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BT-3/D06

Mathernatics-III

\,/ WTotal No. of Fages : 4

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Paper : MATH-201 E, OPtion : I

Time : Three Hours] fMaximumMarks: 100

Notq:-Attempt any FIVE questions in all selecting at least oNE

{ " question from each unit' All questions carry equal marks'

UNIT-I

1. (a) Obtain Fourier series for the function f(x) givenby :

1 -

f ( x )=11 - , -n (x (0 ,

. 1) v

= l_ : : , 0<x<n .+

(b) obtain the Fourier expansion of x sin x as a cosine series rn(0, r).Also show that :

11 ,1 n -2. u 3 _ . . . . . . . , , = - * . i 0

l '3 3.5 5"7 4

2, (a) Solve the integral equation :

.. [r_"[r(e) "ot

cro do = IJ0 L0

hence evaluate

?sinzt .l - r ot 'd^ t -u -

"t-\ ,/. .n"n rv /!'. 1

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is processed,on two machines M, and M'' Product A requires

one minute of processing time on M , atd}Minutes on M' llile

B requires one minute on M, and one minute on M'' Mz

M, is available fornotmorethan.T hours an1].1lftjl

iil* """t"le

for 10 hours during any working day' Findtho

number of units of products A and B to be manufactured to get

maximumProfit, 10

Ror Hc.

Q =Nd. . . : .

oftheril

p is den

weight o

Time : Three Hours

Ir{ote :- Attempt arfrom each

1. (a) A jet of wcurved varL20o to th,is 18o and(i) Vane(ii) Worl

(b) A jet of u25m/sec.the jet anon the plr

(i) Intt(ii) Intt

2. (a) Define tb

(n Fror(i,i) Wel

(b) Using Bconsume

(b) Using Simplex method' soive the following LPP :

Maximize Z = l07xr+ xr* Zxy

subject to the constaints :

14x, * xr- 6x, * 3xo=7

. t -1 6 x , + * * r - 6 x r S 5 ,'/.

3x., - xr- x. S 0

X,, X2: X3: xo 2 0'

Find all the basic soiutions to the following problem :

Maximize Z= xr* 3q 4 3x'

subject to x, * 2x, * 3x, = 4'

2* r * 3x , * 5xr :7 aPd' x , 2 O , x r 2 0 , x r 2 0 .

Which of the basic solutions are

(t uon-degenerate basic feasible,

(ii) optimal"basic feasible ?

Using Dual Simplex method, solve the following problem:

Minimize Z= Zxr* 2xr* 4x,

subject to 2x, + 3xr+ 5r.r22,

3x, * x, - t 7xtS3,

xr* 4xr+ 6x: S 5;

xl, x2' x3 > 0. ,+

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me : Three Hoursl

ET_3/B{}6$tnength of Materiats-X

Faper : ME-2038

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t$4"

fMaximumMarks: 100

(Contd.)

Note :-Attempt any FIVE questions. All questions carryequal marks.

Serive relationship for sharing of load by a composit section. A loadof 2 MN is applied on a short concrete column 500 mm x 500 mm.The column is reinforced with 4 sheet bars of r 0 mm diameter, o* ioeach corner' Find the stresses iu steel bars. Take E for steel as2'l x 105 N/mm2 and bar concrete as 1.4 x 10j N/mm2. Z0At a point in a strained material, the principal tensile stresses acrosstwo perpendicular planes are g0 N/mm2 and 40 N/mm2. Determinenormal stress, shear stress and the resultant stress on a plane inclinedat 20oC with the majorprincipal plane. Determine also the obliquity.what will be the intensity of stress which acting alone will p.oau"e

the same maximum straio if poisson,s ratio = | , Z0

A cantilever beam of5 m span, carries a load of2 kN at 2 m fromfixed end and a load of I klrl at the free end and a road of 4'I*r/muniformly spread over the lst and 2nd metre.of the length m;;;from the free end. Draw the shear force and bending moment d.iagrams.

500 kW power is to be transmitted at 100 r.p.m. Determine thenecessary diameter ofthe (i) solid circular shaft (ii) diameter ofhollowshaft if the inside dian,cter is 0'g times the outside diameter, if theallowable shear stress is 75 N/mmz. what is the saving in materialwhen a hollow shaft is used ? Z0

20

5 .

1l

i. , ,

A beam AB is supported at its ends has span of 2 m ar,d carries. aiu.d.l. of 200 liF{/m over the entire span. The cross-section of thefibeam is a T-section having flange width 125 rnm, flange thicknessi25 mm, web thickness 25 mm and overall depth 200 mm. Calculate Imaxirnum shear stress in the beam. Also draw shear stress distributionfmarking principal values. 20',

6 . Find out the greatest length of amild steel rod 30 mm x 30 mrn whichcan be used as compression member with one end fixed and theotirer end hinged. nt cles a working load of 40 kN with a factar af,

safetv = 4. Take ̂ : =* and o = 300 N/mmz, 20' / ]uu c

,4. cantilever beam of length l0 m and carrying a distributEri' ioaclwhose intensiiy varies uniforrnly from zero at the fixed eird and

1000 N/m at the f,ree end.,Derive equation lbr the rjeflectioir

curve and hence caiqulate the deflection at fiee cnd. E:210 GXJa

and I = 4xlOlo mrn1. 20

A fixed beam of 12irn span oarries two point loads of 20 kN at 3 mfrom each end, JFind support moments. Calculate deflection at c€nhs

I = 108 mma and E = 205 GFa.

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