University of Mumbai 2009-3rd Sem B.E Computer Engineering S.E Applied Maths III - Question Paper
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Con. 5935*09. (REVISED COURSE) SP- 7367
(3 Hours) [Total Marks : 100
N. B. : (1) Question No. 1 is compulsory.
(2) Attempt any four questions out of remaining six questions.
(3) Use of calculator's (Non-Programmable) is allowed.
(4) Write the sub-questions of the main question collectively.
(5) Figures to the right indicates full marks.
1. (a) Find the Z-transform of f(k) = ak, k > 0. 5
(b) Find L (erf Vt) 5
(c) Is the following Matrix orthogonal ? If not, can it be converted into an 5 orthogonal matrix ? If yes how ?
2 2 1 -2 1 2 1 -2 2
A =
(d) Find the half range cosine series for f{x) ** x, 0 < x < 2.
0012 sin 3t
2. (a) Evaluate I
dt.
2t
0 e
(b) Obtain the Fourier Series
n
f(x) = x + ~t:<x<0
=--x 0 < x < n
2
(c) If A is a square matrix of order n and IAI * 0, then prove that
8
(i)
(b) Show that every square matrix can be uniquely expressed as the sum of a Hermitian 6 matrix and Skew-Hermitian matrix.
-ax
e
8
(c) Find Fourier Sine integral representation for f (x) =
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Con, 5935-SP-7367-09 v
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4. (a) Solve g-+ 4y = f(t) with conditions y(O) = 0, y'(0) = 1 6
Ot >. ' .; {; ; <
and f(t) = 1 when Q < t < 1
f(t) = 0 wh.SA .t > 1 . ; >
(b) Examine wholher the vectors X, = ( 3, 1, 1 ], X2 = [2, 0, -1], X3 - [ 4, 2, 1 ] are 7 Linearly independent.
(c) Examine a Fourier series to represent f(x) = x2 in (0, 2n) and hence deduce 7
-2 1 1 1 1 that, p 0 n o .
12 12 2 3 4
5. (a) Find Laplace transform of Sinhat. Sinat. 6
(b) Find the Z - transform of Cos ; + a j , k > 0. g
(c) Find the Fourier expansion of 8
f{x) 2 when, -2 < x < 0 ; . : f(x) = x when 0 < x < 2
6. (a) Show that functions f1 (x) = 1, f2 (x) = x are orthogonal on (-1, 1). 6
Determine the constants a and b such that the function f3 (x) = -1 +ax +bx2 Is orthogonal to both f1 and f2.
(b) State Convolution theorem for Z - transform. 6 Hence find Z - transform of f(k)*g(k) where f(k) = 4k U(k)
(c) Find Inverse L.T. of 8
1
sjs + 4
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