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Applied Mathematics 3 : Question Paper May 2012 - Information Technology (Semester 3) | Mumbai University (MU)
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Applied Mathematics 3 - May 2012

Information Technology (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a) Find adj A, A-1 if A is a matrix as given below. Also find B.
(5 marks)
1(b) Find Laplace transform of
(5 marks)
1(c) A regular function of constant magnitude is constant.(5 marks) 1(d) Find the Fourier series f(x) =1-x2 in (-1,1)(5 marks) 2(a) Expand f(x) with period 2 into a Fourier series.
(6 marks)
2(b) Find the orthogonal trajectories of the family of curves e-x (x siny - y cosy) = c(7 marks) 2(c) Using convolution theorem, prove that,
(7 marks)
3(a) Show that every square matrix A can be uniquely expressed as P+iQ where P and Q are Hermitian matrices.(6 marks) 3(b) Using Cauchy's residue theorem evaluate the following where C is the circle (i) |z| = 1/2 (ii) |z + i| = 3:
(7 marks)
3(c) Solve the following equation by using Laplace transform. Given that y(0) = 1
(7 marks)
4(a) State Laplace equation in polar form and verify it for u = r2cos 2? and also find V and f(z).(6 marks) 4(b) Find the Fourier expansion for
f(x)= ?(1-cosx) ... 0 < x < 2? and hence show that
(7 marks)
4(c) Evaluate the following:
(7 marks)
5(a) Using residue theorem evaluate:
(6 marks)
5(b) Reduce the following matrix to normal form and find its rank
(7 marks)
5(c) (i) Express the function as Heaviside's unit step function and find their Laplace Transforms
f(t) = 0 ... 0 < t < 1
= t2 ... 1 < t < 3
= 0 ... t > 3

(ii) Find L {f(t)} where
f(t) = t ... 0 < t < 1
= 0 ... 1 < t < 2
(7 marks)
6(a) Investigate for what values of ? and ? the equations:
x + 2y + 3z = 4
x + 3y +4z = 5
x + 3y + ?z = ?
have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.
(6 marks)
6(b) Show that the set of functions sin(2n + 1)x where n=0, 1, 2, ... is orthogonal over [0, ?/2]. Hence construct the orthogonal set of functions.(7 marks) 6(c) Find all Laurent's expansion of the function f(z)
(7 marks)
7(a) Find L{cost cos2t cos3t}(6 marks) 7(b) Show that the vectors [1, 0, 2, 1], [3, 1, 2, 1], [4, 6, 2, -4], [-6, 0, -3, -4] are linearly dependent and find the relation between them(7 marks) 7(c) Obtain half range sine series for f(x) where

(7 marks)

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