Visvesvaraya Technological University (VTU) 2005 B.E Engineering Mathematics-2 - Question Paper
First Semester B.E Degree Examination, July/August 2005
Common to all Branches
Engineering Mathematics-2
Full ques. Paper in attachment
Page JVb.;. 1
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VSH M MS |o h |C |s \ I 6 16 Second Semester B.E Degree Examination, July/August 2005
Common to All Branches ,
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.
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 . ,,t .
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'. - . -
2. (a) Evaluate Ja 2 - j , , (a Marks)
,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 or the rectangular box, open at the top, of the maximum , ; . *. capacity whose surface is 432 sq.cm. U (7Mark*)
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 Marks)
0 0 0
(c) Using Beta and Gamma functions evaluate
IT T . : , . .
J __f : ,
f VsinB dO x f M"n * (7 Marks)
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.*..!* r-: 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)
5. (a) Solve 6 + 9y = 6? + 7e~2x - log2 - ,... . ... . .... (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)
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
(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)
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Attachment: |
| Earning: Approval pending. |
