GATE EC question library

Consider the differential equation $\dfrac{\mathrm{d} x }{\mathrm{d} t} = 10 – 0.2x$ with initial condition $x(0) = 1$. The response $x(t)$ for $t>0$ is

The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.

If $C$ denotes the counterclockwise unit circle, the value of the contour integral $$\dfrac{1}{2\pi j}\oint_{C} Re\{z\}dz$$ is __________.

Let the random variable $X$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $X$ is _______.

An $LC$ tank circuit consists of an ideal capacitor $C$ connected in parallel with a coil of inductance $L$ having an internal resistance $R.$ The resonant frequency of the tank circuit is

In the circuit shown, the Norton equivalent resistance $(\text{in}\: \Omega)$ across terminals $a-b$ is _______.

In the circuit shown, the initial voltages across the capacitors $C_{1}$ and $C_{2}$ are $1\: V$ and $3\: V,$ respectively. The switch is closed at time $t = 0$. The total energy dissipated (in Joules) in the resistor $R$ until steady state is reached, is __________.

A dc voltage of $10\: V$ is applied across an $n$-type silicon bar having a rectangular cross-section and a length of $1\:cm$ as shown in figure. The donor doping concentration $N_{D}$ and the mobility of electrons $\mu_{n}$ are $10^{16} cm^{-3}$ and $1000\: cm^{2} V^{-1} s^{-1},$ respectively. The average time $(\text{in}\: \mu s)$ taken by …

In a MOS capacitor with an oxide layer thickness of $10\: nm,$ the maximum depletion layer thickness is $100\: nm.$ The permittivities of the semiconductor and the oxide layer are $\varepsilon_{s}$ and $\varepsilon_{ox}$ respectively. Assuming $\varepsilon_{s}/\varepsilon_{ox} = 3,$ the ratio of the maximum capacitance to the minimum capacitance of …

The energy band diagram and the electron density profile $n(x)$ in a semiconductor are shown in the figures. Assume that $n(x) = 10^{15}e^{\left(\dfrac{q\alpha x}{kT}\right)}\:cm^{-3} ,$ with $\alpha = 0.1\:V/cm$ and $x$ expressed in cm. Given $\dfrac{kT}{q} = 0.026\;V,D_{n} = 36\:cm^{2}s^{-1},$ and $\dfrac{D}{\mu} = \dfrac{kT}{q}.$ The electron …

A function of Boolean variables $X, Y$ and $Z$ is expressed in terms of the min-terms as $$F(X, Y, Z) = \Sigma (1, 2, 5, 6, 7)$$ Which one of the product of sums given below is equal to the function $F(X, Y, Z)?$

The figure shows a binary counter with synchronous clear input. With the decoding logic shown, the counter works as a

A $1$-to-$8$ demultiplexer with data input $D_{in}$ , address inputs $S_{0}, S_{1}, S_{2}$ (with $S_{0}$ as the LSB) and $\overline{Y_{0}}$ to $\overline{Y_{7}}$ as the eight demultiplexed outputs, is to be designed using two $2$-to-$4$ decoders (with enable input $\overline{E}$ and address inputs $A_{0}$ and $A_{1}$) as shown in the …

The diode in the circuit given below has $V_{ON} = 0.7\:V$ but is ideal otherwise. The current $(\text{in}\: mA)$ in the $4\: k\Omega$ resistor is _______.

Assuming that the opamp in the circuit shown below is ideal, the output voltage ܸ$V_{o}$ (in volts) is________.

For the voltage regulator circuit shown, the input voltage $(V_{in})$ is $20 V \pm 20\%$ and the regulated output voltage $(V_{out})$ is $10\: V.$ Assume the opamp to be ideal. For a load $R_L$ drawing $200\: mA,$ the maximum power dissipation in $Q_{1}\:\text{(in Watts)}$ is ________.

In the ac equivalent circuit shown, the two BJTs are biased in active region and have identical parameters with $\beta \gg 1$. The open circuit small signal voltage gain is approximately _______.

Input $x(t)$ and output $y(t)$ of an LTI system are related by the differential equation $y’’(t) – y’(t) – 6y(t) = x(t).$ If the system is neither causal nor stable, the impulse response $h(t)$ of the system is

Consider two real sequences with time-origin marked by the bold value, $$x_{1}[n] = \{\textbf{1},2,3,0\},\:\:x_{2}[n] = \{\textbf{1},3,2,1\}$$ Let ܺ$X_{1}(k)$ and ܺ$X_{2}(k)$ be $4$-point DFTs of $x_{1}[n]$ and $x_{2}[n]$, respectively. Another sequence $x_{3}[n]$ is derived by taking $4$-point inverse DFT of $X_{3}(k) = X_{1}(k)X_{2}(k).$ …

Let $x(t) = \alpha s(t) + s(-t)$ with $s(t) = \beta e^{-4t}u(t),$ where $u(t)$ is unit step function. If the bilateral Laplace transform of $x(t)$ is $$X(s) = \dfrac{16}{s^{2} – 16}\:\: -4 < Re\{s\}<4;$$ then the value of $\beta$ is ______.