Rajiv Gandhi Proudyogiki Vishwavidyalaya 2009-3rd Sem B.E Biomedical Engineering - Question Paper

B. E. third Semester APPLIED MATHEMATICS-III 2009
Time : three Hours Maximum Marks : 60
NOTE:- This paper consist of 3 parts. part A is compulsory. Do any 4 ques.
from part B and any 2 ques. from part C
Section-A Marks : 20
1(a) obtain the locus represented by | z - two i | = 2.
(b) If f(z) = x2 + i y2, obtain the points in the z plane where f'(z) is described. Also obtain its value at these
points.
(c) Distinguish ranging from the zeros and poles of a function w = f(z). Can an analytic function have zeros
and poles ?
(d) describe Jn(x) and write the differential formula which has Jn(x) as its solution. elaborate the values
of Jo(x) and J1(x) ?
(e) State Rodrigue's formula and use it to evaluate P2(x).
(f) Eliminate arbitary functions f and g from u = f(x + i y) + g(x - iy) and classify the resulting partial
differential formula.
(g) describe Fourier sine-cosine over the interval -p to p. Is it possible to write this series for the constant
f(x) = two over this interval.
(h) describe Laplace transform. If f(s) is the Laplace transform of f(t) then what is the laplace transform
of
(i) Write the partial differeantial formula which governs the steady state distribution of temperature
inside a circular plate whose both faces are insulated and the circumference is kept at steady
temperature f(?). Also write its boundary conditions and initial conditions if any.
Section-B Marks:5 every
2. Derive necessary form of C.R. equations for a function w = f(z) to be analytic. Also obtain the image
of the circle | z - one | = one in w plane under the mapping w = z2.
3. Show that with usual notions xnJn(x) is the solution of
4. Solve the subsequent partial differential equations:
(a) (y + z) p + (z + x) q = x + y
5. provided that c is a constant, show that it is possible to write: in the range 0 < x < p.
6. (a) obtain the Laplace transform of
f(t) = t/T, 0 < t <=T = one , t > T
(b) obtain the inverse Laplace transform of (s + 2)/(s-2)3
Section-C Marks : 10 every
7. (a) Evaluate where c is | z - two | = 2.
(b) Use method of contour integration to evaluate:
8. (a) Prove that with usual notations:
(b) Use method of Laplace transform to solve the differential formula :
(D2 + 5D + 6)x = t et provided that x=2, dx/dt = 1, at t = 0.
9. Use method of seperation of variables to solve the wave formula and use this solution to find the
diflection u(x,t) of a vibrating string of length l whose end points are fixed and the string is provided zero
initial velocity and initial deflection: f(x) = 2kx/l; 0 < x < l/2 = (2k/l)(l-x); l/2 < x < l.

Earning:   Approval pending.