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# Visvesvaraya Technological University (VTU) 2005 B.E Engineering Mathematics-2 - exam paper

Wednesday, 12 June 2013 04:25Web

Second Semester B.E Degree Examination, Feb/March 2005
Common to all Branches
Engineering Mathematics-2

Full ques. Paper in attachment

Page JVb.;. 1

: '{ *

VSH M MS |o h |C |s \ I 6 16 Second Semester B.E Degree Examination, July/August 2005

Common to All Branches    ,

Engineering Mathematics n

Time: 3 hrs.]    [Max.Marks : 100

Note: i. Answer any FIVE full questions choosing at least

one question from each part.

2. Att questions carry equal marks.

PART A

1.    (a) For the curve 9 = cos~x - 'nA2-72 prove that = const, (6 Mark)

. (b) State Rolles Theorem and verify the same for s    .

f(x) =    in [o,i]    UVi    (7 Marks)

(c) Find the first four non zero terms iri the expansion of /(x) = using Maclaurins series.    :i'"'    "" ! e ~7 MarJtg)

tan    __'.'.t '. . . i ,

2.    (a) Evaluate Ja 2 - j    , , (6 Mark.)

,l,t (b) Expand /(*, y) = x2y + 3j/ - 2 as a polynomial of powers of (a: - 1) and (y + 2) upto second degree terms using Taylors theorem, 1 - ; (7 Marki)

(c) Find the dimensions of the rectangular box, open at the top, of the maximum , ; _ *. capacity whose surface is 432 sq.cm.    >1    (7 Mark)

PART B

3.    (a) Change the order of integration and evaluate    vi

3 y/l-y        t    a

/ / (x + y)dxdy    ;    (6 Marks)

0 0 :

' ' ' * a2_r2 .

2 asmO T

(b) Evaluate j J f rdrdOdz    (7 Mark)

0 0 0

(c) Using Beta and Gamma functions evaluate

IT    T    . :    ,    .    .

J __f     i-.-nr : ,

f VsinB dO x f M"n    *        (7 Marks)

' I 5 VSllM - : .. : '    :i Ti - ' .

4. (a) Find the directional derivative of <j> = x2yz + 4xz2 at the point (1, -2,-1) in

A    A    *

the direction of the vector 2i - j 2k *,    (6 Marks)

(b) If and v2 be vectors joining the fixed points (x1,y1,zi) and (i2,i/2)2) respectively to a variable point (x, y, z) prove that div(uj x v) = 0 (7 Marks)

Page No. 2    -.yu-s.*..!*    />-;v    MAT21

(c) Verify Green's theorem for fr[(zy + y2)dx 4- x2dy\ where c is bounded by

y = X and y = X2    . W'.-    .    (7 Marks)

PART C

5.    (a) Solve 6 + 9y = 6? + 7e~2x - log2 - ,... . i; . .... (6Marks)

{b| Solve 4- a2y = i<m a#    ; '    (7Marks)

d.i" .

(c) Solve by method of variation of parameters

= (7Marks)

6.    (a) Solve by the method of undetermined coefficients

(D2 - 2D)y = exsinx    (6 Marks)

(b)    Solve x2M- + Axr + 4 = x2loqx    (7 Marks)

i/j;3 i/a:2    J

(c)    Solve the initial value problem + y = sm(: -f a) satisfying the conditions

dx~    ~

y(0)=0 j/(0) = 0    (7 Marks)

PART D

7.    (a) i} Evaluate L{/(im3i - cas3f)}

t ii} Using Laplace transform evaluate

OO

Je *tsin23tdt     g..____ __    (6 Marks)

" 11 ,

*

(b) Find I he Lapiace transform of

/(f) = EsillUit 0 < t < J given f(t + } = /(f)    (7 Marks)

{cl Express the following function in terms of Heavisides unit step function and hence find its Laplace transform

/(f) = i2 0 < f < 2 , t J    7 Marks

= 4f i > 2

8.    {a) Evaluate

i) i-1{Ti3 + -r----sl

y    ,iJ + 6,i + 13 {.<1 2)Jj

) -J ,,.h+?-1

(6 Marks)

(b)    Evaluate

i) jC_1{cof_1s}    ii) L-M ?rn> I    (8Marks)

J

(c)    Using Laplace transform method solve

3& + 3S y = i2ef giveni/(0) = l,yJ(0) = 0!V'(0) = -2 (fiMarks) 