MU Applied Mathematics - 3 - December 2012 Exam Question Paper

MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2012

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)

$A=\frac{1}{7}\ \left[\begin{array}{ccc}2 & 6 & 3 \\6 & -3 & 2 \\3 & 2 & -6\end{array}\right]$
Show that-
(i) |A| = 1
(ii) .adj A=A'

5 M

This Qs paper appeared for Applied Mathematics - 3 of Electronics & Telecomm. (Semester 3)

1 (b) Show that the Fourier cosine transform of-

$f\left(x\right)=\ \frac{1}{\sqrt{x}}\ is\ \frac{1}{\sqrt{s}}$

5 M

1 (c) Show that

$L^{-1}\ {tan}^{-1}\left(\frac{a}{s}\right)=\ \frac{sin?(at)}{t}\$

5 M

1 (d) Show that

$\int_0^{\infty}e^{-t} \sin \left( \frac{t}{2}\right) \sin h{\ \left(\frac{\sqrt{3}t}{2}\right)}\ dt=\ \frac{\sqrt{3}}{2}$

5 M

2 (a) If z is any zero complex number show that -
[A=frac{1}{sqrt{2} vert{}zvert{}}left[egin{array}{cc}z

6 M

2 (b) Find the Fourier series expansion of f(x)=x² in [0,2?]

6 M

2 (c) Show that

$\int_0^{\infty{}}\frac{\sin{2t+\sin{3t}}}{te^t}dt=\ \frac{3\pi{}}{4}$

8 M

3 (a) Find the half range sine series for f(x) = ?x - x² in [ 0, ? ]. Hence find

$\sum_{n=1}^{\infty{}}\frac{1}{{(\ 2n-1)}^6}$

6 M

3 (b) Show that

$L(cos\ at\ cosh\ at)=\frac{s^3}{s^4+\ 4a^4}$

6 M

3 (c) if a,b,c are distinct real numbers such that a+b+c ? 0. Show that the vectors (a,b,c), (b,c,a) and (c,a,b) are linearly independent.

8 M

4 (a) Find Fourier series expansion of -
\[ f(x)=\left\{\begin{matrix}x &-?

6 M

4 (b) Find the all possible ranks of the matrix

$A=\left[\ \begin{array}{ccc}k & -3 & -3 \\-3 & k & -3 \\-3 & -3 & k\end{array}\right]$
where k is any real number

6 M

4 (c) Find

$(i) \ L^{-1}\ \dfrac{se^{-?s}}{s^2+3s+2}; \ (ii) \ L [t. H(t - 4)+t^2 \ ?(t-4)]$

8 M

5 (a) Solve the system of equation using the Gauss- Seidel method :
3x+4y+6z=6
2x+5y+4z=11
2x+y+z+=5

6 M

5 (b) Find

${\ L}^{-1}\frac{s}{{\left(s^2+\ 2\right)}^2}$ using convolution theorem

6 M

5 (c) Show that

$z[\cos\ {(\alpha \ k)]=\ \frac{z^2-z\cos \alpha}{z^2-2(\cos{\alpha)z+1}}}$

8 M

6 (a) Find Lf(t) where f(t) = cos t, 0 < t < ?
sin t, sin t ? ?
Using the Heaviside unit step function.

6 M

6 (b) Find

$Z^{-1}\frac{1}{(z-2)(z-3)}$ for 2<|z|<3 using residues.

6 M

6 (c) Express the matrix

$A=\left[\begin{array}{ccc}\sin{\alpha{}} & 0 & \sin{\alpha{}} \\\left(sin\ \alpha{}\right)\ (sin\beta{}) & cos\beta{} &-\left(sin\beta{}\right)\left(cos\alpha{}\right) \\-(cos\beta{})(sin\alpha{}) & sin\beta{} & (cos\alpha{})(cos\beta{})\end{array}\right]$
as a product of two orthogonal matrices.

8 M

7 (a) Solve the

$DE\ \ y+\int_0^ty\ dt=1-\ e^{-t}$ using Laplace transforms.

6 M

7 (b) if 0 ? x ? ? show that-

$x^2= \ \frac{2}{?}\left[ \left(\frac{\pi^2}{1}-\frac{4}{1^3}\right)\sin x- \ \left(\frac{\pi^2}{2}\right)\ \sin 2x\\ \ \ \ \ \ \ +\ \left(\frac{\pi^2}{3}-\frac{4}{3^3}\right)\ \sin 3x+.....\right]$

6 M

7 (c) Show that {cos nx}_n?1 is an orthogonal family of functions in [ -? , ? ]. Also find the corresponding orthonormal set.

8 M

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