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QUESTION: 1

The Fourier series of a real periodic function has only

P. cosine terms if it is even

Q. sine terms if it is even

R. cosine terms if it is odd

S. sine terms if it is odd

Which of the above statements are correct?

Solution:

Because sine function is odd and cosine is even function.

QUESTION: 2

For the function e^{–x}, the linear approximation around x = 2 is

Solution:

(neglecting higher power of x)

QUESTION: 3

In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)^{2} is

Solution:

QUESTION: 4

The function x(t) is shown in the figure. Even and odd parts of a unit-step function u(t) are respectively,

Solution:

QUESTION: 5

In the Taylor series expansion of e^{x} about x = 2, the coefficient of (x - 2)^{4} is

Solution:

Taylor series expansion of f(x) about a is given by

QUESTION: 6

The Fourier series expansion of a symmetric and even function, f(x) where

Will be

Solution:

f(x) is symmetric and even, it’s Fourier series contain only cosine term. Now.

QUESTION: 7

The Fourier series expansion of the periodic signal shown below will contain the following nonzero terms

Solution:

from the figure, we can say that f(t) is an symmetric and even function of t. as cost is even function so choice (b) is correct.

QUESTION: 8

The Fourier series for the function f(x)=sin^{2}x is

Solution:

Here f(x ) = sin^{2} x is even function, hence f( x ) has no sine term.

QUESTION: 9

X(t) is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency ω = 2π (2k ) /T ; k = 1, 2,.... Also, no sine terms are present. Then x(t) satisfies the equation

Solution:

No sine terms are present.

∴x(t ) is even function.

QUESTION: 10

The residue of the function f(z)

Solution:

QUESTION: 11

The residues of a complex function at its poles are

Solution:

x(z) has simple poles at z = 0,1, 2.

QUESTION: 12

The value of the contour integral in positive sense is

Solution:

QUESTION: 13

The integral evaluated around the unit circle on the complex plane for

Solution:

QUESTION: 14

An analytic function of a complex variable z = x + iy is expressed as f(z) = u (x, y) + i v(x, y) where i =√−1 . If u = xy, the expression for v should be

Solution:

Here u and v are analytic as f(z) is analytic.

∴ u, v satisfy Cauchy-Riemann equation.

QUESTION: 15

The analytic function has singularities at

Solution:

QUESTION: 16

The value of the integral (where C is a closed curve given by |z| = 1) is

Solution:

QUESTION: 17

Roots of the algebraic equation x^{3} +x^{2} +x+ 1 = 0 are

Solution:

QUESTION: 18

The algebraic equation

F (s ) = s^{5} − 3s^{4}+ 5s^{3}− 7s^{2} + 4s + 20 is given F ( s ) = 0 has

Solution:

we can solve it by making Routh Hurwitz array.

We can replace 1^{st} element of s^{1} by 10.

If we observe the 1^{st} column, sign is changing two times.

So we have two poles on right half side of imaginary

Axis and 5s^{2}+20=0

So, s =±2j and1 pole on left side of imaginary axis .

QUESTION: 19

The value of where C is the contour z −i / 2=1 is

Solution:

QUESTION: 20

Let z^{3} = z, where z is a complex number not equal to zero. Then z is a solution of

Solution:

Now by hit and trial method we see the solution being

Z^{4} = 1

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