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

Choose the eigenfunction(s) of stable linear time-invariant continuous-time systems from the following options.

The probability of a student missing a class is 0.1. In a total number of 10 classes, the probability that the student will not miss more than one class is ______ (rounded off to two decimal places).

A metallic strain-gauge (SG) with resistance $R_{SG}$ is connected as shown in the figure, where $R_{L1}$, $R_{L2}$, $R_{L3}$ represent the lead wire resistances. The SG has a gauge factor of 2 and nominal resistance $R_N$ of 125 $\Omega$. When the SG is subjected to a tensile strain of $2\times 10^{-3}$, the resulting change in $R_{SG}$ is $\Delta R$. The $\Delta R$ value is measured as $\Delta R_{MEAS}=R_{EQ2}-R_{EQ1}$. The $R_{EQ1}$ and $R_{EQ2}$ are the equivalent resistances measured between the terminals 1 and 2, and terminals 2 and 3, respectively.
If $R_{L1}=R_{L2}=5\Omega$, and $R_{L3}=4.95\Omega$, the measured value of tensile strain is ______ $\times 10^{-3}$ (rounded off to two decimal places).

Let $X(e^{j\omega })$ represent the discrete time Fourier transform of a 4-length sequence $x[n]$, where $x[0]=1$, $x[1]=2$, $x[2]=2$, and $x[3]=4$. $X(e^{j\omega })$ is sampled at ${\omega }_k=\frac{2\pi k}{3}$ to generate a periodic sequence in $k$ with period 3, where $k$ represents an integer. Let $y[n]$ represent another sequence such that its discrete Fourier transform $Y[k]$ is given as $Y[k]=X(e^{j{\omega }_k})$ for $0\le k\le 2$. The value of $y[0]$ is ______ (in integer).

A schematic of a Michelson interferometer, used for the measurement of refractive index of gas, is shown in the figure. The transparent chamber is filled with a gas of refractive index $n_g$, where $n_g\ne 1$, at atmospheric pressure. If a 532 nm laser beam produces 30 interference fringes on the screen, then the number of fringes produced by a 632.8 nm laser beam will be ______ (rounded off to one decimal place).
Note: Assume that the effect of beamsplitter width is negligible. The setup is placed in air medium with refractive index equal to 1.