Bangalore University 2006 B.E uary/ruary - Question Paper
5/25 Page'No.. 1 [ M/
Reg. No.
Second Semester B.E. Degree Examination January/February 2006
NEW SCHEM
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ance
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Engineering Mathematics - ti
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s. The motor for a od by /larks)
main
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(Max. Marks : 100
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
1. (a) Find the radius of curvature of the curve x3 + y3 = 3xy at (f, f)-
(b) State Lagrange's mean value theorem.
Prove that for 0 < a <b.
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(7 Marks)
(6 Marks)
(7 Marks) (7 Marks)
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ba
+P
(c) Show that
a < tan *6 - tan 1 a < %. l+2 1+a*
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tion
>/l sin 2x 1 + x y ~ t ITT"
2. (a) Evaluate :
lim ( j*> l 'i lim (r. ~\iaii
IJ x0 (coi~x - -jpi (2 - a) 2a
(b) Expand log (1 + x - y) upto third degree terms abouT the origin.
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I
x stan
(c) Rnd the point on the paraboloid z = x* +y which is closest to the point
(3, -6, 4).
(6 Marks)
PART - B
l
3. <a) Evaluate : J J x3y dx dy. o o
(7 Marks)
(7 Marks) (6 Marks)
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(b) Rnd the totcl area of the lemniscate r2 = a2cos 26.
(c) Show that T() = y/n by using the definition of gamma function.
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4. (a) Rnd 1he angle between the tangents to the curve x = t2 + 1, y = At - 3,
z = ?t2 - 6f at i = 1 and t = 2. (7 Marks)
(b) Show that rn r. where f x i 4- y j k, is irrotational. (7 Marks)
(c) Using divergence theorem evaluate
J J F n ds where
s . . *
F = x3 i + y3 j -4- z3 k and S is the surface of the sphere x2 + y2 -r z1 = 4.
(6 Marks)
4-'* - -i.,' - -
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5. (a) Solve the differential, equation tfiy c d%
2 _ ~ e2X SU X x"
(7 Marks)
(7 Marks)
(6 Marks)
dx
(b) Solve :
- 3S- + 4a: = 2. cosh 2t dt3 dt2
(c) Using the method of variation of parameters find the solution of
i) Apoir I Y S 3 4 refere
) Draw end 6
6. (a) Solve the differential equation
|
(7 Marks) |  |
- 5 + 6y = e2x + si
dx
by the method of undetermined coefficients.
(b) Find the solution of the differential eauation
(2x - iff + (2* - 1)| 2y = 8r2 - 2x + 3
.sin a;
prof Mwtienthi Io al the
(7 Marks)
|Arectan i itsfrof
(6 Marks)
||pcrtU*g< ftinciineo g projecti
l-COS3t
in
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(7 Marks)
a*r?s *s i
j|The Ire p of the I
(6 Marks)
(7 Marks) (7 Marks)
-2s
)
(3-3)
m A cone ;r Side2 1 isomet
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<b) Find the Laplace transform of the function /(f) = E sin 0 < t < u. given that f(t + u>) = f(t).
(c) Express the function
r>.\ _ J 7T j 0<i<7T
.sin t > 7T
in terms of unit step function and hence find its Laplace transform.
8. (a) Find the inverse Laplace transform of
(c) Solve the initial-value problem
+ 29x = 0. given (0) = 0, (0) = 15. PART - D
7. (a) Find the Laplace transform of
(b) Using convolution theorem find the inverse Laploce transform of <c) Solve, using Laplace transform, the differential equation
B-3S+2v=,-ti
given that y(0) = 1 and = 1 at x = 0.
0 e2 cos2/
S+2S+Vi
25-1
(6 Marks)
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Attachment: |
| Earning: Approval pending. |
