GATE 2020 EC question paper PDF and answer key

A superadditive function $f(\cdot)$ satisfies the following property

$f(x_1 + x_2) \ge f(x_1) + f(x_2)$

Which of the following functions is a superadditive function for $x > 1$?

$a, b, c$ are real numbers. The quadratic equation $ax^2 - bx + c = 0$ has equal roots, which is $\beta$, then

The current I in the given network is

A finite duration discrete-time signal $x[n]$ is obtained by sampling the continuous-time signal $x(t) = \cos(200\pi t)$ at sampling instants $t = n/400$, $n = 0, 1, \cdots, 7$. The 8-point discrete Fourier transform (DFT) of $x[n]$ is defined as

$X[k] = \sum_{n=0}^{7} x[n]e^{-j\frac{\pi kn}{4}}, \quad k = 0, 1, \cdots, 7.$

Which one of the following statements is TRUE?

$S_{PM}(t)$ and $S_{FM}(t)$ as defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.

$S_{PM}(t) = \cos\left(1000\pi t + K_pm(t)\right)$

and

$S_{FM}(t) = \cos\left(1000\pi t + K_f\int_{-\infty}^{t}m(\tau)d\tau\right)$

where $K_p$ is the phase deviation constant in radians/volt and $K_f$ is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of $S_{PM}(t)$ and $S_{FM}(t)$ are same, then the value of the ratio $\frac{K_p}{K_f}$ is ______ seconds.