Mathematics
Functions and Laplace Transforms
50 Questions
Functions and Laplace transforms involve mapping mathematical domains and ranges to solve complex differential equations. Questions require finding the Laplace transform of exponential, hyperbolic, and trigonometric functions. These advanced topics are typically assessed in engineering and civil services aptitude tests.
Laplace transform exponentialFunction domain rangeHyperbolic function transformsLinear transformation dimensionOptimization functionals
Functions and Laplace Transforms Questions
The Laplace transform of the function $f(t) = t^2$ is:
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$F(s) = \frac{2}{s^3}$
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$F(s) = \frac{2}{s^4}$
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$F(s) = \frac{2!}{s^3}$
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$F(s) = \frac{2!}{s^4}$
C
Correct answer
Explanation
The Laplace transform of $t^2$ is $\frac{2!}{s^3}$.
The Laplace transform of the function $f(t) = \sin(at)$ is:
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$F(s) = \frac{a}{s^2 + a^2}$
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$F(s) = \frac{s}{s^2 + a^2}$
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$F(s) = \frac{1}{s^2 + a^2}$
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$F(s) = \frac{s}{s^2 - a^2}$
A
Correct answer
Explanation
The Laplace transform of $\sin(at)$ is $\frac{a}{s^2 + a^2}$.
The Laplace transform of the function $f(t) = \sinh(at)$ is:
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$F(s) = \frac{a}{s^2 - a^2}$
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$F(s) = \frac{s}{s^2 - a^2}$
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$F(s) = \frac{1}{s^2 - a^2}$
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$F(s) = \frac{s}{s^2 + a^2}$
A
Correct answer
Explanation
The Laplace transform of $\sinh(at)$ is $\frac{a}{s^2 - a^2}$.
The Laplace transform of the function $f(t) = \cosh(at)$ is:
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$F(s) = \frac{s}{s^2 - a^2}$
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$F(s) = \frac{a}{s^2 - a^2}$
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$F(s) = \frac{1}{s^2 - a^2}$
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$F(s) = \frac{s}{s^2 + a^2}$
A
Correct answer
Explanation
The Laplace transform of $\cosh(at)$ is $\frac{s}{s^2 - a^2}$.
The Laplace transform of the function $f(t) = \frac{1}{t}$ is:
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$F(s) = \ln(s)$
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$F(s) = \frac{1}{s}$
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$F(s) = \frac{1}{s^2}$
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$F(s) = \frac{1}{s^3}$
A
Correct answer
Explanation
The Laplace transform of $\frac{1}{t}$ is $\ln(s)$.
What is the Laplace transform of a function?
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An integral transform that converts a function of time into a function of a complex variable
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A Fourier transform that converts a function of time into a function of frequency
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A Hilbert transform that converts a function of time into a function of its conjugate
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A Mellin transform that converts a function of time into a function of its moments
A
Correct answer
Explanation
The Laplace transform is an integral transform that converts a function of time into a function of a complex variable, allowing for the analysis of linear time-invariant systems.
What is the Laplace transform of the function $f(t) = t^2 e^{-at}$?
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$F(s) = \frac{2}{(s+a)^3}$
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$F(s) = \frac{2a}{(s+a)^3}$
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$F(s) = \frac{2}{(s-a)^3}$
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$F(s) = \frac{2a}{(s-a)^3}$
A
Correct answer
Explanation
The Laplace transform of $t^2 e^{-at}$ can be obtained by using the formula $\mathcal{L}{t^n e^{-at}} = \frac{n!}{(s+a)^{n+1}}$. Therefore, $F(s) = \mathcal{L}{t^2 e^{-at}} = \frac{2!}{(s+a)^3} = \frac{2}{(s+a)^3}$.
What is the Laplace transform of the function f(t) = t^2?
Correct answer
Explanation
The Laplace transform of the function f(t) = t^2 is F(s) = 2/s^3, where s is the complex variable.
What is the dimension of the range of the linear transformation f(x) = [x, 2x, 3x]?
A
Correct answer
Explanation
The range of a linear transformation is the set of all vectors that can be obtained by applying the transformation to some vector in the domain. In this case, the range of f(x) = [x, 2x, 3x] is the set of all vectors of the form [x, 2x, 3x], which is a one-dimensional subspace of R^3. Therefore, the dimension of the range is 1.
What is the range of the function (f(x) = 2x + 1) when the domain is ({1, 2, 3})?
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{3, 5, 7}
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{1, 3, 5}
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{2, 4, 6}
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{0, 2, 4}
A
Correct answer
Explanation
The range of a function is the set of all second elements in the ordered pairs.
What is the domain of the function (f(x) = \frac{1}{x})?
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{0}
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All real numbers except 0
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{1}
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{2}
B
Correct answer
Explanation
The domain of a function is the set of all possible input values.
In mathematics, what is the term for a function that maintains a constant value throughout its domain?
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Constant Function
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Linear Function
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Quadratic Function
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Exponential Function
A
Correct answer
Explanation
A constant function is a function whose output is the same for every input value within its domain.
What is the domain of a function?
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The set of all elements in the first set
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The set of all elements in the second set
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The set of all elements that are paired with at least one element in the second set
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The set of all elements that are paired with exactly one element in the second set
A
Correct answer
Explanation
The domain of a function is the set of all elements in the first set that are paired with at least one element in the second set.
What is the range of a function?
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The set of all elements in the first set
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The set of all elements in the second set
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The set of all elements that are paired with at least one element in the second set
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The set of all elements that are paired with exactly one element in the second set
C
Correct answer
Explanation
The range of a function is the set of all elements in the second set that are paired with at least one element in the first set.
What is the domain of a function?
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The set of all possible inputs
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The set of all possible outputs
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The set of all ordered pairs in the function
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The graph of the function
A
Correct answer
Explanation
The domain of a function is the set of all possible values that can be input into the function.