Visvesvaraya Technological University (VTU) 2006 B.E FIRST SEMESTER , ENGINEERING MATHEMATICS II - Question Paper

FIRST SEMESTER B.E. DEGREE EXAMINATION, JANUARY 2006
COMMON TO ALL BRANCHES
ENGINEERING MATHEMATICS II
Time: three hrs. Max. Marks:100
Note: ans any 5 ques.. Choosing lowest 1 ques. from every part.
All ques. carry equal marks.
PART –A
1. (a) obtain the radius of curvature of the curve. x three + y three = three xy at ( 3/2, 3/2) (7 marks)
(b) State Lagrange’s mean value theorem.
Pr ove that for 0 < a < b, ba
/ 1+b2 < tan1
btan1
a < ba
/ 1+a two (7 marks)
(c) Show that
1 + sin two x = one + - x two / two - x three / six + x four / two four ...... (6 marks)
2. (a) Evaluate:
(b) Expand log (1+xy)
up to 3rd degree terms about the origin. (7marks)
(c) obtain the point on the paraboloid z = x two + y two which is nearest to the point
(3, 6,
4) (6 marks)
PART – B
3. (a) Evaluate one 1 - y 2
ò ò
x three ydxdy (7marks)
(b) obtain the total area of the lemniscates r two = a two cos 2?
(c) Show that T(1/2) = p by using the definition of gamma function
(6 marks)
4. (a) obtain the angle ranging from the tangents to the curve
x = t two + 1, y - four t - 3, z = 2t two - 6t at t = one and t = two (7 marks)
(b) Show that rn r, where r = xi + yj + k, is irrotational. (7 marks)
(c) Using divergence theorem evaluates.
F n s
®
ò ò ds where
F = x three i + y three j + z three k and S is the surface of the sphere
x two + y two + z two = four (6 marks)
PART – C
5. (a) Solve the differential formula. d two y / dx two - six dy / dx + 25 y = e two x + sin x + x
(7 marks)
(b) Solve:
d three x / dt three - 3d two x / dt two + four x = two x cosh 2t (7 marks)
(c) Using the method of variation of parameters fined the solution of
d two y / dx two - two dy / dx + y = ex two x e two x sin x (6 marks)
6. (a) Solve the differential formula.
d two y / dx two - five dy / dx + six y = e two x + sin x by the method of undetermined co
efficient. (7 marks)
(b) obtain the solution of the differential formula.
(2 1) two 2 / two ( two 1) / two eight two 2 three x - d y dx + x - dy dx - y = x - x + (7 marks)
(c) Solve the initial value issue.
d two x / dt two + four dx / dt + 29 x = 0, provided x(0) = (0), dx/dt (0) = 15. (6 marks)
PART – D
7. (a) obtain the Laplace transform of
(i) e two t cos two t (ii) 1cos
3t/t (7 marks)
(b) obtain the Laplace transform of the function
f ( t ) = E sin pt / w ,0 < t < w , provided that f(t + w) = f(t) (7 marks)
(c) Express the function
pp p>- < <=t ttf tsin ,1, 0( )
In terms of unit step function and hence obtain its laplace transform. (6 marks)
8. (a) obtain the inverse laplace transform of
(i)2 172 12 + +-s s
s (ii) ( three ) 2
2--se s
(b) Using convolution theorem obtain the inverse laplace transform of 1/s(s two +9)
(7 marks)
(c) Solve, using Laplace transforms the differential formula.
d two y / dx two - 3d y / dx + two y = - e two x
provided that y(0) = one and dy / dx = one at x = 0.

Earning:   Approval pending.