**PART A**

**Question 1 :**

(a)Find Laplace inverse of (1/(s-2)^3)?

(b)Define even and odd functions with one example of each?

(c)Give the region of integration of the areas bounded by ellipse 4x^2 + 9y^2 = 36 and line 2x + 3y = 6?

(d)Find the general value of log(-1+i)?

(e)Find tanhx,if 5sinhx - coshx = 5 ?

**Question 2 :**

(a)solve(D^2 + n^2)x = asin(nt + a),x=Dx=0 at t=0 using laplace transformations?

(b)if x=CiS(p/2^r).show that lim n->infinity x1x2x3x4.....xn = -1.

(c)Using gauss divergence theorem find integration of F.nds:

F = (2x + 3z)i - (xz + y)j + (y^2 + 2z)k and S is the surface of the sphere having centre at (3,-1,2) and radius 3.

**PART B**

**Question 3 :**

Find laplace transform of

i)tsin^2(3t)

ii)t^3e^-3t

iii)t^2e^-2tcost

**Question 4 :**

(a)Find the angle between the surfaces x^2 + y^2 + z^2 = 9 and z= x^2 + y^2 -3 at the point(2,-1,2).

(b)If r=xi+yj+zk, show that del r=nr^(n-2).r

**Question 5 :**

Obtain the fourier series:

f(x) = 2, -2<=x<=0

x, 0<=x<=2

**Question 6 :**

If r=xi+yj+zk and r=|r|!=0. Then prove that

i)div(r/r^3)=0

ii)div(r.r^n)=(n+3)r^n

iii)curl(r.r^n)=0

**PART C**

**Question 7 :**

Sum of series 1+xcosa+x^2cos2a+x^3cos3a+.....to n terms where x is less than unity,Alsofind the sum to infinity.

**Question .8 :**evaluate gauss divergence where A=zi+xj-3y^2zk and S is the surface of cylinder x^2+y^2=16.Included in first octant between z=0 and z=5.