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

The Taylor series expansion of 

The sum of the infinite series,  

The summation of series  

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

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

  are given by 

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   

iⁱ, where i = √−1, is given by

Assuming    and t is a real number 

The modulus of the complex number 

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

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

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

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

The value of the integral of the complex function 

Along the path |s| = 3 is 

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

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

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

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

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:  

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

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 

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  

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

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

 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?  

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