Transform Theory — Part 1 MCQs with Explanations

$$\frac{1}{{{\text{s}} - {\text{a}}}}$$ is the correct answer to “The Laplace transform F(s) of the exponential function. f(t) = e at when t ≥ 0, where a is a constant and (s - a) > 0, is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{\pi }{2}$$ is the correct answer to “Evaluate $$\int\limits_0^\infty {\frac{{\sin {\text{t}}}}{{\text{t}}}{\text{dt}}} $$”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{1 + {{\text{s}}^2}}}{{{{\left( {{{\text{s}}^2} - 1} \right)}^2}}}$$ is the correct answer to “If the Laplace transform of $${{\text{e}}^{\omega {\text{t}}}}$$ is $$\frac{1}{{{\text{s}} - \omega }},$$ the Laplace transform of tcosh t is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{4}{{{{\text{s}}^2} + 1}}$$ is the correct answer to “The function f(t) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{t}}^2}}} + {\text{f}} = 0$$ and the auxiliary conditions, f(0) = 0, $$\frac{{{\text{df}}}}{{{\text{dt}}}}\left( 0 \right) = 4.$$ The Laplace transform of f(t) is given by”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{{\text{s}} + 5{\text{i}}}}{{{{\text{s}}^2} + 25}}$$ is the correct answer to “The Laplace transform of e i5t where $${\text{i}} = \sqrt { - 1} ,$$ is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{\omega }{{{{\text{s}}^2} + {\omega ^2}}}$$ is the correct answer to “Laplace transform of the function sin ωt is”. This question is standard in competitive examination syllabi for Transform Theory.

0 is the correct answer to “For the function \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} { - 2,}&{ - \pi < {\text{x}} < 0} \\ {2,}&{0 < {\text{x}} < \pi } \end{array}} \right.\] The value of a n in the Fourier series expansion of f(x) is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{{\text{s}} + 2}}{{{{\left( {{\text{s}} + 2} \right)}^2} + 16}}$$ is the correct answer to “Laplace transform of cos (ωt) is $$\frac{{\text{s}}}{{{{\text{s}}^2} + {\omega ^2}}}.$$ The Laplace transform of e -2t cos(4t) is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{\text{a}}}{{{{\text{s}}^2} - {{\text{a}}^2}}}$$ is the correct answer to “The Laplace transform of sinh(at) is”. This question is standard in competitive examination syllabi for Transform Theory.

0.5 is the correct answer to “The Fourier cosine series for an even function f(x) is given by $${\text{f}}\left( {\text{x}} \right) = {{\text{a}}_0} + \sum\limits_{{\text{n}} = 1}^\infty {{{\text{a}}_{\text{n}}}\cos \left( {{\text{nx}}} \right)} $$ The value of the coefficient a 2 for the function f(x) = cos 2 (x) in [0, π] is”. This question is standard in competitive examination syllabi for Transform Theory.

e-2t u(t) is the correct answer to “A solution for the differential equation \[{\rm{\dot x}}\left( {\rm{t}} \right) + 2{\rm{x}}\left( {\rm{t}} \right) = \delta \left( {\rm{t}} \right)\] with initial condition x(0 - ) = 0 is”. This question is standard in competitive examination syllabi for Transform Theory.

1 is the correct answer to “Consider the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}\left( {\text{t}} \right)}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dy}}\left( {\text{t}} \right)}}{{{\text{dt}}}} + {\text{y}}\left( {\text{t}} \right) = \delta \left( {\text{t}} \right)$$ with $${\left. {{\text{y}}\left( {\text{t}} \right)} \right|_{{\text{t}} = 0}} = - 2$$ and $${\left. {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}} = 0.$$ The numerical value of $${\left. {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}}$$ is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{{{\text{e}}^{{\text{as}}}}}}{{\text{a}}}$$ is the correct answer to “A delayed unit step function is defined as \[{\text{u}}\left( {{\text{t}} - {\text{a}}} \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{for t}} < {\text{a}}} \\ {1,}&{{\text{for t}} \geqslant {\text{a}}} \end{array}} \right..\] Its Laplace transform is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{5}{{{{\text{s}}^2} - 4{\text{s}} + 29}}$$ is the correct answer to “The Laplace Transform of f(t) = e 2t sin(5t) u(t) is”. This question is standard in competitive examination syllabi for Transform Theory.

