Transform Theory - Study Mode
[#16] If f(t) is a function defined for all t ≥ 0, its Laplace transform F(s) is defined as
Correct Answer
(B) $$int_0^infty {{{ ext{e}}^{ - { ext{st}}}}{ ext{f}}left( { ext{t}}
ight){ ext{dt}}} $$
[#17] Laplace transform for the function f(x) = cosh(ax) is
Correct Answer
(B) $$frac{{ ext{s}}}{{{{ ext{s}}^2} - {{ ext{a}}^2}}}$$
[#18] If F(s) is the Laplace transform of function f(t), then Laplace transform of $$intlimits_0^{ ext{t}} {{ ext{f}}left( au
ight){ ext{d}} au } $$ xa0 is
Correct Answer
(A) $$frac{1}{{ ext{s}}}{ ext{F}}left( { ext{s}}
ight)$$
[#19] The inverse Laplace transform of $$frac{1}{{left( {{{ ext{s}}^2} + { ext{s}}}
ight)}}$$ xa0is
Correct Answer
(C) 1 - e -t
[#20] The Fourier series of the function, [x08egin{array}{*{20}{c}}
{{ ext{f}}left( { ext{x}}
ight) = 0,}&{ - pi < { ext{x}} leqslant 0} \
{,,,,,,,,,,,,,,,,,,,,, = pi - { ext{x,}}}&{0 < { ext{x}} < pi }
end{array}] xa0 xa0 xa0in the interval $$left[ { - pi ,,pi }
ight]$$ xa0is $${ ext{f}}left( { ext{x}}
ight) = frac{pi }{4} + frac{2}{pi }left[ {frac{{cos { ext{x}}}}{{{1^2}}} + frac{{cos { ext{3x}}}}{{{3^3}}} + ,...}
ight] + left[ {frac{{sin { ext{x}}}}{1} + frac{{sin { ext{2x}}}}{2} + frac{{sin { ext{3x}}}}{3} + ,...}
ight]$$ The convergence of the above Fourier series at x = 0 gives
Correct Answer
(C) $$sumlimits_{{ ext{n}} = 1}^infty {frac{1}{{{{left( {{ ext{2n}} - 1}
ight)}^2}}} = frac{{{pi ^2}}}{8}} $$