Electronics & Communication Engineering (EC) 2017 [Session 1]

Consider the 5 × 5 matrix \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 2 & 3 & 4 \\ 4 & 5 & 1 & 2 & 3 \\ 3 & 4 & 5 & 1 & 2 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix} \] It is given that \( A \) has only one real eigenvalue. Then the real eigenvalue of \( A \) is

The rank of the matrix \( \mathbf{M} = \begin{bmatrix} 5 & 10 & 10 \\ 1 & 0 & 2 \\ 3 & 6 & 6 \end{bmatrix} \) is

Consider the following statements about the linear dependence of the real valued functions \( y_1 = 1, y_2 = x \) and \( y_3 = x^2 \), over the field of real numbers. I. \( y_1, y_2 \) and \( y_3 \) are linearly independent on \( -1 \le x \le 0 \) II. \( y_1, y_2 \) and \( y_3 \) are linearly dependent on \( 0 \le x \le 1 \) III. \( y_1, y_2 \) and \( y_3 \) are linearly independent on \( 0 \le x \le 1 \) IV. \( y_1, y_2 \) and \( y_3 \) are linearly dependent on \( -1 \le x \le 0 \) Which one among the following is correct?

Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place) ______.

Consider the following statements for continuous-time linear time invariant (LTI) systems. I. There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane. II. There is no causal and BIBO stable system with a pole in the right half of the complex plane. Which one among the following is correct?

Consider a single input single output discrete-time system with \( x[n] \) as input and \( y[n] \) as output, where the two are related as \[ y[n] = \begin{cases} n|x[n]|, & \text{for } 0 \le n \le 10 \\ x[n] - x[n-1], & \text{otherwise}. \end{cases} \] Which one of the following statements is true about the system?