Applied Mathematics 4 - May 2015

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) Evaluate $$ int_c |z| dz $$ where c is the left half of unit circle |z|=1 from z=-i to z=i(5 marks) 1 (b) If λ is an Eigen value of the matrix A with corresponding Eigen vector X. Prove that λⁿ is an Eigen value of Aⁿ with corresponding Eigen vector X.(5 marks) 1 (c) Find the external of $$ \int^{x_2}_{x_1} \dfrac {\sqrt{1+y'^2}}{x} dx $$(5 marks) 1 (d) Find the unit vector orthogonal to both [1, 1, 0] & [0, 1, 1](5 marks) 2 (a) Find the curve on which the functional $$ \int^1_0 [y'^2 +12 x y$$ dx $$ with y(0)=0 & y(1)=1 can be Extremised.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the Eigen values and Eigen vectors for the matrix $$ \begin{bmatrix} 2 &2 &1 \1 &3 &1 \1 &2 &2 \end{bmatrix} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c)\lt/b\gt Obtain two distinct Laurent's series expansions of $$ f(z) = \dfrac {2z-3} {z^2 -4z +3} $$ in power of (z-4) indicating the region of convergence in each case.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt $$ if \ A= \begin{bmatrix} 2 &1 \1 &2 \end{bmatrix} \ find \ A^{20} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Evaluate $$ \int_c \dfrac {\sin Π z^2 - \cos \pi z^2}{(z-1)(z-2)}dz, $$ where c is the circle |z|=3.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c)\lt/b\gt Using Reyleigh-Ritz method, find an approximate solution for the external of the functional $$ I(y) = \int^1_0 (y'^2-2y-2xy) dx $$ subject to y(0)=2, y(1)=1 $$(8 marks) 4 (a) Find the vector orthogonal to both [-6, 4, 2] & [3, 1, 5](6 marks) 4 (b) Show that the matrix $$ A= \begin{bmatrix} 7 &4 &-1 \\4 &7 &-1 \\-4 &-4 &4 \end{bmatrix} $$ is derogatory and find is minimal polynomial.(6 marks) 4 (c) Reduce the matrix of the quadratic form [ 6x^2_1 + 3x^2_2 + 3x^2_3 - 4x_1x_2-2x_2x_3 ] to canonical form through congruent transformation and find its rank, signature, and value class.(8 marks) 5 (a) Find the external of $$ \int^{x_1}_{x_0} (2xy-y''^2) dx $$(6 marks) 5 (b) Show that the set W={[x,y,z] | y=x+z} is a subspace of Rⁿ under the usual addition and scalar multiplication.(6 marks) 5 (c) Show that the following matrix $$ A= \begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix} $$ is diagonalisable. Also find the diagonal form and a diagonalising matrix.(8 marks) 6 (a) $$ If f(a) = \int_c \dfrac {3z^2 = 7z +1}{z-a} dz $$ where c is a circle |z|=2, find the values of i) f(-3), ii) f(i), iii) f'(1-i)(6 marks) 6 (b) Evaluate $$ \int^{2\pi}_0 \dfrac {d\theta}{13+5 \sin \theta } $$(6 marks) 6 (c) Verify Caylex-Hamilton theorem for the matrix A and hence find A^-1 and A⁴ where $$ A= \begin{bmatrix} 1 &2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix} $$(8 marks)