University of Mumbai 2008-3rd Sem B.E Electrical and Electronics Engineering Applied Maths-III - Question Paper
Applied Maths-III Sem III June 2008
P4/RT-E*0*-711
CO-9523 [Total Marks : 100Con. 3528-08.
v\P
N.B.; (1) Question No 1 is compulsory.
(2) Attempt any four questions from remaining six questions
FVoMu. ITT
[REVISED COURSE]
(3 Hours)
<3) Figures to the right indicate full marks. -pr-
\ \rcei) Lam \3r~w rt\ouH-w t y\
1, '"(a) State and prwe ParsevaTi
's identity over (c, c + 21)*
(b) Find orthogonal trajectories of the family of curves e-x cos y + xy = const.
(s + (s2 + 4)
-1
(c) Fmd L
(d) S.T every square matrix can be uniquely expressed as a sum of Hermilian and Skew- 5 Hermitian matrix.
2. (a) Solve using Laplace transform -
6
-y + 2y + fydl * sint. given y(0) = 1. dl 0
sin2x
6
4
(b) Find analytic function 1(2) whose real part is cos2y + cos2x *
(c) (i) If 'A' is a nonsingular square mainx of order VT trrnn S.T.
adj (ad) - A) = lAP-A.
(ii) Verify A(adj A) IAI I
1-2 3'
Where A * 2 3-1 ~3 1 2
6
3. 'a) Use Laplace transform to evaluate -
u
0 0
(b) S.T. every bilinear transformation is the resultant of three basic transformations.
(c) Find Fourier series for -
over (0, 2n).
f(x)
(*) 12 * 12 5? + 32 42
4. (a) Find A~1 by using elementary transformations
2 3 1
1 2 3
Where A
3 1 2
Con. 3528-CO-9523-08. r~ , . 2 ,
(b) Find Fourier series for - 6
f(x) = x cos x ; n < X < n.
(o) S.T. the relation W = transforms the real axis in z-plane into a circle in w-plane. 8
Find its centre and radius. Also find the point in z-plane which is mapped on the centre of the circle in w-plane.
5. (a) For what values of X and m the system 6
x + y + z = 6 x + 2y + 3z * 10 x + 2y + kz = m
has (i) no solution (ii) unique solution (iii) more than one solutions.
Also find parametric solution.
(b) Find hall range Cosine series for 6
[1 0 < x < 1
1 < x < 2
(c) Find (i) L
s -2s -2J
1 Os t S 1
6 (a) If f(t)
0 1 < t < 2 and f(t) = f(t + 2) then
1
(1 + 8"*)
(b) Define orthogonal and orthonormal set of functions. S.T. { sin n x ]n 2 3....... s
orthogonal set of functions over [0, it]. Hence construct orthonormal set of functions.
<c) Find Fourier series for
f(x) = 2x x2 ; 0 < x < 2.
(a) Find complex form of Fourier series for -f(x) = ; K, L).
... . ... . JVl+sin4t Fmd (1) L
(b)
(ii) L {(t sin2t)2 J.
-1
(c) State convolution theorem and use it to find L*
(s2+4)
OR
(c) Define cross ratio. Find Bilinear transformation which transforms point z * I, 1,-1 into 8 w = 1, 0, respectively.
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