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QUESTION: 1

Choose the function f(t); –∞ < t < ∞, for which a Fourier series cannot be defined.

Solution:

QUESTION: 2

The trigonometric Fourier series for the waveform f(t) shown below contains

Solution:

From figure it’s an even function. so only cosine terms are present in the series and for DC value,

So DC take negative value.

QUESTION: 3

Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x = 0?

Solution:

QUESTION: 4

The Taylor series expansion of

Solution:

We know.

QUESTION: 5

Solution:

QUESTION: 6

The sum of the infinite series,

Solution:

QUESTION: 7

The summation of series

Solution:

QUESTION: 8

Fourier series for the waveform, f (t) shown in fig. is

Solution:

From the figure, we say f (x) is even functions. so choice (c) is correct.

QUESTION: 9

The Fourier Series coefficients, of a periodic signal x (t), expressed as

are given by

Solution:

QUESTION: 10

Solution:

QUESTION: 11

For the function of a complex variable W = ln Z (where, W = u + jv and Z = x + jy), the u = constant lines get mapped in Z-plane as

Solution:

Given,

QUESTION: 12

i^{i}, where i = √−1, is given by

Solution:

QUESTION: 13

Assuming and t is a real number

Solution:

QUESTION: 14

The modulus of the complex number

Solution:

QUESTION: 15

Using Cauchy’s integral theorem, the value of the integral (integration being taken in counter clockwise direction)

Solution:

Here f (z) has a singularities at z i / 3

QUESTION: 16

Which one of the following is NOT true for complex number Z_{1}and Z_{2} ?

Solution:

(a) is true since

(b) is true by triangle inequality of complex number.

(c) is not true since |Z_{1} − Z_{2} |≥|Z_{1} |– |Z_{2} |

(d) is true since

QUESTION: 17

For the equation, s^{3} - 4s^{2} + s + 6 =0

The number of roots in the left half of s-plane will be

Solution:

Constructing Routh-array

Number of sign changes in the first column is two, therefore the number of roots in the left half splane is 2

QUESTION: 18

The value of the integral of the complex function

Along the path |s| = 3 is

Solution:

f (s) has singularities at s =−1, −2 which are inside the given circle

QUESTION: 19

For the function of a complex variable z, the point z = 0 is

Solution:

Therefore the function has z = 0 is a pole of order 2.

QUESTION: 20

The polynomial p(x) = x^{5} + x + 2 has

Solution:

As, P(x) of degree 5 .So other four roots are complex.

QUESTION: 21

If z = x + jy, where x and y are real, the value of |e^{jz}| is

Solution:

QUESTION: 22

The root mean squared value of x(t) = 3 + 2 sin (t) cos (2t) is

Solution:

x(t) = 3 + 2 sin t cos 2t

x(t) = 3 + sin 3t – sin t

∴ Root mean square value

QUESTION: 23

We wish to solve x^{2} – 2 = 0 by Netwon Raphson technique. Let the initial guess b x0 = 1.0 Subsequent estimate of x(i.e.x1) will be:

Solution:

QUESTION: 24

The order of error is the Simpson’s rule for numerical integration with a step size h is

Solution:

QUESTION: 25

The table below gives values of a function F(x) obtained for values of x at intervals of 0.25.

The value of the integral of the function between the limits 0 to 1 using Simpson’s rule is

Solution:

QUESTION: 26

A differential equation has to be solved using trapezoidal rule of integration with a step size h=0.01s. Function u(t) indicates a unit step function. If x(0-)=0, then value of x at t=0.01s will be given by

Solution:

QUESTION: 27

Consider the series obtained from the Newton-Raphson method. The series converges to

Solution:

The series converges when X _{n+1}= X_{n} =α

QUESTION: 28

Given a>0, we wish to calculate its reciprocal value 1/a by using Newton-Raphson method :

for f(x) = 0. For a=7 and starting wkith x_{0} = 0.2. the first 2 iterations will be

Solution:

QUESTION: 29

With a 1 unit change in b, what is the change in x in the solution of the system of equations x + y = 2, 1.01 x + 0.99 y = b?

Solution:

Given x + y = 2 …………….. (i)

1.01 x + 0.99 y = b …………….. (ii)

Multiply 0.99 is equation (i), and subtract from equation (ii), we get

QUESTION: 30

The accuracy of Simpson's rule quadrature for a step size h is

Solution:

### Complex Fourier Transform

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