Bangalore University 2004 B.E Engineering Mathematics-II, y/ - Question Paper
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NEW SCHEME
' USN
Second Semester B.E Degree Examination, July/August 2004
Common to all Branches
Engineering Mathematics n
Page No... 1 MAT21
IMax.Marks : 100
Page No... 1 MAT21
Time: 3 hrs.] Note:
1. Answer any FIVE full questions choosing at least one question from each part.
2. All questions carry equal marks.
Part A
(a) Obtain the formula for the radius of curvature in polar form. (7 Marks)
(b) If P11P2 be the radii of curvature at the extremities of any focal chord of the cardioid 7 = a(l + cos6) show that
p\-T p\ = 16 - (7 Marks)
(c) State and prove Cauchy's mean value theorem. (6 Marks)
2. (a) Find the values of a and b such that lim x(l+acosx)bsinx __ 1
(7 Marks)
_ r>-5-
(b) Find the maximum and minimum distances from the origin to the curve
6x3/ + 5 y2 8
5a:2
(7 Marks)
(c) Expand x2y - 3y - 2 in powers of (a: - l) and (y 4- 2) using Taylors theorem.
(6 Marks)
Part B
3. (a) Find the value of
J Jxy(x + y)dxdy taken over the region enclosed by the curves y = x and y = x2
(7 Marks)
(b) Evaluate by change of order of integration
/ I
=dydx
x+y
o
(7 Marks)
(c) With usual notation show that
d/ \ _ r(m)-r(n) B{m,n) r(m+n)
(6 Marks)
4. (a) Find the directional derivative of 0 = x2yz 4- 4.xz2 at the point (1,-2,-1) in the direction of the vector 21 - J - 2K (7 Marks)
(b) Given that (f> is a scalar point function and A is a vector point function, prove that
' turl(ct>A) = <f>(curlA) - (grad<f)) x A
(7 Marks)
(6 Marks)
(c) Evaluate j J F-vds given S
b = xl + yJ + zk over the sphere x2 4- y2 + z1 = a2
Contd.... 2
Page No... 2 MAT21
PART C
d2
5. (a) Solve + 4y x- 4- cos2x + 2~x
(7 Marks) 17 Marks')
(6 Marks)
(7 Marks) (7 Marks)
(b) Solve - y = zsinx + (1 -f x2)ex
(c) Using the method of variation of parameters solve
+ Ay = tan2x
6. (a) Using the method of undetermined coefficients, solve
+ 2+4 = 2*>+3e-*
(b) Solve x3 -j- 3x2 + x + 8y 65cos(logx)
(c) Solve the initial value problem &
-f- 4 + 5y + 2coshx = 0
given y = 0, =. 1 at x = 0
(6 Marks} " J
'v/
(7 Marks)
PART D
i- * *T: *
7. (ai Find the Laplace Transforms of
i) t2e~3isin2t
CO!2tCOSZt . . i-
Ui -r--j- ...
(b) Find the Laplace Transform of the full-wave rectifier
fit) Esinuft, 0<t< having period ~
|
(7 Marks? (6 Marks) (7 Marks) (7 Maitas) |
 |
, ;..gi
(c) Find
i) L[e~*u{t - 2)]
ii) L[t2u(t 3)]
8. (a) Find the inverse Laplace Transform of
(5+1)
ii) cof-f)
-{-rV}
(b) Evaluate: L 1 <(-- using convolution theorem.
(s2+a2)
(c) Using Laplace Transforms solve
given y = 0, = 0 when t = 0
\
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Attachment: |
| Earning: Approval pending. |
