Rajasthan Technical University 2009 B.Tech Mechanical Engineering Advanced EnggMathematics (mechanical Engg.) - Question Paper

RTU Advanced Engg. Mathematics (mechanical Engg.)

Rajasthan tech. University
B.tech three sem (Main/back)
Year 2009

Roll No.:_ _ Total Printed Pages : 4

3E1416

B.Tech. (Sem.lll) (Main/Back) Examination, January - 2009 (3ME6) Advanced Engg. Mathematics (Mechanical Engg.)

(3PI6) Advanced Engg. Mathematics (Prod. & Indus. Engg.) (3AE6) Advanced Engg. Mathematics (Automobile Engg.)

Time : 3 Hours]    [Total Marks : 80 [Min. Passing Marks : 24 Attempt five questions in all. Schematic diagrams must be shown wherever necessary. Any data you fee! missing may suitably be assumed and stated clearly.

Use of following supporting material is permitted during examination. (Mentioned in form No. 205)

1-_____-Nil__-    2. _Nil

1 (a) Find the Fourier series for /(x) = + x², tt < jc < jc -

Hence show that ⁼ l + t + ~t + .

⁶ 2 3

(b) State and prove convolution theorem of Laplace transforms.

OR

1    (a) Find half range sine series for the function

f(x) = x in the interval 0 < x < 2 -

(b) Solve the integral equation

0

J f(x)cos px dx = e~^p

o

2    (a) A tightly stretched string with fixed end points x = 0 and

* - *    *     7UX

x = l is initially in a position given by y = J₀ sin -. It is

v

released from rest from this position. Find the displacement

->2 _2 & y ,2 d y

wave eqn. = C

dT dx²

du d²

(b) Solve ~T7 =2 by Laplace transform method, given 8x

u(x, 0) = 3sin2icx_f u(0, i) = 0, u(l, t) = 0. where

0 <x < 1 > > 0-

OR

2    (a) An insulated rod of length I has its ends A and B kept at

0C and 100C respectively until steady state conditions prevail. If the temperature of B is then suddenly reduced

to 0C and kept so, while that of end A is maintained, find the temperature u(x, t) at a distance * from A at time t.

2

du 2 d u [Diffusion equation -C ].

vt    dx

d²x

(b) Solve by Laplace transform method " 2 ⁺ x = * ^cos given x(0) = x'(0) = 0.

3    (a) If /(z) is an analytic function of z, prove that

l/(z)l³ = 4|r(*>[².

,    , _ z ~%z

(b) Find the residues of / ~ 22 at all its poles

(z + \y (z+4)    ^F

in the finite plane of z.

OR

2z + 3

3 (a) Show that the transformation iv =- maps the circle

z-4

x² +y² -4x = 0 into the straight line 4u + 3 = 0

^3E14161 llll II I I III III II 2    [Contd...

* 3 E 1 k 1 6 *

(b) Expand _z(_z -1) (z - 2) ^{is Laurent}'^{s seri}es for

(i)    \s\ > 2

(ii)    |z-l|<l

d²u d²u

4 (a) Solve TT ⁺ ~7Y when it(0, ;y) = 0 u(l

ox oy

ICC

u(x, 0) = 0 and u(x, a) = sin.

(b) Show that

f 48 81    ( 24

₄(x> = [ -J-- _e/₁(x) + jl--2- J₀(x),

,X" ^x J

OR

4 (a) Show that

x ⁿJ_n(x)\ = -x ⁿJ_n+l(x), n>0

(b) Show that

P_n(x) = -_rr(x²- l)ⁿ

\n2ⁿ dxⁿ Hence find P_g(x).

5 (a) Prove that

XU-, x u₀ x it,

+-r-^L + r+r+'

⁰ H [2 [3

2

x 2

Uq + x A u₀ + - A Mq +,

(b) Evaluate J 2 by (i) Simpson's rule and

Q 1 X    

(ii) Trapezoidal rule. Hence obtain the value of n by result obtained from (i) [Take six intervals],

OR

5 (a) From the following table find v(0.25), v(0.62) and v(0.43)

X	0.0	0.2	0.4	0.6	0.8
y	0.3989	0.3910	0.3683	0.3332	0.2897

(b) Using Lagranges interpolation formula, find y for x = 10 from the following table :

X	5	6	9	11
y	12	13	14	16

^3E14161 I 111 111 III 111 11 lllll lllllll    I 5000 ]

^k 5 E 1 k 1 6 *

Attachment: rajasthan-technical-university-2009-b-tech-mechanical-engineering-36592-1.pdf

Earning:   Approval pending.