GATE EC question library

The pole-zero map of a rational function $G(s)$ is shown below. When the closed contour $\Gamma$ is mapped into the $G(s)$-plane, then the mapping encircles

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous if at least one of its bits is decoded wrongly. The probability that the received block erroneous is

The impedances $Z=jX,$ for all $X$ in the range $\left ( -\infty ,\infty \right )$, map to the Smith chart as

Which one of the following pole-zero plots corresponds to the transfer function of an $\text{LTI}$ system characterized by the input-output difference equation given below? $$y\left [ n \right ]=\sum ^{3}_{k=0}\left ( -1 \right )^{k}x\left [ n-k \right ]$$

In the given circuit, the two-port network has the impedance matrix $\begin{bmatrix} Z \end{bmatrix}=\begin{bmatrix} 40 & 60\\ 60& 120 \end{bmatrix}$. The value of $Z_{L}$ for which maximum power is transferred to the load is _____________$\Omega$.

The current in the $\text{RL}$-circuit shown below is $i\left ( t \right )=10\cos\left ( 5t-\pi /4 \right )A$. The value of the inductor $\text{(rounded off to two decimal places)}$ is _______ $\text{H}$.

In the circuit shown below, all the components are ideal and the input voltage is sinusoidal. The magnitude of the steady-state output $V_{o}$ ( rounded off to two decimal places) is ______ $V$.

In the circuit shown below, all the components are ideal. If $V_{i}$ is $+2\:V$, the current $I_{o}$ sourced by the op-amp is __________ $\text{mA}$.

In an $8085$ microprocessor, the number of address lines required to access a $16$ K byte memory bank is ____________ .

A $10$-bit D/A converter is calibrated over the full range from $0$ to $10$ V. If the input to the D/A converter is $13 \:A$ (in hex), the output ( rounded off to three decimal places) is __________ $V$.

A transmission line of length $3\lambda /4$ and having a characteristic impedance of $50$ $\Omega$ is terminated with a load of $400 \Omega$. The impedance (rounded off to two decimal places) seen at the input end of the transmission line is __________ $\Omega$.

A binary random variable $X$ takes the value $+2$ or $-2$. The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is __________.

The loop transfer function of a negative feedback system is $$G\left ( s \right )H\left ( s \right )=\frac{K(s+11)}{s(s+2)(s+8)}.$$ The value of $K$, for which the system is marginally stable, is ___________.

The random variable $$Y=\int_{-\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}$$ and $W(t)$ is a real white Gaussian noise process with two-sided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. …

The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.

Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ Which one of the following conditions ensures that a solution exists for the above system?

Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y-1 \right )x?$

The current $I$ in the given network is

A finite duration discrete-time signal $x[n]$ is obtained by sampling the continuous-time signal $x\left ( t \right )=\cos\left ( 200\pi t \right )$ at sampling instants $t=n/400, n=0, 1, \dots ,7.$ The $8$-point discrete Fourier transform $\text{(DFT)}$ of $x[n]$ is defined as …

For the given circuit, which one of the following is correct state equation?