Question Paper: Applied Mathematics 4 : Question Paper May 2015 - Electronics Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - May 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) 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 $$\int^1_0 [y'^2 +12 x y \mathrm {dx}$$ with y(0)=0 & y(1)=1 can be Extremised.(6 marks) 2 (b) Find the Eigen values and Eigen vectors for the matrix $$\begin{bmatrix} 2 &2 &1 \\1 &3 &1 \\1 &2 &2 \end{bmatrix}$$(6 marks) 2 (c) 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.(8 marks) 3 (a) $$if \ A= \begin{bmatrix} 2 &1 \\1 &2 \end{bmatrix} \ find \ A^{20}$$(6 marks) 3 (b) Evaluate $$\int_c \dfrac {\sin Π z^2 - \cos \pi z^2}{(z-1)(z-2)}dz,$$ where c is the circle |z|=3.(6 marks) 3 (c) 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 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Find the vector orthogonal to both [-6, 4, 2] & [3, 1, 5]\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Show that the matrix$$ A= \begin{bmatrix} 7 &4 &-1 \4 &7 &-1 \-4 &-4 &4 \end{bmatrix} $$is derogatory and find is minimal polynomial.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Reduce the matrix of the quadratic form [ 6x^2_1 + 3x^2_2 + 3x^2_3 - 4x_1x_2-2x_2x_3 ] to canonical form through congruent transformation and find its rank, signature, and value class.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Find the external of$$ \int^{x_1}_{x_0} (2xy-y''^2) dx $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Show that the set W={[x,y,z] | y=x+z} is a subspace of R\ltsup\gtn\lt/sup\gt under the usual addition and scalar multiplication.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Show that the following matrix$$ A= \begin{bmatrix} 6 &-2 &2 \-2 &3 &-1 \2 &-1 &3 \end{bmatrix} $$is diagonalisable. Also find the diagonal form and a diagonalising matrix.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt$$ If f(a) = \int_c \dfrac {3z^2 = 7z +1}{z-a} dz $$where c is a circle |z|=2, find the values of i) f(-3), ii) f(i), iii) f'(1-i)\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Evaluate$$ \int^{2\pi}_0 \dfrac {d\theta}{13+5 \sin \theta } $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Verify Caylex-Hamilton theorem for the matrix A and hence find A\ltsup\gt-1\lt/sup\gt and A\ltsup\gt4\lt/sup\gt where$$ A= \begin{bmatrix} 1 &2 &-2 \-1 &3 &0 \0 &-2 &1 \end{bmatrix} (8 marks)

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