Applied Mathematics - 3 - Dec 2012

Electronics & Telecomm. (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.

This Qs paper appeared for Applied Mathematics - 3 of

Electronics Engineering . (Semester 3) 1 (a) Prove that $$w=\frac{x}{x^2+y^2}\ -i\frac{y}{x^2+y^2}$$ is analytical find f(z) in terms of z.
(5 marks) 1 (b) Find the fourier expansion for f(x)=x in (0,2?).(5 marks) 1 (c) Find the Laplace Transform of
Sint. H $$\left(t-\frac{\pi{}}{2}\right)-h\left(t-\frac{3\pi{}}{2}\right)$$(5 marks) 1 (d) Find Z-Transform of {k ? e^-ak} k?0.(5 marks) 2 (a) Evaluate $$\int_0^{\infty{}}\frac{\cos{at-\cos{bt}}}{t}dt$$(6 marks) 2 (b) Find the Fourier series for f(x)= 4-x² in (0,2).Hence deduce that
$$\frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+$$(7 marks) 2 (c) Find the inverse of A if -
$$\left[\begin{array}{ccc}1 & 0 & 0 \\2 & -1 & 0 \\-2 & 1 & 1\end{array}\right]$$A $\ \left[\begin{array}{ccc}1 & -2 & 9 \\0 & 1 & -6 \\0 & 0 & 1\end{array}\right]$ = $\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$}(7 marks) 3 (a) Find Laplace transform of the following-
(i) $$e^{-t}\int_0^te^ucoshu\ du\ \ \ \ \ \$$\ltbr\gt (ii) $$e^{-et}\bullet{}erf\sqrt{t}$$
(6 marks) 3 (b) Find non singular matrices P and Q such that PAQ is in normal form. Also find rank fo A and A^-1 if it exist-
{\raggedrightA=$\ \left[\ \begin{array}{ccc}3 & 2 & -1 \\5 & 1 & 4 \\1 & -4 & 11\end{array}\ \ \ \ \ \begin{array}{ccc}5 \\-2 \\-19\end{array}\ \right]$}(7 marks) 3 (c) Evaluate by Green's Theorem
? _cF.dr where F= -xy(xi-yj)and 'C' is r=a(1+cos ?).(7 marks) 4 (a) Obtain Complex form of Furier Series for the function f(x)=sin ax in (-?,?) where 'a' is not an integer.(6 marks) 4 (b) Investigate for what value of ? and ? the equations.
x+2y+3z=4, x+3y+4z=5, x+3y+?z= ?
have (I) no solution
(II) a unique solution
(III) an infinite no. of solutions.(7 marks) 4 (c) Find Inverse Laplcae Transform of following?-
(I) 2tanh^-1s
(II) s+29/(s+4)(s²+9)(7 marks) 5 (a) Prove that u=1/2 log (x²+y²) is harmonic.(6 marks) 5 (b) Examine whether the following vectors are Linearly independent or dependent.
X₁ = [1,1,-1]
X₂ = [2,-3,5]
X₃ [2,-1,4] (7 marks) 5 (c) Express the function
f(x) = -e^kx, for x < 0
f(x) = -e^kx, for x > 0.
as Fourier integral and prove that -
$$\int_0^{\infty{}}\frac{wSin\ wx}{w^2+k^2}$$ dw = ?/2 e^-kx, if x > 0, k > 0(7 marks) 6 (a) Obtain half range cosine series for f(x)=sin(?x/l) in 0<x<l.< a="">

</x<l.<></span>(6 marks) 6 (b) Under the transformation W=z-1/z+1 show that the map of the straight line y=x is a circle and find its centre and radius.(7 marks) 6 (c) Verify Stoke's Theorem for-
F = yzi + zxj +xyk and C is the boundary of the circle x²+y²+z²=1, z=0.(7 marks) 7 (a) Find inverse Z-transform of F(z)=z/(z-1)(z-2), 1<|z|<2(6 marks) 7 (b) Find the analytic function whose real part is a=sin?2x/cosh?2y-cos?2x (7 marks) 7 (c) Using Laplace Transform.Solve the following differential equation with given condition (D²-4)y=3e^t, y(0)=0, y'(0)=3 (7 marks)