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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.

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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.

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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).

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12. a) Find the volume of the tetrahedran bounded by the co-ordinate planes and the plane

x y z + + = 1. a b c

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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.

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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.

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MODULE – II 13. a) Expand 2x – x2 in a Fourier series in the interval (0, 3).

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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

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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 .

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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).

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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).

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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

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c) Solve the heat equation

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