GATE EC question library

Let $X(t)$ be a wide sense stationary random process with the power spectral density $S_{X}(f)$ as shown in Figure (a), where $f$ is in Hertz$(Hz)$. The random process $X(t)$ is input to an ideal lowpass filter with the frequency response …

A continuous time signal $x(t)=4 \cos(200\pi t)+8 \cos(400\pi t)$, where $t$ is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response $$h(t)=\begin{cases} \frac{2 \sin (300\pi t)}{\pi t},& t\neq 0 \\ 600, & t=0. \end{cases}$$ Let $y(t)$ be the output of this filter. The maximum value of $\mid y(t) \mid$ is …

An optical fiber is kept along the $\hat{z}$ direction. The refractive indices for the electric fields along $\hat{x}$ and $\hat{y}$ directions in the fiber are $n_{x}=1.5000$ and $n_{y}=1.5001$, respectively ($n_{x} \neq n_{y}$ due to the imperfection in the fiber cross-section). The free space wavelength of a light wave propagating in the fiber …

The expression for an electric field in free space is , $\textbf{E}=E_{0}( \hat{x}+\hat{y}+j2 \hat{z} )e^{-j( \omega t-kx+ky)}$ where $x,y,z$ represent the special coordinates, $t$ represents time, and $\omega,k$ are constants.This electric field,

A half wavelength dipole is kept in the $x-y$ plane and oriented along $45^{\circ}$ from the $x$-axis. Determine the direction of null in the radiation pattern for $0\leq \phi \leq \pi$. Here the angle $\theta (0\leq \theta \leq \pi )$ is measured from the $z$-axis and the angle$\phi (0\leq \theta \leq 2\pi )$ is measured from the $x$-axis in …

The ninth and the tenth of this month are Monday and Tuesday________.

It is ___________ to read this year’s textbook ________ the last year’s.

A rule states that in order to drink beer, one must be over $18$ years old. In a bar, there are $4$ people. $P$ is $16$ years old. $Q$ is $25$ years old. $R$ is drinking milkshake and $S$ is drinking a beer. What must be checked to ensure that the rule is being followed?

Fatima starts from point $P$, goes North for $3$ km, and then East for $4$ km to reach point $Q$. She then turns to face point $P$ and goes $15$ km in that direction. She then goes North for $6$ km. How far is she from point $P$, and in which direction should she go to reach point $P$?

$500$ students are taking one or more course out of Chemistry, Physics, and Mathematics. Registration records indicate course enrollment as follows: Chemistry $(329)$, Physics $(186)$, and Mathematics $(295)$. Chemistry and Physics $(83)$, Chemistry and Mathematics $(217)$, and Physics and Mathematics $(63)$. How many students are taking all …

“If you are looking for a history of India, or for an account of the rise and fall of the British Raj, or for all reason of the cleaving of the subcontinent into two mutually antagonistic parts and the effects this mutilation will have in the respective sections, and ultimately on Asia, you will not find it in these pages; for though I have spent a lifetime …

Each of $P, Q, R, S, W, X, Y$and $Z$ has been married at most once . $X$ and $Y$ are married and have two children $P$ and $Q$. $Z$ is the grandfather of the daughter $S$ of $P$. Further , $Z$ and $W$ are married and are parents of $R$. Which one of the following must necessarily be FALSE?

$1200$ men and $500$ women can build a bridge in $2$ weeks. $900$ men and $250$ women will take $3$ weeks to build the same bridge. How many men will be needed to build the bridge in one week?

The number of $3$-digit numbers such that the digit $1$ is never to the immediate right of $2$ is

A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at $25$ m intervals in this plot. Which of the following is the steepest path leaving from $P$?

The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0& 0& 1& -1& \end{bmatrix}$ is ________.

The general solution of the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}-5y=0$ in terms of arbitrary constants $K_1$ and $K_2$ is

The smaller angle (in degrees) between the planes $x+y+z=1$ and $2x-y+2z=0$ is ________.

The residues of a function $$f(z)=\frac1{(z-4)(z+1)^3 }$$ are

In the circuit shown, V is a sinusoidal voltage source. The current $I$ is in phase with voltage V. The ratio$\frac{\text{amplitude of voltage across the capacitor}}{\text{amplitude of voltage across the resistor}}$ is ___________.