*8023*
8203
(Pages : 3)
Reg. No. : ..................................... Name : ..........................................
Third Semester B.Tech. Degree Examination, November 2009 (2008 Scheme) 08.301 : ENGINEERING MATHEMATICS – II (CMPUNERFHBTA) Time : 3 Hours
Max. Marks : 100 PART – A
Answer all questions. Each question carries 4 marks. 1 x2
y
1. Evaluate ∫ ∫ e dydx x
0
0
∞∞
2. Evaluate
∫∫ 0x
e − y dydx by changing the order of integration. y
3. If S is any closed surface enclosing a volume V and F = x ˆi + 2 yˆj + 3 z kˆ , Find
∫∫ F ⋅ nˆds .
s
4. Express F(x) = | x |, − π < x < π as a Fourier series. 5. Obtain the half-range sine series for ex in 0 < x < 1. 6. Find the Fourier cosine transform of e–2x. 7. Obtain the p.d.e by eliminating arbitrary constants from the relation 1 Z = ax e y + a 2e 2 y + b . 2
8. Obtain the p.d.e by elimination of arbitrary function from the relation z = y F (y/x). ⎛q ⎞ 9. Find the singular solution of z = px + qy + ⎜⎜ − p ⎟⎟ . ⎝p ⎠ 10. State the assumptions involved in the derivation of one dimensional wave equation.
P.T.O.
8203
*8023*
-2-
PART – B Answer one full question from each Module. Each question carries 20 marks. MODULE – I 11. a) Find by double integration, the smallest of the areas bounded by the circle x2 + y2 = 9 and the line x + y = 3. b) Evaluate
∫∫ xy dx dy taken over the positive quadrant of the circle x2 + y2 = a2.
6 7
c) Apply Stokes theorem to evaluate ∫ ( x + y) dx + ( 2 x − z ) dy + ( y + z ) dz where c
C is the boundary of the triangle with vertices (0, 0, 0), (2, 0, 0) and (0, 3, 0).
7
12. a) Find the volume of the tetrahedran bounded by the co-ordinate planes and the plane
x y z + + = 1. a b c
6
b) Use Green’s theorem in a plane to evaluate ∫ ( 2 x − y) dx + ( x + y) dy where C is c
the boundary of the circle
x2
+ y2 = a2 in
c) Use divergence theorem to evaluate
∫ ∫s
the xy – plane.
7
F ⋅ nˆds where F = 4 xˆi − 2 y 2ˆj + z 2kˆ and
S is the surface bounding x2 + y2 = 4, z = 0 and z = 3.
7
MODULE – II 13. a) Expand 2x – x2 in a Fourier series in the interval (0, 3).
6
b) Find the Fourier series expansion of the function F( x) = − π, = x,
− π< x <0 0<x <π
1 1 1 π2 Hence deduce that 2 + 2 + 2 + ... = . 1 3 5 8 ∞
c) Using Fourier integral show that ∫ 0
cos xλ π dλ = e −x , x ≥ 0 . 2 1+ λ 2
7
7
*8023*
8203
-3-
6
14. a) Obtain the Fourier series for F(x) = 0, − π ≤ x ≤ 0 = sin x, 0 ≤ x ≤ π b) Obtain the Fourier cosine series for K in the range (0, π ). 2 c) Find the Fourier cosine transform of e–x .
7 7
MODULE – III 6
15. a) Solve (y2 + z2)p – xyq + xz = 0. 2
b) Solve ( D 3 + D 2D′ − D D′ − D′3 ) z = cos ( x + y).
7
c) A tightly stretched string with fixed end points x = 0 and x = l is initially displaced in a sinusoidal arch of height y0 and then released from rest. Find the displacement function y(x, t).
7 6
16. a) Solve p2x2 + q2y2 = z2. b) Using the method of separation of variables, solve the equation subject to u(x, 0) = 6e–3x.
∂u ∂u =2 +u ∂x ∂t
2 ∂u 2 ∂ u subject to the conditions u(0, t) = 0 = u(l, t), =c ∂t ∂x 2 n πx for t ≥ 0 ; u (x, 0) = 3 sin , 0 < x < l. l
7
c) Solve the heat equation
––––––––––––––––
7