3 is the correct answer to “Let $${\text{X}}\left( {\text{s}} \right) = \frac{{3{\text{s}} + 5}}{{{{\text{s}}^2} + 10{\text{s}} + 21}}$$ be the Laplace Transform of a signal x(t). Then, x(0 + ) is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\int_0^\infty {{{\text{e}}^{ - {\text{st}}}}{\text{f}}\left( {\text{t}} \right){\text{dt}}} $$ is the correct answer to “If f(t) is a function defined for all t ≥ 0, its Laplace transform F(s) is defined as”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{\text{s}}}{{{{\text{s}}^2} - {{\text{a}}^2}}}$$ is the correct answer to “Laplace transform for the function f(x) = cosh(ax) is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{1}{{\text{s}}}{\text{F}}\left( {\text{s}} \right)$$ is the correct answer to “If F(s) is the Laplace transform of function f(t), then Laplace transform of $$\int\limits_0^{\text{t}} {{\text{f}}\left( \tau \right){\text{d}}\tau } $$ is”. This question is standard in competitive examination syllabi for Transform Theory.

1 - e-t is the correct answer to “The inverse Laplace transform of $$\frac{1}{{\left( {{{\text{s}}^2} + {\text{s}}} \right)}}$$ is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\sum\limits_{{\text{n}} = 1}^\infty {\frac{1}{{{{\left( {{\text{2n}} - 1} \right)}^2}}} = \frac{{{\pi ^2}}}{8}} $$ is the correct answer to “The Fourier series of the function, \[\begin{array}{*{20}{c}} {{\text{f}}\left( {\text{x}} \right) = 0,}&{ - \pi < {\text{x}} \leqslant 0} \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \pi - {\text{x,}}}&{0 < {\text{x}} < \pi } \end{array}\] in the interval $$\left[ { - \pi ,\,\pi } \right]$$ is $${\text{f}}\left( {\text{x}} \right) = \frac{\pi }{4} + \frac{2}{\pi }\left[ {\frac{{\cos {\text{x}}}}{{{1^2}}} + \frac{{\cos {\text{3x}}}}{{{3^3}}} + \,...} \right] + \left[ {\frac{{\sin {\text{x}}}}{1} + \frac{{\sin {\text{2x}}}}{2} + \frac{{\sin {\text{3x}}}}{3} + \,...} \right]$$ The convergence of the above Fourier series at x = 0 gives”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{1}{{{{\text{S}}^2} - 1}}$$ is the correct answer to “Laplace transform of sin ht is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{\text{s}}}{{{{\text{s}}^2} + {\omega ^2}}}$$ is the correct answer to “Laplace transform of cos (ωt) is”. This question is standard in competitive examination syllabi for Transform Theory.

t - 1 + e-1 is the correct answer to “The Laplace transform of a function f(t) is $$\frac{1}{{{{\text{s}}^2}\left( {{\text{s}} + 1} \right)}}.$$ The function f(t) is”. This question is standard in competitive examination syllabi for Transform Theory.

2t e-t + e-t is the correct answer to “The inverse Laplace transform of $${\text{H}}\left( {\text{s}} \right) = \frac{{{\text{s}} + 3}}{{{{\text{s}}^2} + 2{\text{s}} + 1}}$$ for t ≥ 0 is”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{4}{5}$$ is the correct answer to “Let \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} { - \pi ,\,\,{\text{if}}}&{ - \pi < {\text{x}} \leqslant {\text{0}}} \\ {\pi ,\,\,{\text{if}}}&{0 < {\text{x}} \leqslant \pi } \end{array}} \right.\] be a periodic function of period 2π. The coefficient of sin5x in the Fourier series expansion of f(x) in the interval [-π, π] is”. This question is standard in competitive examination syllabi for Transform Theory.

