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 And f(t) = f(t+2a)
5 M
1 (b)
Find the Fourier series of f(x) = cos ?x in (-?,?), where ? is not an integer.
5 M
1 (c)
Find the value of P for which the following matrix A will be of rank one, rank two, rank three where
[A=3ppp3ppp3]
[A=3ppp3ppp3]
5 M
1 (d)
Find Z-transform of {k2 - 2k + 3}k ? 0
5 M
2 (a)
Solve by using L.T.
dydt+2y+∫t0y dt=sint when y (0)=1
dydt+2y+∫t0y dt=sint when y (0)=1
8 M
2 (b)
Find Fourier series for f(x) = x + x2 in (-? , ?) Hence deduce that
112+132+152+172+....=π28
112+132+152+172+....=π28
6 M
2 (c)
Show that vectors X1, X2, X3 are linearly independent and vector X4 depends upon them where
X1= (1,2,4)X2=(2,−1,3)X3=(0,1,2) andX4=(−3,7,2)
X1= (1,2,4)X2=(2,−1,3)X3=(0,1,2) andX4=(−3,7,2)
6 M
3 (a)
Find the Fourier integral representation of the function f(x)={1−x2, when |x| ≤1x, when |x| >1
And hence evaluate ∫∞0[xcosx−sinxx3]cosx2 dx
And hence evaluate ∫∞0[xcosx−sinxx3]cosx2 dx
8 M
3 (b)
Find matrix A if adj [A=−213−2−31121−5]
6 M
3 (c)
FindL {1−costt2}
6 M
4 (a)
(i)FindL−1{tan−1(s+2)2}(ii)FindL{t2H(t−2)−cosht δ(t−4)}
6 M
4 (b)
Find inverse Z-transform of z(z+1)(z−1)(z2+z+1)
6 M
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=[ 32−15514−21−41119]
A=[ 32−15514−21−41119]
6 M
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} hence find the sum ∞∑n=11n4
\[ f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}
8 M
5 (b)
Discuss the consistancy of the following system of equation and if consistant solve them
3x+3y+2z=1
x+2y+=4
10y+3z= -2
2x-3y-z=5
3x+3y+2z=1
x+2y+=4
10y+3z= -2
2x-3y-z=5
6 M
5 (c)
Evaluate by using L.T. ∫∞0t3e−t sin t dt
6 M
6 (a) (i)
(i) Find complex form of Fourier series for f(x) = cosh3x + sinh3x in (-?,?)
4 M
6 (a) (ii)
(ii) Show that the functions {sin(2n−1)}∞n=0 are orthogonal on [0,π2]hence construct an orthonormal set of functions from this.
4 M
6 (b)
Apply Gauss- Seidal itterative method to solve the equations upto three itteratism
3x+20y-z=-18
2x-3y+20z=25
20x+y-2z=17
3x+20y-z=-18
2x-3y+20z=25
20x+y-2z=17
6 M
6 (c)
Find Z-transform of {k2 ak−1 ∪ (k−1)}
6 M
7 (a) (i)
By using cinvolution theorem finf
L−1{1(s−4)4(s+3)}
L−1{1(s−4)4(s+3)}
5 M
7 (a) (ii)
Find : - L(sin2t cost cosh2t)
3 M
7 (b)
Find inverse Z-transform of 2z2−10z+13(z−3)2(z−2)
if R.O.C. is 2|z|<3.
if R.O.C. is 2|z|<3.
6 M
7 (c)
Find Fourier sin a integral of f(x) where
\[ f(x)=\left\{\begin{matrix}x & ; &02\end{matrix}\right.\]
\[ f(x)=\left\{\begin{matrix}x & ; &0
6 M
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