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

Let $C$ be the closed curve in the $xy$ -plane, traversed in the counterclockwise direction along the boundary of the rectangle with vertices at $(0,0),(2,0),(2,1),(0,1)$. The value of the line integral
${\oint }_C(-e^{y}dx+e^{x}dy)$ is

In the figure shown, assume
• $\alpha$ is the phase angle between the load current and the load voltage
• $\beta$ is the phase angle by which pressure coil current lags the pressure coil voltage of the wattmeter
• $\gamma$ is the phase angle between currents in the pressure coil and the current coil of the wattmeter
• $\delta$ is the phase angle of the voltage transformer
• $\theta$ is the phase angle of the current transformer
When the load has a lagging phase angle of $\alpha$, which one of the following options is correct?

Consider an ultrasonic measurement system shown in the figure. The ultrasonic transmitter (T) sends a continuous wave signal $x(t)=\cos (2\pi f_{1}t)$ volts towards an object whose vibration is modeled as $m(t)=0.5\sin (2\pi f_{2}t)$ volts. Neglecting the phase shift due to any other effect, the received signal at the receiver (R) is $y(t)=\cos (2\pi f_{1}t+\beta \cos (2\pi f_{2}t))$ volts.
Assuming the frequency sensitivity factor as 500 Hz/volt, $f_1=40$ kHz, $f_2=1$ kHz, the modulation index $(\beta )$ and the frequency deviation in $y(t)$, respectively, are

The complex functions $f(z)=u(x,y)+iv(x,y)$ and $\overline{f(z)}=u(x,y)\text{–}iv(x,y)$ are both analytic in a given domain. Choose the correct option(s) from the following.

The readings recorded from a 20-psig pressure gauge are given in the Table. The regression line obtained for the data is $y=0.04x+10.32$. The regression coefficient of determination, $R^2$ = __________ (rounded off to three decimal places).