GATE EC 2018 Question Paper (10-Feb-2018) (Shift 1)

The contour C given below is on the complex plane z = x + jy, where $\mathrm{j}=\sqrt{-1}$



The value of the integral $\frac{1}{\pi \mathrm{j}}\underset{\mathrm{c}}{\oint }\frac{\mathrm{dz}}{{\mathrm{z}}^2-1}$ is _______

Let r = x² + y - z and z³ -xy + yz + y³ = 1. Assume that x and y are independent variables. At (x, y, z) = (2, -1, 1), the value (correct to two decimal places) of $\frac{\partial \mathrm{r}}{\partial \mathrm{x}}$ is _________ .

Consider the network shown below with 𝑅₁ = 1 Ω, 𝑅₂ = 2 Ω and 𝑅₃= 3 Ω. The network is connected to a constant voltage source of 11 V.



The magnitude of the current (in amperes, accurate to two decimal places) through the source is ________.

A band-limited low-pass signal x(𝑡) of bandwidth 5 kHz is sampled at a sampling rate 𝑓${}_{\text{s}}$. The signal x(𝑡) is reconstructed using the reconstruction filter H(𝑓) whose magnitude response is shown below:



The minimum sampling rate fₛ(in kHz) for perfect reconstruction of x(t) is _______.

Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n], where

$X\left[k\right]=\sum _{n=0}^{N-1}x\left[n\right]e^{-j2\pi nk/N}$

The value (correct to two decimal places) of $\sum _{n=0}^{3}x\left[2n\right]$ is ______