MORE IN Applied Mathematics - 3
MU Electronics 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

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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