1 (a)
Find L{t e31sint}
5 M
1 (b)
Show that every square matrix can be uniquely expressed as the sum of a Hermitian and skew-Hermitian matrix.
5 M
1 (c)
Find Z-transform and region of convergence of f(k)=3k, k≥0.
5 M
1 (d)
Find the Fourier expansion of f(x)= x2 where -π ≤x ≤π.
5 M
2 (a)
Prove that following matrix is orthogonal and hence find its inverse.
A=19[−84114−8474]
A=19[−84114−8474]
6 M
2 (b)
Find L−1{s+2(s2+4s+5)2}
6 M
2 (c)
Obtain the Fourier expansion of f(x)=(π−x2)2 in the internal and 0≤x≤2π and f(x+2π)=f(x). Also deduce that,
(i) π26=112+122+132+142+∙∙∙∙∙∙∙∙
(ii)π490=114+124+134+144+∙∙∙∙∙∙∙∙
(i) π26=112+122+132+142+∙∙∙∙∙∙∙∙
(ii)π490=114+124+134+144+∙∙∙∙∙∙∙∙
8 M
3 (a)
Investing for what values of λ and μ the equations.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
6 M
3 (b)
Obtain complex form of Fourier series for f(x)=eax (-π,π) where is not integer.
6 M
3 (c)
Solve (D2 - D - 2) y=20 sin 2t with y(0)=1, y'(0)=2.
8 M
4 (a)
Find Laplace transform of
f(t)=asinpt 0<t≤πpf(t)=0πP<t≤2πPand f(t)=(t+2πP)
f(t)=asinpt 0<t≤πpf(t)=0πP<t≤2πPand f(t)=(t+2πP)
6 M
4 (b)
Find the inverse Z-transform for
f(z)=1(z−3)(z−2)
for 2<|z|<3.
f(z)=1(z−3)(z−2)
for 2<|z|<3.
6 M
4 (c)
Find inverse Laplace transform of
(i) e4−3s(s+4)52(ii)tan−12s
(i) e4−3s(s+4)52(ii)tan−12s
8 M
5 (a)
Examine whether the following vectors are linearly independent or dependent [2,1,1], [1,3,1], [1,2,-1]
6 M
5 (b)
Using Convolution theorem prove that
l−1[1s ln (s+1s+2)]= ∫t0(e−2u−e−uu)du
l−1[1s ln (s+1s+2)]= ∫t0(e−2u−e−uu)du
6 M
5 (c)
Using Fourier cosine Integral prove that
e−xcosx=1π∫∞0w2+2w4+4 coswx dw
e−xcosx=1π∫∞0w2+2w4+4 coswx dw
8 M
6 (a)
Find the Fourier Transform 0+f(x)=e-1x1
6 M
6 (b)
Find z[f(x)] where
f(k)=cos(kπu+a) where k≥0.
6 M
6 (c)
Find Fourier expansion of f(x)=2x - x2 where 0 ≤ x ≤ 3 and period is 3.
8 M
7 (a)
Reduce the following matrix to noraml form and find its rank.
A=[1−13613−3−453311]
A=[1−13613−3−453311]
6 M
7 (b)
Evalute ∫∞0cos6t−cos4ttdt
6 M
7 (c)
Show that the set of function
sin(πx2L),sin(3πx2L),sin(5πx2L),∙∙∙∙∙∙∙∙
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.
sin(πx2L),sin(3πx2L),sin(5πx2L),∙∙∙∙∙∙∙∙
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.
8 M
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