GATE EC question library

In the circuit shown, $W$ and $Y$ are MSBs of the control inputs. The output $F$ is given by

If $X$ and $Y$ are inputs and the Difference $(D=X-Y)$ and the Borrow $(B)$ are the outputs, which one of the following diagrams implements a half-subtractor?

Let $H_{1}(z)= (1-pz^{-1})^{-1},H_{2}(z)= (1-qz^{-1})^{-1},H(z)=H_{1}(z)+rH_{2}(z).$ The quantities $p,$ $q$, $r$ are real numbers. Consider $p=\frac{1}{2},q=-\frac{1}{4},\mid r \mid < 1.$ If the zero of $H(z)$ lies on the unit circle, then $r$ $=$ _________

Let $h(t)$ denote the impulse response of a causal system with transfer function $\frac{1}{s+1}.$ Consider the following three statements. $S1$: The system is stable. $S2$: $\frac{h(t+1)}{h(t)}$ is independent of $t$ for $t > 0$. $S3$: A non-causal system with the same transfer function is stable. For the above system,

The $z$-transform of the sequence $x[n]$ is given by $X(z)=\frac{1}{(1-2z^{-1})^{2}},$ with the region of convergence $\mid z \mid >2$. Then, $x[2]$ is _________.

The steady state error of the system shown in the figure for a unit step input is _________.

The state equation of a second-order linear system is given by $$\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$$ For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix},$ and for $x_{0}= \begin{bmatrix} 0\\ 1 \end{bmatrix},$ $x(t) \begin{bmatrix} e^{-t}-e^{-2t}\\ -e^{-t}+2e^{-2t} \end{bmatrix}$. When …

In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following transfer functions has this root locus?

Let $X(t)$ be a wide sense stationary $(WSS)$ random process with power spectral density $S_{X}(f).$ If $Y(t)$ is the process defined as $Y(t)= X(2t-1)$, the power spectral density $S_{Y}( f )$ is

A real band-limited random process $X(t)$ has two-sided power spectral density $$S_{X}(f)= \begin{cases} 10^{-6} (3000-\mid f \mid) \text{Watts/Hz} & \text{for } \mid f \mid \leq 3 kHz \\ 0 & \text{otherwise} \end{cases}$$ where $f$ is the frequency expressed in Hz.The signal $X(t)$ modulates a carrier $\cos16000\pi t$ and the resultant …

In a $PCM$ system, the signal $m(t)=\left \{\sin(100\pi t)+\cos(100\pi t) \right \}V$ is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step size $0.75$ $V$. The minimum data rate of the $PCM$ system in bits per second is ______.

A binary random variable $X$ takes the value of $1$ with probability $1/3$. $X$ is input to a cascade of $2$ independent identical binary symmetric channels (BSCs) each with crossover probability $1/2$. The output of BSCs are the random variables $Y_{1}$ and $Y_{2}$ as shown in the figure. The value of $H( Y_{1} )+H( Y_{2} )$ in bits is …

Given the vector $\textbf{A}= ( \cos x ) ( \sin y )\hat{a_{x}}+( \sin x )( \cos y )\hat{a_{y}},$ where $\hat{a_{x}},$ $\hat{a_{y}}$ denote unit vectors along $x$, $y$ directions, respectively. The magnitude of curl of $\textbf{A}$ is __________

A region shown below contains a perfect conducting half-space and air. The surface current $\overrightarrow{K_{s}}$ on the surface of the perfect conductor is $\overrightarrow{K_{s}}= \hat{x}2$ amperes per meter. The tangential $\overrightarrow{H}$ field in the air just above the perfect conductor is

Assume that a plane wave in air with an electric field $\overrightarrow{E}= 10\cos ( \omega t-3x-\sqrt{3}z )\hat{a_{y}}V/m$ is incident on a non-magnetic dielectric slab of relative permittivity $3$ which covers the region $z > 0$. The angle of transmission in the dielectric slab is _________ degrees.

Which of the following options is the closest in meaning to the word underlined in the sentence below? In a democracy, everybody has the freedom to disagree with the government.

After the discussion, Tom said to me, 'Please revert!'. He expects me to _________.

While receiving the award, the scientist said, "I feel vindicated". Which of the following is closest in meaning to the word ‘vindicated’?

Let $f(x,y)=x^n y^m=P$. If $x$ is doubled and $y$ is halved, the new value of $f$ is

In a sequence of $12$ consecutive odd numbers, the sum of the first $5$ numbers is $425$. What is the sum of the last $5$ numbers in the sequence?