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Test: Fourier Series, Numerical Methods & Complex Variables- 2


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30 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Fourier Series, Numerical Methods & Complex Variables- 2

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Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 1

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 1

Fourier series is defined for periodic function and constant.

Fourier series can’t be defined for non-periodic functions.

The sum of two periodic function is also a periodic function.

3 sin (25 t) is a periodic function

4 cos (20 t + 3) and 2 sin (710 t) both are periodic function. So, 4 cos (20t + 3) + 2 sin (710 t) is also a periodic function.

e-|t| sin (25 t) is not a periodic function. So, Fourier series is can’t be defined.

1 is a constant. So, Fourier series is defined.

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 2

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 2

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. 

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 3

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 3

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 4

The Taylor series expansion of 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 4

 We know.  

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 5

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 5

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 6

The sum of the infinite series,  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 6

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 7

The summation of series  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 7

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 8

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

 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 8

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 9

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

  are given by 

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 10

  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 10

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - 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   

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 11

Given, 

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 12

ii, where i = √−1, is given by

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 12

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 13

Assuming    and t is a real number 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 13

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 14

The modulus of the complex number 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 14

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 15

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 15

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 16

Which one of the following is NOT true for complex number Z1and Z2

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 16

(a) is true since 

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

(c) is not true since  |Z1 − Z2 |≥|Z1 |– |Z2

(d) is true since 

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 17

For the equation, s3 - 4s2 + s + 6 =0 

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 17

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 18

The value of the integral of the complex function 

Along the path |s| = 3 is 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 18

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 19

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 19

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 20

The polynomial p(x) = x5 + x + 2 has    

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 20

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 21

 If z = x + jy, where x and y are real, the value of |ejz| is  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 21

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 22

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 22

x(t) = 3 + 2 sin t cos 2t  

x(t) = 3 + sin 3t – sin t 

∴ Root mean square value  

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 23

We wish to solve x2 – 2 = 0 by Netwon Raphson technique. Let the initial guess b x0 = 1.0 Subsequent estimate of x(i.e.x1) will be:  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 23

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 24

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

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - 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 

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 25

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - 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  

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 27

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

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 27

The series converges when X n+1= Xn

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - 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 x0 = 0.2. the first 2 iterations will be

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 28

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - 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?  

Detailed Solution for Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 29

 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  

Test: Fourier Series, Numerical Methods & Complex Variables- 2 - Question 30

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

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