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(3x²y - 2xy)dx + (x³ - x²)dy along y² = 2x³ 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 - z²)j + (3x - 2y + 4z)k and 'S' is the surface of the cylinder x² + y² = 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^xycos z)j - (e^xysin 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 A¹⁰⁰
(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 A³+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 2A⁵ - 3A⁴ + A² - 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=z²+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)