Transform Theory MCQs Question and Answer in Engineering Maths

1

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

A) $$\frac{1}{{{\text{s}} + {\text{a}}}}$$ B) $$\frac{1}{{{\text{s}} - {\text{a}}}}$$ C) $$\frac{1}{{{\text{a}} - {\text{s}}}}$$ D) $$\infty $$

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Answer: Option B

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2

Evaluate $$\int\limits_0^\infty {\frac{{\sin {\text{t}}}}{{\text{t}}}{\text{dt}}} $$

A) $$\pi $$ B) $$\frac{\pi }{2}$$ C) $$\frac{\pi }{4}$$ D) $$\frac{\pi }{8}$$

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Answer: Option B

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3

If the Laplace transform of $${{\text{e}}^{\omega {\text{t}}}}$$ is $$\frac{1}{{{\text{s}} - \omega }},$$ the Laplace transform of tcosh t is

A) $$\frac{{1 + {{\text{s}}^2}}}{{{{\left( {{{\text{s}}^2} - 1} \right)}^2}}}$$ B) $$\frac{{{\text{st}}}}{{\left( {{{\text{s}}^2} - 1} \right)}}$$ C) $$\frac{{1 - {{\text{s}}^2}}}{{{{\left( {{{\text{s}}^2} - 1} \right)}^2}}}$$ D) $$\frac{{1 + {{\text{s}}^2}}}{{1 - {{\text{s}}^2}}}$$

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Answer: Option A

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4

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

A) $$\frac{2}{{{\text{s}} + 1}}$$ B) $$\frac{4}{{{\text{s}} + 1}}$$ C) $$\frac{4}{{{{\text{s}}^2} + 1}}$$ D) $$\frac{2}{{{{\text{s}}^2} + 1}}$$

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Answer: Option C

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5

The Laplace transform of e^i5t where $${\text{i}} = \sqrt { - 1} ,$$ is

A) $$\frac{{{\text{s}} - 5{\text{i}}}}{{{{\text{s}}^2} - 25}}$$ B) $$\frac{{{\text{s}} + 5{\text{i}}}}{{{{\text{s}}^2} + 25}}$$ C) $$\frac{{{\text{s}} + 5{\text{i}}}}{{{{\text{s}}^2} - 25}}$$ D) $$\frac{{{\text{s}} - 5{\text{i}}}}{{{{\text{s}}^2} + 25}}$$

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Answer: Option B

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6

Laplace transform of the function sin ωt is

A) $$\frac{{\text{s}}}{{{{\text{s}}^2} + {\omega ^2}}}$$ B) $$\frac{\omega }{{{{\text{s}}^2} + {\omega ^2}}}$$ C) $$\frac{{\text{s}}}{{{{\text{s}}^2} - {\omega ^2}}}$$ D) $$\frac{\omega }{{{{\text{s}}^2} - {\omega ^2}}}$$

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Answer: Option B

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7

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

A) 2 B) 4 C) 0 D) -2

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Answer: Option C

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8

Laplace transform of cos (ωt) is $$\frac{{\text{s}}}{{{{\text{s}}^2} + {\omega ^2}}}.$$ The Laplace transform of e^-2t cos(4t) is

A) $$\frac{{{\text{s}} - 2}}{{{{\left( {{\text{s}} - 2} \right)}^2} + 16}}$$ B) $$\frac{{{\text{s}} + 2}}{{{{\left( {{\text{s}} - 2} \right)}^2} + 16}}$$ C) $$\frac{{{\text{s}} - 2}}{{{{\left( {{\text{s}} + 2} \right)}^2} + 16}}$$ D) $$\frac{{{\text{s}} + 2}}{{{{\left( {{\text{s}} + 2} \right)}^2} + 16}}$$

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Answer: Option D

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9

The Laplace transform of sinh(at) is

A) $$\frac{{\text{s}}}{{{{\text{s}}^2} + {{\text{a}}^2}}}$$ B) $$\frac{{\text{s}}}{{{{\text{s}}^2} - {{\text{a}}^2}}}$$ C) $$\frac{{\text{a}}}{{{{\text{s}}^2} - {{\text{a}}^2}}}$$ D) $$\frac{{\text{a}}}{{{{\text{s}}^2} + {{\text{a}}^2}}}$$

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Answer: Option C

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10

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 a₂ for the function f(x) = cos²(x) in [0, π] is

A) -0.5 B) 0.0 C) 0.5 D) 1.0

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Answer: Option C

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