If {x} is a continuous, real valued random variable defined over the interval (-$$\infty $$, +$$\infty $$) and its occurrence is defined by the density function given as: $${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$ where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$

$$\frac{{\frac{1}{3}.\frac{1}{3}.\frac{1}{3} + \frac{1}{4}.\frac{1}{4}.\frac{1}{4} - 3.\frac{1}{3}.\frac{1}{4}.\frac{1}{5} + \frac{1}{5}.\frac{1}{5}.\frac{1}{5}}}{{\frac{1}{3}.\frac{1}{3} + \frac{1}{4}.\frac{1}{4} + \frac{1}{5}.\frac{1}{5} - \left( {\frac{1}{3}.\frac{1}{4} + \frac{1}{4}.\frac{1}{5} + \frac{1}{5}.\frac{1}{3}} \right)}}{\text{ is?}}$$

A

$$\frac{2}{3}$$

B

$$\frac{3}{4}$$

C

$$\frac{{47}}{{60}}$$

D

$$\frac{{49}}{{60}}$$

Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $${a_k}$$ , that is
$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$
The same function x(t) can also be considered as a periodic function with period T' = 40. Let b_k be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16,$$ then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} $$ is equal to

A

256

B

64

C

16

D

4

Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below

The value of $$I$$ is

A

$$\frac{1}{2}{\text{i}}$$

B

$$\frac{2}{3}{\text{i}}$$

C

$$\frac{3}{4}{\text{i}}$$

D

$$\frac{4}{5}{\text{i}}$$

For a function g(t), it is given that
$$\int\limits_{ - \infty }^{ + \infty } {g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$ for any real value $$\omega $$ . If $$y\left( t \right) = \int\limits_{ - \infty }^t {g\left( \tau \right)} d\tau ,\,{\rm{then}}\,\int\limits_{ - \infty }^{ + \infty } {y\left( t \right)} dt$$ is. . . . . . . .

A

0

B

-j

C

$$ - {j \over 2}$$

D

$${j \over 2}$$

Mean Value theorem is applicable to the and continuous in open interval (a, b) b) Functions continuous in closed interval only and having same value at point ‘a’ and ‘b’ c) Functions continuous in closed interval and differentiable in open interval (a, b) d) Functions differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’

A

Functions differentiable in closed interval and continuous in open interval (a,

B

a, b

C

a, b

D

a, b

The simplified value of $$\frac{{\left( {1 + \frac{1}{{1 + \frac{1}{{100}}}}} \right)\left( {1 + \frac{1}{{1 + \frac{1}{{100}}}}} \right) - \left( {1 - \frac{1}{{1 + \frac{1}{{100}}}}} \right)\left( {1 - \frac{1}{{1 + \frac{1}{{100}}}}} \right)}}{{\left( {1 + \frac{1}{{1 + \frac{1}{{100}}}}} \right) + \left( {1 - \frac{1}{{1 + \frac{1}{{100}}}}} \right)}} = ?$$

A

100

B

$$\frac{{200}}{{101}}$$

C

200

D

$$\frac{{202}}{{100}}$$

Let x be a continuous variable defined over the interval $$\left( { - \infty ,\,\infty } \right)$$ , and f(x) = e^{-x-e^-x} . The integral $${\text{g}}\left( {\text{x}} \right) = \int {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$ is equal to

A

ee-x

B

e-e-x

C

e-ex

D

e-x

Rolle’s theorem is applicable to the and continuous in open interval (a, b) only and having same value at point ‘a’ and ‘b’ b) Functions continuous in closed interval only and having same value at point ‘a’ and ‘b’ c) Functions continuous in closed interval and differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’ d) Monotonically Increasing funtions

A

Functions differentiable in closed interval and continuous in open interval (a,

B

a, b

C

a, b

D

a, b

With respect to the numerical evaluation of the definite integral $${\text{K}} = \int_{\text{a}}^{\text{b}} {{{\text{x}}^2}{\text{dx,}}} $$ where a and b are given, which of the following statements is/are TRUE?
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.

A

I only

B

II only

C

Both I and II

D

Neither I nor II

A normal random variable X has following probability density function $${{\text{f}}_{\text{x}}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {8\pi } }}{{\text{e}}^{ - \left\{ {\frac{{{{\left( {{\text{x}} - 1} \right)}^2}}}{8}} \right\}}},\, - \infty Then $$\int\limits_1^\infty {{{\text{f}}_{\text{x}}}\left( {\text{x}} \right){\text{dx}}} $$ is

A

0

B

$$\frac{1}{2}$$

C

$$1 - \frac{1}{{\text{e}}}$$

D

1