MCQs of Fourier Integral (Mathematics - II - 3110015)

Showing 11 to 20 out of 21 Questions

11.

The Fourier integral of

f (x)

is represented as

(a)	$f (x) = \int_{- \infty}^{\infty} [A (λ) \cos (λx) + B (λ) \sin (λx)] dλ; \forall x \in ℝ$
(b)	$f (x) = \int_{0}^{\infty} [A (λ) \cos (λx) + B (λ) \sin (λx)] dλ; \forall x \in ℝ$
(c)	$f (x) = \int_{0}^{L} [A (λ) \cos (λx) + B (λ) \sin (λx)] dλ; \forall x \in ℝ$
(d)	$f (x) = \frac{1}{L} \int_{- \infty}^{\infty} [A (λ) \cos (λx) + B (λ) \sin (λx)] dλ; \forall x \in ℝ$

12.

The integral of

f (x) = \frac{2}{π} \int_{0}^{\infty} {[\int_{0}^{\infty} f (u) \sin (λu) \cos (λu) du]}_{0}^{\infty} dx

is called ______.

13.

The integral of

f (x) = \int_{0}^{\infty} B (ω) \sin (ωx) dω

is a fourier sine integral, then

B (ω)

is ________.

14.

The integral of

f (x) = \frac{1}{π} \int_{0}^{\infty} [\int_{- \infty}^{\infty} f (t) \cos (t - x) dt] dω

is called __________ .

15.

Fourier sine integral of

f (x) = e^{- kx}, x > 0, k > 0

is ______.

(a)	$\frac{1}{π} \int_{0}^{\infty} \frac{ksin (ωx)}{k^{2} + ω^{2}} dω$
(b)	$\frac{1}{π} \int_{0}^{\infty} \frac{ωsin (ωx)}{k^{2} + ω^{2}} dω$
(c)	$\frac{2}{π} \int_{0}^{\infty} \frac{ωsin (ωx)}{k^{2} + ω^{2}} dω$
(d)	$\frac{2}{π} \int_{0}^{\infty} \frac{ksin (ωx)}{k^{2} + ω^{2}} dω$

16.

Find Fourier cosine integral of

f (x) = e^{- 2 bx}, x > 0, b > 0

(a)	$\frac{2 b}{π} \int_{0}^{\infty} \frac{\cos ωx}{4 b^{2} + ω^{2}} dω$
(b)	$\frac{4}{π} \int_{0}^{\infty} \frac{\cos ωx}{4 b^{2} + ω^{2}} dω$
(c)	$- \frac{4 b}{π} \int_{0}^{\infty} \frac{\cos ωx}{4 b^{2} + ω^{2}} dω$
(d)	$\frac{4 b}{π} \int_{0}^{\infty} \frac{\cos ωx}{4 b^{2} + ω^{2}} dω$

17.

Fourier integral of the function

f (x) = \{\begin{matrix} 0 & , x < 0 \\ e^{- x} & , x > 0 \end{matrix}

is given by

f (x) = \int_{0}^{\infty} [A (ω) \cos (ωx) + B (ω) \sin (ωx)] dω

, then

A (ω) =

_________ .

18.

Let

f (x) = \{\begin{cases} e^{- x}, x > 0 \\ 0, x < 0 \end{cases} \begin{array}{l}  \end{array}

. If Fourier integral of the function is

f (x) = \int_{0}^{\infty} [A (ω) \cos (ωx) + B (ω) \sin (ωx)] dω

, then

B (ω)

is ________ .

19.

f (x) = \{\begin{matrix} 0, 0 < x < 1 \\ 1, 1 < x < 2 \\ 0, 0 < x > 2 \end{matrix}

, then find Fourier sine integral.

(a)	$\int_{0}^{\infty} \frac{2}{πω} (\cos ω - \cos 2 ω) \cdot \sin (ωx) dω$
(b)	$\int_{0}^{\infty} \frac{1}{πω} (\cos ω - \cos 2 ω) \cdot \sin (ωx) dω$
(c)	$\int_{0}^{\infty} \frac{2}{πω} (\cos 2 ω - \cos ω) \cdot \sin (ωx) dω$
(d)	$\int_{0}^{\infty} \frac{1}{πω} (2 \cos ω - \cos 2 ω) \cdot \sin (ωx) dω$

20.

Using Fourier cosine integral of the function

f (x) = e^{- 3 x}, x > 0

, find the value of

\int_{0}^{\infty} \frac{\cos ω}{9 + ω^{2}} dω

Showing 11 to 20 out of 21 Questions