Mathematics

Functions and Laplace Transforms

59 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

Multiple choice logarithm and its uses basic mathematical concepts physics

The domain of the function $f(x)=[log _{10}(\frac{5x-x^2}{4})]^{{1}/{2}}$ is 

  1. $- \infty < x < \infty $
  2. $1\le x \le 4$
  3. $4\le x \le 16$
  4. $-1\le x \le 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$log _{10}\dfrac{5x-x^2}{4}>0$
$\dfrac{5x-x^2}{4}\geq 1$
$5x-x^2\geq 4$
$x^2-5x+4 \leq 0$
$(x-4)(x-1) \leq 0$
$x $ belongs to $[1,4]$
So the domain is $1 \leq x \leq 4$
Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii
The domain of $f(x)=\dfrac 1{\sqrt {x-[x]}}$ is 
  1. $R$
  2. $Z$
  3. $R-Z$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The function is defined as 

$f(x)=\dfrac 1{\sqrt {x-[x]}}$

The function is not defined if

$\sqrt {x-[x]}=0$

$\implies x-[x]=0$

$\implies x=[x]$

This happens only in the case of Integers 

So The function is not defined at Integer Values

Hence , The Domain of function is $R-Z$

Multiple choice

What is the domain of the function (f(x) = \sqrt{x})?

  1. All real numbers

  2. All nonnegative real numbers

  3. All positive real numbers

  4. All rational numbers
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The domain of a function is the set of all possible values of the independent variable for which the function is defined. Since the square root function is defined only for nonnegative real numbers, the domain of (f(x) = \sqrt{x}) is all nonnegative real numbers.

Multiple choice

What is the range of the function (f(x) = x^2)?

  1. All real numbers

  2. All nonnegative real numbers

  3. All positive real numbers

  4. All rational numbers
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The range of a function is the set of all possible values of the dependent variable. Since the square function always produces a nonnegative real number, the range of (f(x) = x^2) is all nonnegative real numbers.

Multiple choice

The inverse Laplace transform of $F(s) = \frac{1}{s^2 + 4}$ is:

  1. $f(t) = \frac{1}{2} \sin(2t)$
  2. $f(t) = \frac{1}{2} \cos(2t)$
  3. $f(t) = \frac{1}{2} e^{2t}$
  4. $f(t) = \frac{1}{2} e^{-2t}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse Laplace transform of $\frac{1}{s^2 + 4}$ is $\frac{1}{2} \sin(2t)$.

Multiple choice

The Laplace transform of the function $f(t) = t^2$ is:

  1. $F(s) = \frac{2}{s^3}$
  2. $F(s) = \frac{2}{s^4}$
  3. $F(s) = \frac{2!}{s^3}$
  4. $F(s) = \frac{2!}{s^4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Laplace transform of $t^2$ is $\frac{2!}{s^3}$.

Multiple choice

The Laplace transform of the function $f(t) = \sin(at)$ is:

  1. $F(s) = \frac{a}{s^2 + a^2}$
  2. $F(s) = \frac{s}{s^2 + a^2}$
  3. $F(s) = \frac{1}{s^2 + a^2}$
  4. $F(s) = \frac{s}{s^2 - a^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laplace transform of $\sin(at)$ is $\frac{a}{s^2 + a^2}$.

Multiple choice

The Laplace transform of the function $f(t) = \sinh(at)$ is:

  1. $F(s) = \frac{a}{s^2 - a^2}$
  2. $F(s) = \frac{s}{s^2 - a^2}$
  3. $F(s) = \frac{1}{s^2 - a^2}$
  4. $F(s) = \frac{s}{s^2 + a^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laplace transform of $\sinh(at)$ is $\frac{a}{s^2 - a^2}$.

Multiple choice

The Laplace transform of the function $f(t) = \cosh(at)$ is:

  1. $F(s) = \frac{s}{s^2 - a^2}$
  2. $F(s) = \frac{a}{s^2 - a^2}$
  3. $F(s) = \frac{1}{s^2 - a^2}$
  4. $F(s) = \frac{s}{s^2 + a^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laplace transform of $\cosh(at)$ is $\frac{s}{s^2 - a^2}$.

Multiple choice

The Laplace transform of the function $f(t) = \frac{1}{t}$ is:

  1. $F(s) = \ln(s)$
  2. $F(s) = \frac{1}{s}$
  3. $F(s) = \frac{1}{s^2}$
  4. $F(s) = \frac{1}{s^3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laplace transform of $\frac{1}{t}$ is $\ln(s)$.

Multiple choice

What is the Laplace transform of a function?

  1. An integral transform that converts a function of time into a function of a complex variable

  2. A Fourier transform that converts a function of time into a function of frequency

  3. A Hilbert transform that converts a function of time into a function of its conjugate

  4. A Mellin transform that converts a function of time into a function of its moments
Reveal answer Fill a bubble to check yourself
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.

Multiple choice

What is the Laplace transform of the function $f(t) = t^2 e^{-at}$?

  1. $F(s) = \frac{2}{(s+a)^3}$
  2. $F(s) = \frac{2a}{(s+a)^3}$
  3. $F(s) = \frac{2}{(s-a)^3}$
  4. $F(s) = \frac{2a}{(s-a)^3}$
Reveal answer Fill a bubble to check yourself
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, $F(s) = \mathcal{L}{t^2 e^{-at}} = \frac{2!}{(s+a)^3} = \frac{2}{(s+a)^3}$.

Multiple choice

What is the Laplace transform of the function f(t) = t^2?

  1. 2/(s^3)

  2. 2/s

  3. 2s

  4. 2s^2
Reveal answer Fill a bubble to check yourself
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.

Multiple choice

What is the dimension of the range of the linear transformation f(x) = [x, 2x, 3x]?

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
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.

Multiple choice

What is the range of the function (f(x) = 2x + 1) when the domain is ({1, 2, 3})?

  1. {3, 5, 7}

  2. {1, 3, 5}

  3. {2, 4, 6}

  4. {0, 2, 4}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The range of a function is the set of all second elements in the ordered pairs.