MU Applied Mathematics 4 - December 2014 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Applied Mathematics 4
December 2014

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M

1 (b)

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

5 M

1 (c)

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

5 M

1 (d) Verify Cauchy-Schwartz inequality for the vectors.
u=(-4, 2, 1) & V=(8, -4, -2)

5 M

2 (a) Determine the function that gives the shortest distance between two given points

6 M

2 (b)

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

6 M

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 M

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 M

3 (b)

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

6 M

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 M

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 M

4 (b)

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

6 M

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 M

4 (c) (ii) Is the set W={[a,1,1]|a∈R} a subspace of R³ under the usual addition and scalar multiplication?

4 M

5 (a) Find the plane curve of fixed perimeter and maximum area.

6 M

5 (b) Construct an orthonormal basis of R² by applying Gram schmidt orthogonalization to S={[3,1],[2,2]}

6 M

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 M

6 (a)

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

6 M

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 M

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

8 M

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