GATE IN 2024 Question Paper (04-Feb-2024) (Shift 2)

In the figure shown, $R=4.5$ kΩ, $\Delta R=1.5$ kΩ, and INA is assumed to be ideal. The equivalent resistance between A and B is ___________ kΩ (rounded off to nearest integer).

Consider the capacitive sensor circuit and its output voltage shown in the figure. The circuit is switched ON at $t=0$. Assuming the opamp to be ideal, the frequency of the output voltage $V_o$ is _____________ kHz (rounded off to two decimal places).

The 4-point DFTs of two sequences $x[n]$ and $y[n]$ are $X[k]=[1,-j,1,j]$ and $Y[k]=[1,3j,1,-3j]$, respectively. Assuming $z[n]$ represents the 4-point circular convolution of $x[n]$ and $y[n]$, the value of $z[0]$ is ______________ (rounded off to nearest integer).
Note: The DFT of a N-point sequence $x[n]$ is defined as
$X[k]={\sum }_{n=0}^{N-1}x[n]e^{\frac{-j2\pi nk}{N}}$

Consider the figure shown. For zero deflection in the galvanometer, the required value of resistor $R_x$ is ______________ Ω (rounded off to nearest integer).

Consider a unity negative feedback system with its open-loop pole-zero map as shown in the figure. If the point $s=j\alpha$, $\alpha >0$, lies on the root locus, the value of $\alpha$ is ______________ (rounded off to nearest integer).
Note: The poles are marked with $\times$ in the figure.