Applied Mathematics 4 - Dec 2015

Electronics Engineering (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) Find the extremal of $ \int_{x_4}^{x_1} (2xy - y^{-/2}) dx $(5 marks) 1 (b) Find an orthonormal basis for the subspaces of R³ by applying gram-Schmidt process where S={(1, 2, 0) (0, 3, 1)}.(5 marks) 1 (c) Show that Eigen values of unitary matrix are of unit modulus.(5 marks) 1 (d) Evaluate $ \int \dfrac {dz}{z^3 (z+4)} \text { where }|z|=4 $(5 marks) 2 (a) Find the complete solution of $ \int^{x_1}_{z_0} (2xy - y^{1/2})dx $(6 marks) 2 (b) Find the Eigen value and Eigen vectors of the matrix A^3 where $ A=\begin{bmatrix} 4 &6 &6 \\\\1 &3 &2 \\\\-1 &-5 &-2 \end{bmatrix} $(6 marks) 2 (c) Find expansion of $ f(z) = \dfrac {1} {(1+z^2)(z+2)} $ indicating region of convergence.(8 marks) 3 (a) Verify Cayley-Hamilton Theorem and find the value A⁶⁴ for the matrix $ A= \begin{bmatrix} 1 &2 \\\\2 &-1 \end{bmatrix}$(6 marks) 3 (b) Using Cauchy's Residue Theorem evaluate $ \int^\infty_{-\alpha} \dfrac {x^2}{x^6 +1}dx $(6 marks) 3 (c) Show that a closed curve 'C' of given fixed length (perimeter) which encloses maximum area is a circle.(8 marks) 4 (a) State and prove Cauchy-Schwartz inequality. Verify the inequality for vector u=(-4, 2, 1) and v=(8, -4, 2).(6 marks) 4 (b) Reduce the quadratic form xy+yz+zx to diagonal form through congruent transformation.(6 marks) 4 (c) If $ A= \begin{bmatrix} \frac {3}{2} & \frac {1}{2} \\\\ \frac {1}{2} & \frac {3}{2} \end{bmatrix} $ then find e^A and 4^A with the help of Modal Matrix.(8 marks) 5 (a) Solve the boundary value problem $ \int^1_0 (2xy+y^2 - y^2) dx, \ 0\le x \le 1, \ y(0)=0, \ y(1)=0 $ by Rayleigh - Ritz Method.(6 marks) 5 (b) If W={∝; ∝∈Rⁿ and a₁ ≥ 0} a subset of V=Rⁿ with ∝=(a₁, a₂ ....... a_n) in Rⁿ (n≥3.). Show that W is not a subspace of V by giving suitable counter example.(6 marks) 5 (c) Show that the matrix $ A=\begin{bmatrix} 8 &-8 &-2 \\\\4 &-3 &-2 \\\\3 &-4 &1 \end{bmatrix} $ is similar to diagonal matrix. Find the diagonalising matrix and diagonal form.(8 marks) 6 (a) State and prove Cauchy's integral Formula for the simply connected region and hence evaluate $$ \int \dfrac {z+6}{z^2-4}dz, |z-2|=5 $$(6 marks) 6 (b) Show that $ \int^{2\pi}_0 \dfrac {\sin^2 \theta}{a+ b\cos \theta}d \theta = \dfrac {2\pi}{b^2} (a- \sqrt{a^2 - b^2}), \ 0 < b< a. $(6 marks) 6 (c) Find the Singular value decomposition of the following matrix $ A=\begin{bmatrix} 1 &2 \\\\1 &2 \end{bmatrix} $(8 marks)