written 8.1 years ago by
teamques10
★ 64k
|
modified 16 months ago
by
sagarkolekar
★ 11k
|
|
written 8.1 years ago by
teamques10
★ 64k
|
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 Eigen values of a hermitian matrix are real.(5 marks)
1(b) Evaluate$$ \displaystyle \int_{c}\dfrac{e^{kx}}{z}dz $$ over the circle |z|=1 and k is real.hence prove that$$ \displaystyle\int_{0}^{\pi}c^{k cos \theta}cos(k sin \theta)d \theta =\pi $$
(5 marks)
1(c) Find the extremal of $$ \displaystyle \int_{x_{2}}^{x_{1}}(16y^{2}-(y^n)^{2}+x^{2})dx $$(5 marks)
1(d) Find a vector orthogonal to both u =(-6,4,2)and v=(3,1,5).(5 marks)
2(a) Find the curve y=f(x)for which $$ \displaystyle \int_{x_{1}}^{x_{2}}y\sqrt{1+(y')^{2}}dx $$ is minimum subject to the constraint $$ \displaystyle \int_{x_{1}}^{x_{2}}y\sqrt{1+(y')^{2}}dx=l.$$ (6 marks)
2(b) Find eigen values and eigen vectors of the matrix $$ A=\begin{bmatrix}
-2 & 5 & 4\\
5& 7 &5 \\
4& 5& -2
\end{bmatrix}$$
(6 marks)
2(c)
Obtain Taylors series and two distinct Laurents series expansion of $f(z)=\dfrac{z^{2}-1}{z^{2}+5z+6}$about z=0, indicating region of convergence.
(8 marks) 3(a)State Cayley-Hamilton Theorem,hence deduce that A8=6251,where $ A=\begin{bmatrix} 1&2 \\\\ 2 &-1 \end{bmatrix}$
(6 marks) 3(b) Using calculus of Residues,prove that $$ \displaystyle \int_{0}^{2pi}e^{cos \theta}\cos(sin \theta -n \theta)d \theta=\dfrac{2\pi}{n!}$$ (6 marks) 3(c) Find the plane curve of fixed perimeter and maximum area.(8 marks) 4(a) State Cauchy-Schwartz inequality and hence show that $$ (x^{2}+y^{2}+z^{2})^{1/2} \ge\dfrac{1}{13}(3x+4y+12z),x,y,z$$ are positive. (6 marks) 4(b) Reduce the quadratic form Q =x2+y2-2z2-4xy-2yz+10xz to to Canonical form using congruent transformation.(6 marks) 4(c) (ii)Show that the matrix $A=\begin{bmatrix} 5 &-6 &-6 \\\\ -1&4 &2 \\\\ 3 &-6 &-4 \end{bmatrix}$is Derogatory.
(4 marks) 4(c)(i) If $$A=\begin{bmatrix} \pi/2 &3\pi/2 \\ \pi&\pi \end{bmatrix}$$, find Sin A(4 marks) 5(a) Using Rayleigh-Ritz method,find an appropriate solution for the extremal of the functional $$ I \left [ y(x) \right ]=\int_{0}^{1}\left [ xy+\dfrac{1}{2}(y^{'})^{2}\right]dx $$ subject to y(0)=y(1)=0. (6 marks) 5(b) Find an orthonormal basis of the following subspace of R3,S ={[1,2,0][0,3,1]}.(6 marks) 5(c)Is the matrix $A=\begin{bmatrix} 2 & 1 &1 \\\\ 1& 2 &1 \\\\ 0 & 0 & 1 \end{bmatrix}$diagonalizable.If so find diagonal form and transforming matrix.
(8 marks) 6(a) Find f(3),f'(1+i),f"(1-i),if $$ \displaystyle f(a) = (\int _{c}\dfrac{3x^{2}+11z+7}{z-a}dz $$c:|z|=2 (6 marks) 6(b) Evaluate$$ \displaystyle \int_{0}^{\infty}\dfrac{x^{3}sin x}{(x^{2}+z^{2})^{2}}$$ using contour integration.(6 marks) 6(c) Find the singular value decomposition of the matrix $$ A = \begin{bmatrix} 1 & 1\\ 1 & 1\\ 1 &-1 \end{bmatrix}$$ (8 marks)
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