GATE EC question library

For the circuit shown in the following figure, the capacitor $C$ is initially uncharged. At $t=0$, the switch $S$ is closed. The voltage $V_{C}$ across the capacitor at $t=1$ millisecond is In the figure shown above, the $\text{OP AMP}$ is supplied with $\pm 15 \mathrm{~V}$ and the ground has been shown by the symbol $\nabla$.

A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-$5$ number is represented by tis corresponding $3$-bit binary code. For example, the base-$5$ number $24$ will be represented by its BCP code $010100.$ In this numbering system, the BCP code $100010011001$ corresponds to the following number in base-$5$ …

An $\frac{\mathrm{I}}{\mathrm{O}}$ peripheral device shown in Figure $(b)$ below is to be interfaced to an $8085$ microprocessor. To select the $\frac{\mathrm{I}}{\mathrm{O}}$ device in the $\frac{\mathrm{I}}{\mathrm{O}}$ address range $\mathrm{D} 4 \mathrm{H}-\mathrm{D} 7 \mathrm{H}$, its chip-select $(\overline{C S})$ should be connected to …

For the circuit shown in figure below, two $4$-bit parallel-in serial-out shift registers loaded with the data shown are used to feed the data to a full adder. Initially, all the flip-flops are in clear state. After applying two clock pulses, the outputs of the fulladder should be

Two $\text{D}$-flip-flops, as shown below, are to be connected as a synchronous counter that goes through the following $\mathrm{Q}_{1} \mathrm{Q}_{0}$ sequence $00 \longrightarrow 01 \longrightarrow 11 \longrightarrow 10 \longrightarrow 00 \longrightarrow \dots$ The inputs $D_{0}$ and $D_{1}$ respectively should be connected as

Following is the segment of a $8085$ assembly language program $\text{LXISP,EFFFH}$ $\text{CALL 3000 H}$ $3000 \mathrm{H}: \mathrm{LXIH}, 3 \mathrm{CF} 4 \mathrm{H}$ $\text{PUSH PSW}$ $\text{SPIH}$ $\text{POP PSW}$ $\text{RET}$ On completion of $\text{RET}$ execution, the contents of $\mathrm{SP}$ is

The point $\text{P}$ in the following figure is stuck-at-$1$. The output $f$ will be

A signal $m(t)$ with bandwidth $500 \mathrm{~Hz}$ is first multiplied by a signal $g(t)$ where $$g(t)=\sum_{\mathrm{R}=-\infty}^{\infty}(-1)^{k} \delta\left(t-0.5 \times 10^{-4} k\right)$$ The resulting signal is then passed through an ideal lowpass filter with bandwidth $1 \; \mathrm{kHz}$. The output of the lowpass filter would be

The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion $$x(t)=5\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{3}+7\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{2}$$ would be

A uniformly distributed random variable $X$ with probability density function $$f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5))$$ where $u(.)$ is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable $Y$ would be

A system with input $x[n]$ and output $y[n]$ is given as $y[n]=\left(\sin \frac{5}{6} \pi n\right) x(n)$. The system is

The unit-step response of a system starting from rest is given by $\qquad C(t)=1-e^{-2 t} \text{ for } t \geq 0$ The transfer function of the system is

The Nyquist plot of $\mathrm{G}(j \omega) \mathrm{H}(j \omega)$ for a closed loop control system, passes through $(-1, j 0)$ point in the $\mathrm{GH}$ plane. The gain margin of the system in $\mathrm{dB}$ is equal to

The positive values of $\text{“K”}$ and “$a$” so that the system shown in the figure below oscillates at a frequency of $2 \mathrm{rad} / \mathrm{sec}$ respectively are

The unit impulse response of a system is $$h(t)=e^{-t}, t \geq 0$$ For this system, the steady-state value of the output for unit step input is equal to

The transfer function of a phase-lead compensator is given by $$G_{c}(s)=\frac{1+3 T s}{1+T s} \text{ where } T>0$$ The maximum phase-shift provided by such a compensator is

A linear system is described by the following state equation $$X(t)=\mathrm{AX}(t)+\mathrm{BU}(t), \mathrm{A}=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right]$$ The state-transition matrix of the system is

The minimum step-size required for a Delta-Modulator operating at $32 \mathrm{~K}$ samples/ $\mathrm{sec}$ to track the signal (here $u(t)$ is the unit-step function) $\begin{array}{l}x(t)=125 t(u(t)-u(t-1))+(250-125 t)(u(t-1) -u(t-2)) \end{array}$ so that slope-overload is avoided, would be

A zero- mean white Gaussian noise is passed through an ideal lowpass filter of ban width $10 \; \mathrm{kHz}$. The output is the uniformly sampled with sampling period $t_{s}=0.03 \; \mathrm{msec}$. The samples so obtained would be

A source generates three symbols with probabilities $0.25,0.25,0.50$ at a rate of $3000$ symbols per second. Assuming independent generation of symbols, the most efficient source encoder would have average bit rate as