Applied Mathematics 4 - May 2012

Electronics & Telecomm. (Semester 4)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Prove that:
(5 marks) 1 (b) Show that:
is derogatory (5 marks) 1 (c) Evaluate the following:
where S is the surface of the plane

in the first octant and (5 marks) 1 (d) Evaluate the following:
where C is the curve |z-2| + |z+2| = 6.(5 marks) 2 (a) Prove that for any positive integer n, J_-n(x) = (-1)ⁿ J_n(x)(7 marks) 2 (b) Show that the matrix
is diagonalizable.
Also find the transforming and diagonal matrix.(7 marks) 2 (c) Show that the area bounded by simple closed curve C is given by
Find the area of ellipse (6 marks) 3 (a) Verify Cauchy's theorem for function f(z) = 3z² + iz - 4 if 'C' is the perimeter of square with vertices at 1±i, -1±i(7 marks) 3 (b) Prove that
(7 marks) 3 (c) Show that F = (2xy+z³)i+(x²)j+3xz²k is irrotaional and find its field from (1, -2, 1) to (3, 1, 4)(6 marks) 4 (a) Define analytic function. State and prove Cauchy-Reimann equation in polar co-ordinates(7 marks) 4 (b) Verify Divergence Theorem; evaluate for F = xi - 3y²j + zk over the region bounded by the cylinder x² + y² = 16, z = 0, z = 5(7 marks) 4 (c) Find e^At If
(6 marks) 5 (a) Define conformal mapping. Find bilinear transofrmation which maps the points z = 2, I, -2 onto w = 1, I, -1(7 marks) 5 (b) Evaluate the following:
(7 marks) 5 (c) Show that cos(x sinθ)=J0(x) + 2cos2θJ2(x) + 2cos4θJ4(x)-�(6 marks) 6 (a) Find all the possible Laurent's expansion of the function
about z = -1 indicating the region of convergence(7 marks) 6 (b) Prove that there does not exist any analytic function whose real part is 3x²-2x² y+y²(7 marks) 6 (c) Verify Cayley Hamilton Theorem for

$A=\begin{bmatrix} 3 &1 \\\\-1 &2 \end{bmatrix}$
and hence find the matrix 2A⁵ - 3A⁴ + A² - 4I (6 marks) 7 (a) Explain removable singularity with example.
Evaluate ∫_C tan z dz, where C is the circle |z| = 2, using residue theorem(7 marks) 7 (b) Find the analytic funstion whose real part is: e^-x {(x² - y²)cosy + 2xy siny} (7 marks) 7 (c) Reduce the following quadratic equation to canonical form and find its rank and signature x² + 4y² + 9z² + t² - 4xy + 6zx - 12yz - 2xt - 6zt(6 marks)