List of top Questions asked in GATE EC

Consider a real-valued base-band signal \( x(t) \), band limited to 10 kHz. The Nyquist rate for the signal \( y(t) = x(t) \cdot \left( 1 + \frac{t}{2} \right) \) is

Consider two 16-point sequences \(x[n]\) and \(h[n]\). Let the linear convolution of \(x[n]\) and \(h[n]\) be denoted by \(y[n]\), while \(z[n]\) denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of \(x[n]\) and \(h[n]\). The value(s) of \(k\) for which \(z[k] = y[k]\) is/are

A bar of silicon is doped with boron concentration of \( 10^{16} \, \text{cm}^{-3} \) and assumed to be fully ionized. It is exposed to light such that electron-hole pairs are generated throughout the volume of the bar at the rate of \( 10^{20} \, \text{cm}^{-3} \, \text{s}^{-1} \). If the recombination lifetime is 100 µs, intrinsic carrier concentration of silicon is \( 10^{10} \, \text{cm}^{-3} \) and assuming 100\% ionization of boron, then the approximate product of steady-state electron and hole concentrations due to this light exposure is

Consider the differential equation given below.
\[ \frac{dy}{dx} + \frac{x}{1 - x^2} y = x \sqrt{y}. \] The integrating factor of the differential equation is

Two continuous random variables \( X \) and \( Y \) are related as
\( Y = 2X + 3 \).
Let \( \sigma_X^2 \) and \( \sigma_Y^2 \) denote the variances of \( X \) and \( Y \), respectively. The variances are related as