4s

b. A jet ofl water, 50mm in diameter, strikes a curved vane at its centre with a veiocity of18na/s. The curved vane is rnoving with a velocity of 6mls in the direction of the jet. The 1et

is deflected through an angle of 165o. Assurning the plate to be srnooth, Find :

i) Thrust on the plate in the direction ofjetii) Fcwer: of the jet, andiii) Efficieney of ttrre jet (trO l{arks)

With the help of velocity triangles derive an expression for work done and rnaximumhydraulic efficiency of a pelton wheel. {10 Marks)

A Pelton wheel is receiving water from a penstock with a gross head of 510rn. One third ofgross head is los1" in friction in the penstock. The rate of flow through the nazzle fitted at the

end oflthe penstock is 2.2ri ls. the angle of deflection of the jet is 165o. Detennine :

6a.i^t_r.

7a.b.

o^oa,

b.

C.

0 Fower given by water to the ruflner,ii) Hydraulic efficiency of the pelton wheel.

Take Cv: 1'0 and sPeed ratio : 0'45'

Draw the neat sketch of Kaplan turbine and mention the parts.

Explain with the help of neat sketches the different types of draft tubes.

7Sa/o. Deterrnine the vane angle at the outer periphery of the impeller.

A Kaplan turbine develops 22000kW at an average head of 35m. Assriming a speed ratio of2, flow ratio of 0.6, diameter of the boss equal to 0.35 times the diameter of the runner andan overall effici.enicy of 88%, calculatethe diarneter, speed and specific speed of the turbine.

({}6 Marks)

Explain brriefly the various types of efficiencies of a centrifugal pump. (08 Marks)

(10 Marks)

(08 Marks)(S6 Marks)

(05 Marks)

Distinguish bertween pumps in series and pumps in parallel. (07 Marks)A centrifligal pump is to discharge 0.118m3/s at a speed of i450rpm against a head of 25m.The impelier diarneter is 250mrn, its width at outlet is 50mm and manometric efficiency is

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