GATE EC question library

The ordinary differential equation $$\frac{dx}{dt}=-3x+2, \text{ with }x(0) = 1$$ is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is _________

Suppose $C$ is the closed curve defined as the circle $x^{2}+y^{2}=1$ with $C$ oreinted anti-clockwise. The value of $\oint$ ( $xy^{2}$ $dx$ + $x^{2}y$ $dy$ )over the curve $C$ equals _________

Two random variables $X$ and $Y$ are distributed according to $$f_{X,Y}(x,y)=\begin{cases} (x+y),& 0\leq x\leq 1,&0\leq y\leq 1\\ 0, & \text{otherwise.} \end{cases}$$ The probability $P(X+Y\leq 1)$ is ________

The matrix $A=\begin{bmatrix} a & 0 &3 &7 \\ 2& 5&1 &3 \\ 0& 0& 2 &4 \\ 0&0 & 0 &b \end{bmatrix}$ has $\text{det}(A) = 100$ and $\text{trace}(A) = 14$. The value of $\mid a-b \mid$ is ________

In the given circuit, each resistor has a value equal to $1\Omega$. What is the equivalent resistance across the terminals $a$ and $b$?

In the circuit shown in the figure, the magnitude of the current (in amperes) through $R_{2}$ is ____

A continuous-time filter with transfer function $H\left ( s \right )= \frac{2s+6}{s^{2}+6s+8}$ is converted to a discrete-time filter with transfer function $G\left ( z\right )= \frac{2z^{2}-0.5032 \: z}{z^{2}-0.5032 \: z+k}$ so that the impulse response of the continuous-time filter, sampled at $2$ $Hz$, is identical at the sampling instants to the …

The Discrete Fourier Transform (DFT) of the $4$-point sequence $x\left [ n \right ]=\left \{ x\left [ 0 \right ],x\left [ 1 \right ], x\left [ 2 \right ], x\left [ 3 \right ] \right \}= \left \{ 3,2,3,4 \right \}$ is …

The switch $S$ in the circuit shown has been closed for a long time. It is opened at $t = 0$ and remains open after that. Assume that the diode has zero reverse current and zero forward voltage drop. The steady state magnitude of the capacitor voltage $V_{c}$ (in volts) is ______

A voltage $V_{G}$ is applied across a $MOS$ capacitor with metal gate and $p$-type silicon substrate at $T=300$ $K$. The inversion carrier density (in number of carriers per unit area) for $V_{G}= 0.8$ $V$ is $2\times 10^{11} cm^{-2}.$ For $V_{G}= 1.3$ $V$, the inversion carrier density is $4\times10^{11} cm^{-2}.$ What is the value of …

Consider avalanche breakdown in a silicon $p^{+}n$ junction. The $n$-region is uniformly doped with a donor density $N_{D}.$ Assume that breakdown occurs when the magnitude of the electric field at any point in the device becomes equal to the critical field $E_{crit}.$ Assume $E_{crit}$ to be independent of $N_{D}$. If the built-in voltage of the …

Consider a region of silicon devoid of electrons and holes, with an ionized donor density of $N_{d}^{+}=10^{17} cm^{-3}.$ The electric field at $x = 0$ is $0\: V/cm$ and the electric field at $x=L$ is $50$ $kV/cm$ in the positive $x$ direction, Assume that the electric field is zero in the $y$ and $z$ directions at all points. Given …

Consider a long-channel $NMOS$ transistor with source and body connected together. Assume that the electron mobility is independent of $V_{GS}$ and $V_{DS}.$ Given, $g_{m}=0.5\mu A/V \text{ for } V_{DS}=50\:mV \text{ and } \: V_{GS}= 2 \: V$, $g_{d}=8\mu A/V \text{ for } V_{GS}=2\:V \text{ and }V_{DS}=0 \:V$, where …

The figure shows a half-wave rectifier with a $475 \: \mu F$ filter capacitor. The load draws a constant current $I_{o}= 1 \: A$ from the rectifier. The figure also shows the input voltage $V_{i}$, the output voltage $V_{C}$ and the peak-to-peak voltage ripple $u$ on $V_{C}$. The input voltage $V_{i}$ is a triangle-wave with an amplitude of …

In the opamp circuit shown, the Zener diodes $Z1$ and $Z2$ clamp the output voltage $V_{0}$ to $+5V$ or $-5 V$. The switch $S$ is initially closed and is opened at time $t=0$. The time $t=t_{1}$ (in seconds) at which $V_{0}$ changes state is ________

An opamp has a finite open loop voltage gain of $100$. Its input offset voltage $V_{ios}(=+5mV)$ is modeled as shown in the circuit below. The amplifier is ideal in all other respects. $V_{\text{input}}$ is $25 \: mV$. The output voltage (in millivolts) is _________

An $8$ Kbyte ROM with an active low Chip Select input $\left (\overline{CS}\right )$ is to be used in an $8085$ microprocessor based system. The ROM should occupy the address range $1000H$ to $2FFFH$. The address lines are designated as $A_{15}$ to $A_{0}$, where $A_{15}$ is the most significant address bit. Which one of the following logic …

In an $N$ bit flash ADC, the analog voltage is fed simultaneously to $2^{N}-1$ comparators. The output of the comparators is then encoded to a binary format using digital circuits. Assume that the analog voltage source $V_{in}$ (whose output is being converted to digital format) has a source resistance of $75 \: \Omega$ as shown in the circuit …

The state transition diagram for a finite state machine with states $A$, $B$ and $C$, and binary inputs $X$, $Y$ and $Z$, is shown in the figure. Which one of the following statements is correct?

In the feedback system shown below $G\left ( s \right )=\frac{1}{\left ( s^{2}+2s \right )}$. The step response of the closed-loop system should have minimum settling time and have no overshoot. The required value of gain $k$ to achieve this is __________