## Applied Mathematics 4 - May 2013

### 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 matrix

is derogatory(5 marks)
**1 (c) ** Evaluate the following:

where C is |z-1| = 1(5 marks)
**1 (d)** Evaluate _{A}∫^{B}(3x^{2}y - 2xy)dx + (x^{3} - x^{2})dy along y^{2} = 2x^{3} from A(0,0) and B(2,4)(5 marks)
**2 (a)** Prove that: xJ'_{n}(x) = -nJ_{n}(x) + xJ_{n-1}(x)(6 marks)
**2 (b)** Show that the matrix:

is diagonalizable.

Also find the transforming and diagonal matrix.(7 marks)
**2 (c) ** Evaluate ∬(∇×F). ds where F = (2x - y + z)i + (x + y - z^{2})j + (3x - 2y + 4z)k and 'S' is the surface of the cylinder x^{2} + y^{2} = 4 bounded by the plane z = 9 and open at the other end.(6 marks)
**3 (a)** Evaluate the following:

where C is:

(i)|z - 2 - i| = 2

(ii)|z - 1 - 2i| = 2

(7 marks)
**3 (b)** Show that F = (ye^{xy} cos z)i + (xe^{xy}cos z)j - (e^{xy}sin z)k is irrotational and find the scalar potential for F and evaluate ∫F.ds along the curve joining the points (0, 0, 0) and (-1, 2, π)(7 marks)
**3 (c) ** Prove that:

(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 = 2xi + xyj - zk over the region bounded by the cylinder x2 + y2 = 4, z = 0, z = 6(7 marks)
**4 (c)** find A^{100}

(6 marks)
**5 (a)** Define conformal mapping. Find Bilinear transformation which maps the points z = 0, i, -1 onto w = i, 1, 0.(7 marks)
**5 (b)** Evaluate the following:

(7 marks)
**5 (c) ** Find the characteristic roots and characteristic vectors of A^{3}+1 (6 marks)
**6 (a)** Expand the following:

about z = 0 for

(i) |z| < 1; (ii) 1 < |z| < 2; (iii) |z| > 2(7 marks)
**6 (b)** if f(z) = u + iv is analytic and find f(z)(7 marks)
**6 (c) ** Verify Cayley Hamilton Theorem for

and hence find the matrix 2A^{5} - 3A^{4} + A^{2} - 4I(6 marks)
**7 (a)** Prove that the circle |z| = 1 in the z-plane is mapped onto the cardiode in the w-plane under the transformation w=z^{2}+2z(7 marks)
**7 (b) ** Reduce the following quadratic form to canonical form and find its rank and signature:

(7 marks)
**7 (c) ** Verify Green's Theorem for

where c is boundary of the region defined by x = 1, x = 4, y = 1, y = √x(6 marks)