Question Paper: Applied Mathematics 4 : Question Paper Dec 2014 - Electronics & Telecomm. (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - Dec 2014

### 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)

Find the value of μ which satisfy the equation A100x=μ X, where $A= \begin{bmatrix}2 &1 &-1 \\\\0 &-2 &-2 \\\\ 1 &1 &0 \end{bmatrix}$

(5 marks) 1 (b)

Evaluate $$\int^{1+i}_{0} (x^2+iy)dz\ along\ y=x\ and\ y=x^2$$

(5 marks)
1 (c)

Find the external of the function  $\int^{x_2}_{x_1} [y^2-y^{'2} -2ycosh x]dx$

(5 marks)
1 (d) Verify Cauchy-Schwartz inequality for the vectors.
u=(-4, 2, 1) & V=(8, -4, -2)
(5 marks)
2 (a) Determine the function that gives the shortest distance between two given points(6 marks) 2 (b)

Find eigen values and eigen vectors of :- $A= \begin{bmatrix} 2 & 1 &1 \\\\2 &3 &2 \\\\3 &3 &4 \end{bmatrix}$

(6 marks)
2 (c)

Obtain Taylor's and two distinct Laurent's series expansion of $f(z) = \dfrac {z-1}{z^2-2z-3}$ about z=0 indicating the region of convergence.

(8 marks)
3 (a)

Verify Caley-Hamilton theorem for $A= \begin{bmatrix}1 &2 &0 \\\\2 &-1 &0 \\\\0 &0 &-1 \end{bmatrix}$  hence find $A^{-2}$

(6 marks)
3 (b)

Evaluate by using Residue theorem. $\int^{2\pi}_0 \dfrac {d\theta}{(2+cos \theta )^2}$

(6 marks)
3 (c)

Solve the boundary value problem: $I=\int_0^1(2xy-y^{2}-y'^{2})dx$

given y(0)=y(1)=0 by Rayleigh Ritz method.

(8 marks) 4 (a)

Reduce the following Quadrature form  $Q=3x^2_1 + 5x_2^2 + 3x^2_3 -2x_1x_2-2x_2x_3 +2x_3x_1$  into canonical form. Hence find its rank index and signature.

(6 marks)
4 (b)

Show that the matrix  $A= \begin{bmatrix} 7 &4 &-1 \\\\4 &7 &-1 \\\\-4 &-4 &4 \end{bmatrix}$ is derogatory .

(6 marks)
4 (c) (i) Show that the set W={(1,x)|x∈R} is a subspace of R2 under operations [1,x]+[1,y]=[1, x+y]; k[1,x]=[1,kx]; k is any scalar:(4 marks) 4 (c) (ii) Is the set W={[a,1,1]|a∈R} a subspace of R3 under the usual addition and scalar multiplication?(4 marks) 5 (a) Find the plane curve of fixed perimeter and maximum area.(6 marks) 5 (b) Construct an orthonormal basis of R2 by applying Gram schmidt orthogonalization to S={[3,1],[2,2]}(6 marks) 5 (c)

Show that the matrix $A= \begin{bmatrix} -9 &4 &4 \\\\-8 &3 &4 \\\\-16 &8 &7 \end{bmatrix}$ is diagnosable. Also find diagonal form and diagonalising matrix.

(8 marks)
6 (a)

Evaluate  $\int^{\infty}_{-\infty} \dfrac {cos 3x}{(x^2+1)(x^2+4)} dx$ using Cauchy Residue Theorem.

(6 marks)
6 (b)

[ If $\phi(\alpha)= \oint_{c}\dfrac {ze^z}{z-\alpha}dz$ where $c \space is \space|z-2i|=3$  find $\phi (1), \phi '(2), \phi ' (3), \phi ' (4)$

(6 marks)
6 (c) Show that the set V of position real numbers with operations Addition : x+y=xy, Scalar multiplication: kx=xk is a vector space where x, y are any two real numbers and k is any scalar.(8 marks)