GATE IN 2025 Question Paper (01-Feb-2025) (Shift 2)

A 2-pole, 50 Hz, 3-phase induction motor supplies power to a certain load at 2970 rpm. The torque-speed curve of this machine follows a linear relationship between synchronous speed and 95 % of synchronous speed. Assume mechanical and stray losses to be zero. If the load torque of the motor is doubled, the new operating speed of the motor, in rpm, is

The figure shows a closed-loop system with a plant $G(s)=\frac{1}{s^2}$ and a lead compensator $C(s)$. The compensator is designed to place the dominant closed-loop poles at $-1.5\pm j\frac{\sqrt{27}}{2}$. From the following options, choose the phase lead that the compensator needs to contribute.

Let $f(t)$ and $g(t)$ represent continuous-time real-valued signals. If $h(t)$ denotes cross-correlation between $f(t)$ and $g(-t)$, its continuous-time Fourier transform $H(j\omega )$ equals
Note: $F(j\omega )$ and $G(j\omega )$ denote the continuous-time Fourier transforms of $f(t)$ and $g(t)$, respectively.

Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration of ${\oint }_{C}f(z)dz$ where $C$ is a simple closed path in a simply connected domain $D$.

The plant in the feedback control system shown in the figure is $P(s)=\frac{a}{s^2-b^2}$, where $a>0$ and $b>0$. The type(s) of controller $C(s)$ that CANNOT stabilize the plant is/are