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MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
December 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M
1 (b) Find the fourier expansion for f(x)=x in (0,2?).
5 M
1 (c) Find the Laplace Transform of
Sint. H $\left(t-\frac{\pi{}}{2}\right)-h\left(t-\frac{3\pi{}}{2}\right)$
5 M
1 (d) Find Z-Transform of {k ? e-ak} k?0.
5 M

2 (a) Evaluate $\int_0^{\infty{}}\frac{\cos{at-\cos{bt}}}{t}dt$
6 M
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 M
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 M

3 (a) Find Laplace transform of the following-
(i) $e^{-t}\int_0^te^ucoshu\ du\ \ \ \ \ \$
(ii) $e^{-et}\bullet{}erf\sqrt{t}$
6 M
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 M
3 (c) Evaluate by Green's Theorem
? cF.dr where F= -xy(xi-yj)and 'C' is r=a(1+cos ?).
7 M

4 (a) Obtain Complex form of Furier Series for the function f(x)=sin ax in (-?,?) where 'a' is not an integer.
6 M
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 M
4 (c) Find Inverse Laplcae Transform of following?-
(I) 2tanh-1s
(II) s+29/(s+4)(s2+9)
7 M

5 (a) Prove that u=1/2 log (x2+y2) is harmonic.
6 M
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 M
5 (c) Express the function
f(x) = -ekx, for x < 0
f(x) = -ekx, 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 M

6 (a) Obtain half range cosine series for f(x)=sin(?x/l) in 0
6 M
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 M
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 M

7 (a) Find inverse Z-transform of F(z)=z/(z-1)(z-2), 1<|z|<2
6 M
7 (b) Find the analytic function whose real part is a=sin?2x/cosh?2y-cos?2x
7 M
7 (c) Using Laplace Transform.Solve the following differential equation with given condition (D2-4)y=3et, y(0)=0, y'(0)=3
7 M

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