1 (a)
Find Laplace of {t5 cosht}.
5 M
1 (b)
Find Fourier series for f(x)=t-x2 in (-1, 1).
5 M
1 (c)
Find a, b, c, d, e, if,
f(z)=(ax4+bx2y2+cy4+dx2-2y2)+i (4x3-exy3+4xy) is analytic.
f(z)=(ax4+bx2y2+cy4+dx2-2y2)+i (4x3-exy3+4xy) is analytic.
5 M
1 (d)
Prove that Δ=(1r)=rr3
5 M
2 (a)
If f(z)= u + iv is analytic and u+v=2sin2xe2y+e−2y−2cos2x Find f(z).
6 M
2 (b)
Find inverse Z-transform of f(z)=z+2z2−2z+1 for |z|>1.
6 M
2 (c)
Find Fourier series for f(x)=√1−cosx in (0,2π) Hence, deduce that 12=∑∞114n2−1
8 M
3 (a)
Find L^{-1} \left { \dfrac {1}{(s-3)+(s+3)} \right } using Convolution theorem.
6 M
3 (b)
Prove that f1(x)=1, f2(x)=x, f3(x)=(3x2-1)/2 are orthogonal over (-1, 1).
6 M
3 (c)
Verify Green's theorem for \( \int_c \overline {F} \cdot \overline{dr} \text { where } \overline {F} = (x2-y2)i+(x+y)j \) and c is the triangle with vertices (0, 0), (1, 1), (2, 1).
8 M
4 (a)
Find Laplace Transform of f(t)=|sinpt|, t≥0.
6 M
4 (b)
Show that F = (ysinz-sinx)i+(xsinz+2yz)j+(xycosz+y2)k is irrotational. Hence, find its scalar potential.
6 M
4 (c)
Obtain Fourier expansion of f(x)=x+π2 where −π<x<0=π2−x where 0<x<π
Hence, deduce that i) π28=112+132+152+⋯ ⋯ii) π496=114+134+154+⋯ ⋯
Hence, deduce that i) π28=112+132+152+⋯ ⋯ii) π496=114+134+154+⋯ ⋯
8 M
5 (a)
Using Gauss Divergence theorem to evaluate ∬ and S is the region bounded by x2 + y2 = 4, z=0, z=3.
6 M
5 (b)
Find Z{2k cos (3k+2)}, k≥0.
6 M
5 (c)
Solve (D2+2D+5)y=e-t sint, with y(0)=0 and y'(0)=1.
8 M
6 (a)
Find L^{-1} \left { \tan^{-1} \left ( \dfrac {2}{s^2} \right ) \right }
6 M
6 (b)
Find the bilinear transformation which maps the points 2, i, -2 onto points 1, i, -1 by using cross ratio property.
6 M
6 (c)
Find Fourier Sine integral representation for f(x) = \dfrac {e^{-ax}}{x}
8 M
More question papers from Applied Mathematics 3