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MU Information Technology (Semester 3)
Applied Mathematics 3
December 2014
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) Find the Laplace Transform of sint cos2t cosht.
5 M
1 (b) Find the Fourier series expansion of f(x)=x2 (-?, ?)
5 M
1 (c) Find the z-transform of (13)1K1
5 M
1 (d) Find the directional derivative of 4xz2+x2yz at (1, -2, -1) in the direction of 2i-j-2k
5 M

2 (a) Find an analytic function f(z) whose real part is ex (xcosy-ysiny)
6 M
2 (b) Find inverse Laplace Transform by using convolution theorem 1(s3)(s+4)2
6 M
2 (c) Prove that ¯F=(6xy22z3)¯i+(6x2y+2yz)¯j+(y26z2x)¯k is a conservative field. Find the scalar potential ? such that ??=F. Hence find the workdone by F in displacing a particle from A(1,0,2) to B(0,1,1) along AB.
8 M

3 (a) Find the inverse z-transform of f(z)=z3(z3)(z2)2
i) 2<|z|<3 ii) |z|>3
6 M
3 (b) Find the image of the real axis under the transformation w=2z+i
6 M
3 (c) Obtain the Fourier series expansion of f(x)=πx;0x1=π(2x);1x2 Here deduce That 112+132+ =π28
8 M

4 (a) Find the Laplace Transform of f(t)=E;0tp/2=E;p/2tp,f(t+p)=f(t)
6 M
4 (b) Using Green's theorem evaluate \[ \int_c \dfrac {1}{y} dx + \dfrac {1}{x} dy where c is the boundary of the region bounded by x=1, x=4, y=1, y=?x
6 M
4 (c) Find the Fourier integral for f(x)=1x2.0x1=0   x>1 Hence Evaluate 0λcosλsinλλ3cos(λ2)dλ
8 M

5 (a) If F =x2 i + (x-y)j+ (y+z)k moves a particular from A(1,0,1) to B(2,1,2) along line AB. Find the work done.
6 M
5 (b) Find the complex form of Fourier series f(x)= sinhax(-l,l).
6 M
5 (c) Solve the differential equation using Laplace Transform. (D2+2D+5)y=e-t sint y(0)=0 y'(0)=1
8 M

6 (a) If int0e2tsn(t+α)cos(tα)dt=38 find the value of ?.
6 M
6 (b) where is the hemisphere x2+y2+z2=1 above xy-plane and bounded by this plane.
6 M
6 (c) Find Half range sine series for f(x)=lx-x2 (0, l) Hence prove that \dfrac {1}{1^6}+ \dfrac {1}{3^6}+ \cdots \cdots = \dfrac {\pi ^6}{960}
8 M



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