0 is the correct answer to “Given the Fourier series in (-π, π) for f(x) = x cosx, the value of a 0 will be”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{2 - 2{{\text{e}}^{ - {\text{s}}}}}}{{\text{s}}}$$ is the correct answer to “Laplace transform of the function f(t) is given by $${\text{F}}\left( {\text{s}} \right) = {\text{L}}\left\{ {{\text{f}}\left( {\text{t}} \right)} \right\} = \int_0^\infty {{\text{f}}\left( {\text{t}} \right){{\text{e}}^{ - {\text{st}}}}{\text{dt}}{\text{.}}} $$ Laplace transform of the function shown below is given by”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{\text{a}}}{{{{\text{s}}^2} + {{\text{a}}^2}}}$$ is the correct answer to “If L defines the Laplace Transform of a function, L [sin (at)] will be equal to”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{\pi }{4}$$ is the correct answer to “The Fourier series expansion of the saw-toothed waveform f(x) = x in (-π, π) of period 2π gives the series, $$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \,....$$ The sum is equal to”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{{48}}{{{{\text{S}}^4}}}$$ is the correct answer to “Laplace transform of 8t 3”. This question is standard in competitive examination syllabi for Transform Theory.

f(t) = 1 - e-t is the correct answer to “The inverse Laplace transform of the function $${\text{F}}\left( {\text{s}} \right) = \frac{1}{{{\text{s}}\left( {{\text{s}} + 1} \right)}}$$ is given by”. This question is standard in competitive examination syllabi for Transform Theory.

Both A and B is the correct answer to “Laplace transform analysis gives”. This question is standard in competitive examination syllabi for Transform Theory.

$$ - {e^{ - t}} + 2{e^{ - 2t}}$$ is the correct answer to “Inverse Laplace transform of the function $$\frac{s}{{{s^2} + 3s + 2}},$$ is”. This question is standard in competitive examination syllabi for Transform Theory.

All of the above is the correct answer to “Which of the following is the advantage of using Laplace transform techniques”. This question is standard in competitive examination syllabi for Transform Theory.

Delta function is the correct answer to “The initial value theorem does not hold good for which of the following functions”. This question is standard in competitive examination syllabi for Transform Theory.

$$\left( {\frac{1}{{10}} + \frac{1}{6}{e^{ - 4t}} + \frac{{11}}{{15}}{e^{5t}}} \right)u\left( t \right)$$ is the correct answer to “Find the inverse Laplace transform of $$F\left( s \right) = \frac{{{s^2} + 2s - 2}}{{s\left( {s + 4} \right)\left( {s - 5} \right)}}$$”. This question is standard in competitive examination syllabi for Transform Theory.

-45° is the correct answer to “Give transfer function $$H\left( s \right) = \frac{{s + 2}}{{{s^2} + s + 4}},$$ under steady state condition, the sinusoidal input and output are, respectively x(t) = cos 2t, y(t) = cos(2t + $$\phi $$), then angle $$\phi $$ will be”. This question is standard in competitive examination syllabi for Transform Theory.

$$L\left\{ {f\left( t \right)} \right\} = \int_{{0^ - }}^\infty {f\left( t \right){e^{ - st}}dt} $$ is the correct answer to “Which of the following correctly defines Laplace transform of a function in the time domain”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{5}{2}$$ is the correct answer to “The Laplace transform of $$I\left( t \right)$$ is given by $$I\left( s \right) = \frac{5}{{s\left( {{s^2} + 2} \right)}}.$$ As $$t \to \infty $$ the value of $$I\left( t \right)$$ tends to”. This question is standard in competitive examination syllabi for Transform Theory.

$$\frac{1}{s}F\left( s \right)$$ is the correct answer to “Given that $$F\left( s \right)$$ is the one-side Laplace transform of $$f\left( t \right),$$ the Laplace transform of $$\int_0^t {f\left( \tau \right)d\tau } $$ is”. This question is standard in competitive examination syllabi for Transform Theory.