Reg.
No
MANIPAL INSTITUTE OF TECHNOLOGY
(A constituent college of Manipal University, Manipal –576 104)
THIRD SEMESTER BE DEGREE END SEMESTER EXAMINATIONS – 2007
SUB: ENGG.MATHEMATICS – III (MAT –ECE – 201 )
(REVISED CREDIT SYSTEM)
Time : 3 Hrs. Max.Marks : 50
" Note : Answer any FIVE full questions.
1A. Find the Fourier series of
⎧ π π
⎪⎪ x, − ≤x≤
2 2
f (x) = ⎨ ; f(x + 2π) = f(x) for all x.
⎪π − x, π 3π
≤x≤
⎪⎩ 2 2
( ) ( ) ( )
ur ur ur ur ur ur
1B. Show that ∇g A × B = Bg ∇ × A − Ag ∇ × B
1C. If f(z) = u + iv is an analytic function, Prove that
2
⎡∂ ⎡∂ ⎤
2
⎤
⎢⎣ ∂x | f (z) |⎥⎦ + ⎢ ∂y | f (z) |⎥ =| f ′ ( z ) |
2
⎣ ⎦
(4+3+3)
2A. Using the indicated transformations, solve :
uxx + 2uxy + uyy = 0 with v = x, z = x – y .
x
2B. If f (z) = u + iv is analytic and u + v = , find f(z).
x + y2
2
2C. Prove that F = ( y 2 cos x + z3 ) i + ( 2ysin x − 4 ) j + (3xz 2 + 2)k is a
conservative force field. Find the scalar potential for F.
(4+3+3)
⎧1 + 2x, −3 ≤ x ≤ 0
3A. Find the Fourier series of f (x) = ⎨ and T = 6.
⎩1 − 2x, 0 < x < 3
3B. Verify Cauchy’s Integral theorem for f(z) = z2 over the simple closed curve
given by y = 0, x = 0 and y2 = 8 – 4x in the first quadrant.
P.T.O
3C. Solve the following partial differential equations by the method of
separation of variables : x2 uxy + 3y2 u = 0.
(4+3+3)
4A. Find the half range cosine series for the function
⎧ kx, 0 < x <1
f (x) = ⎨
⎩k(2 − x), 1 < x < 2
4B. Show that every straight line or circle maps onto a circle or straight line
1
under the mapping w = .
z
∫ ( 3x + 2y )dx − ( x + 3cos y ) dy around the parallelogram
2
4C. Evaluate : Ñ
having vertices at (0,0), (2,0), (3,1) and (1,1)
(4+3+3)
5A. Find the linear fractional transformation which maps
z1 = 1, z 2 = i, z3 = −1 onto w1 = i, w 2 = −1, w 3 = −i respectively.
ur
5B. Evaluate : ∫∫
S
A.n ds , where A = z i + xj − 3y 2 zk and S is the surface of the
cylinder x + y2 = 16 included in the first octant between z = 0 and z = 5.
2
1
5C. Find all possible series expansion of f (z) =
z − 3z + 2
2
(4+3+3)
6A. Evaluate : ∫∫ ( ∇ × A )gnds
S
where A = xzi – yj + x2yk and S is the surface
of the region bounded by x = 0, y = 0, z = 0, 2x + y +2z = 8 which is not
include in the xz – plane.
2π
sin 2 θ − 2cos θ
6B. Evaluate : ∫ dθ ,
0
2 + cos θ
using Contour integration.
∞
dx
6C. Evaluate: ∫
(x + 3)( x 2 + 2 )
2 2
0
(4+3+3)
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