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Applied Mathematics - 3 : Question Paper Dec 2011 - Electronics Engineering (Semester 3) | Mumbai University (MU)
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Applied Mathematics - 3 - Dec 2011

Electronics Engineering (Semester 3)

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.


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

1 (a) Find L.T. of $$f\left(t\right)=\ f\left(x\right)= \Bigg\{\begin{align*}{}1,\ \ \ \ \ 0\ltt\lta \\-1,\="" \="" a\ltt\lt2a\end{align*}="" $$<br=""> And f(t) = f(t+2a)</t<a>(5 marks) 1 (b) Find the Fourier series of f(x) = cos ?x in (-?,?), where ? is not an integer.(5 marks) 1 (c) Find the value of P for which the following matrix A will be of rank one, rank two, rank three where
$$\left[A=\begin{array}{ccc}3 & p & p \\p & 3 & p \\p & p & 3\end{array}\right]$$
(5 marks)
1 (d) Find Z-transform of {k2 - 2k + 3}k ? 0(5 marks) 2 (a) Solve by using L.T.
$$ \dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1$$
(8 marks)
2 (b) Find Fourier series for f(x) = x + x2 in (-? , ?) Hence deduce that
$$ \dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8} $$
(6 marks)
2 (c) Show that vectors X1, X2, X3 are linearly independent and vector X4 depends upon them where
$$X_1=\ \left(1,2,4\right)$$$$X_2=\left(2,-1,3\right)$$$$X_3=\left(0,1,2\right)\ and \\ X_4=\left(-3,7,2\right)$$
(6 marks)
3 (a) Find the Fourier integral representation of the function $$f\left(x\right)=\left\{\begin{array}{l}1-x^2,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \leq{}1 \\ x,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \ >{}1\end{array}\right.$$
And hence evaluate $$ \int_0^{\infty{}}\left[\frac{x\cos x-\sin x}{x^3}\right]\cos{\frac{x}{2}}\ dx\ $$
(8 marks)
3 (b) Find matrix A if adj $$\left[A=\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]$$(6 marks) 3 (c) $$ Find L \ \left\{\frac{1-cost}{t^2}\right\} $$(6 marks) 4 (a) $${(i) Find L^{-1}\left\{{tan}^{-1}{\left(s+2\right)}^2\right\} } \\ {(ii) Find L\left\{t^2H\left(t-2\right)-cos h t\ \delta{}\left(t-4\right)\right\}} $$(6 marks) 4 (b) Find inverse Z-transform of $$\frac{z\left(z+1\right)}{\left(z-1\right)\left(z^2+z+1\right)}$$(6 marks) 4 (c) Find the nm singular matrices P and Q such that PAQ is in the normal form and hence find rank of A and rank of (PAQ) where
$$A=\left[\ \begin{array}{ccc}3 & 2 & -1 & 5 \\5 & 1 & 4 & -2 \\1 & -4 & 11 & 19\end{array}\right]$$
(6 marks)
5 (a) Find half range cosine series for
$$ f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}\ltx\lt\pi{}\end{array}\right.$$
hence find the sum $$ \sum^{\infty}_{n=1} \dfrac {1}{n^4} $$</x<\pi{}\end{array}\right.$$\ltbr\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Discuss the consistancy of the following system of equation and if consistant solve them \ltbr\gt 3x+3y+2z=1\ltbr\gt x+2y+=4\ltbr\gt 10y+3z= -2\ltbr\gt 2x-3y-z=5\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c) \lt/b\gt Evaluate by using L.T. $$\int_0^{\infty{}}{t^3e}^{-t}\ \ sin\ t\ dt$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (i)\lt/b\gt (i) Find complex form of Fourier series for f(x) = cosh3x + sinh3x in (-?,?)\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (ii)\lt/b\gt (ii) Show that the functions $${\left{\sin{\left(2n-1\right)}\right}}_{n=0}^{\infty{}}$$ are orthogonal on $$\left[0,\frac{\pi{}}{2}\right]$$hence construct an orthonormal set of functions from this.\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Apply Gauss- Seidal itterative method to solve the equations upto three itteratism \ltbr\gt 3x+20y-z=-18\ltbr\gt 2x-3y+20z=25\ltbr\gt 20x+y-2z=17\ltbr\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c) \lt/b\gt Find Z-transform of $${k^2\ a^{k-1}\ \cup{}\ \left(k-1\right)}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (i)\lt/b\gt By using cinvolution theorem finf \ltbr\gt $$L^{-1}\left{\frac{1}{{\left(s-4\right)}^4(s+3)}\right}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (ii)\lt/b\gt Find : - L(sin2t cost cosh2t)\lt/span\gt\ltspan class='paper-ques-marks'\gt(3 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (b)\lt/b\gt Find inverse Z-transform of $$\frac{2z^2-10z+13}{{\left(z-3\right)}^2\left(z-2\right)}$$ \ltbr\gt if R.O.C. is 2|z|<3.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (c) \lt/b\gt Find Fourier sin a integral of f(x) where \ltbr\gt $$ f(x)=\left{\begin{matrix}x & ; &0<x <1="" \\="" 2-x&;="" &1<x<2="" 0&;="" &="" x="">2\end{matrix}\right.$$</x>
(6 marks)

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